Numerical Solution ofBoundary Value Problems
with an Essential Singularity
Winfried Auzinger
Othmar KochJelena Petrickovic
Ewa Weinmuller
Technical Report ANUM Preprint No. 3/03
Institute for Applied Mathematicsand Numerical Analysis
Contents
1 Introduction 5
2 Theoretical foundations 82.1 Problem class . . . . . . . . . . . . . . . . . . . . . . . . . . . 82.2 Analytical properties . . . . . . . . . . . . . . . . . . . . . . . 9
2.2.1 Linear systems . . . . . . . . . . . . . . . . . . . . . . 92.2.2 Nonlinear case . . . . . . . . . . . . . . . . . . . . . . . 13
2.3 Numerical approach . . . . . . . . . . . . . . . . . . . . . . . . 172.3.1 Collocation methods for regular problems . . . . . . . . 172.3.2 Collocation for α = 1 . . . . . . . . . . . . . . . . . . . 192.3.3 Collocation for α > 1 . . . . . . . . . . . . . . . . . . . 20
3 Test problems 21
4 Numerical tests based on the collocation solver sbvpcol 264.1 Convergence results for the global error . . . . . . . . . . . . . 264.2 Convergence results for first and second derivatives . . . . . . 30
5 A posteriori error estimate and conditioning 325.1 A posteriori error estimate and conditioning in sbvp . . . . . . 33
5.1.1 Conditioning of the collocation equations . . . . . . . . 335.1.2 A posteriori error estimates . . . . . . . . . . . . . . . 34
5.2 The backward Euler method . . . . . . . . . . . . . . . . . . . 355.2.1 Backward Euler by means of sbvpcol . . . . . . . . . . 365.2.2 An implementation of backward Euler with precondi-
tioning . . . . . . . . . . . . . . . . . . . . . . . . . . . 365.3 Midpoint rule . . . . . . . . . . . . . . . . . . . . . . . . . . . 38
5.3.1 Midpoint rule by means of sbvpcol . . . . . . . . . . . 385.3.2 An implementation of the midpoint rule with precon-
ditioning . . . . . . . . . . . . . . . . . . . . . . . . . . 38
2
6 Error estimation based on a h-h/2 strategy 41
7 On the stability of the preconditioned midpoint rule 44
8 Lobatto distribution 48
3
Appendix
Figures:
1. Figure 1.1 51
2. Figure 2.1 96
3. Figure 3.1 109
4. Figure 4.1 114
5. Figure 5.1 119
6. Figure 6.1 126
7. Figure 7.1 150
8. Figure 9.1 158
Tables:
1. Tables 1.0 - 1.43 52-95
2. Tables 2.1 - 2.12 97-108
3. Tables 3.1 - 3.4 110-113
4. Tables 4.1 - 4.4 115-118
5. Tables 5.1 - 5.6 120-125
6. Tables 6.1 - 6.23 127-149
7. Tables 7.1 - 7.4 151-154
8. Tables 8.1 - 8.3 155-157
9. Tables 9.1 - 9.3 159-161
4
Chapter 1
Introduction
Singular boundary value problems (BVPs) with a singularity of the first kindfrequently occur in applications, e.g. when a partial differential equation(PDE) is reduced to an ordinary differential equation (ODE) in the presenceof symmetries. More difficult problems featuring a singularity of the secondkind (essential singularity) arise, for instance, when a boundary value ODE istransformed from an infinite interval to a finite one. They are also very com-mon in quantum physics, mechanics, investigations of ferromagnetic systems(e.g. Ginzburg-Landau equations) etc.
In this work we present a study of certain numerical techniques – whichare well known to perform efficiently and accurately in the case of a singu-larity of the first kind – applied to problems with an essential singularity.In particular, collocation methods are considered together with a posteriorierror estimates intended to provide the basis for a grid selection strategy.We concentrate on the methods implemented in the MATLAB code sbvp.This code was originally intended to solve regular BVPs and problems witha singularity of the first kind. An extension to problems with an essentialsingularity is in preparation, and our work provides important prerequisitesconcerning the proper choice of algorithmic components.
This document is organized as follows:
• In Chapter 2, we recapitulate the analytical theory concerning the well-posedness of boundary value problems with an essential singularity,and we give a short review of some relevant results from numericalanalysis. Here we note that, for the methods implemented in sbvp, afairly complete theoretical foundation exists for regular problems andproblems with a singularity of the first kind; however, hardly any resultsare available in the presence of an essential singularity.
• This latter fact motivated our extensive numerical experiments, which
5
are based on a set of test examples specified in Chapter 3, for collocationbased on equidistant or Gaussian nodes.
• In Chapter 4, we present our numerical results obtained with the col-location solver sbvpcol (which is at the core of sbvp). Due to theseresults, collocation seems to be a very accurate and reliable method forproblems with an essential singularity.
• In Chapter 5, we report the results obtained by direct application ofsbvp, which is based on sbvpcol and uses an a posteriori error estimateimplemented in sbvperr for global error estimation. This estimate isbased on a defect correction approach and uses the backward Eulermethod as an auxiliary algorithmic component. Our results show thatthis estimate is completely unreliable in the essentially singular case.Furthermore, we observed that the condition numbers of the algebraicsystems representing the Euler equations are extremely large; an at-tempt to remedy this situation by preconditioning was not successful.
We have also considered the implicit midpoint rule as an alternative tothe backward Euler scheme. For this method, theoretical convergenceresults are available which are consistent with our numerical results:The convergence order is at least equal to the stage order and mayeven be as high as the classical convergence order p = 2. Moreover,an appropriate diagonal preconditioning yields optimal condition num-bers for the corresponding algebraic system (not worse than for regularproblems).
If we consider the implicit midpoint rule as an auxiliary scheme for thepurpose of error estimation, the corresponding results are significantlybetter but not really convincing.
• These results motivated us to test a classical, but computationally moreexpensive error estimation strategy based on grid refinement (‘h-h/2estimate’). The results presented in Chapter 6 show that this methodof error estimation is very successful and robust w.r.t. singularities ofthe first or second kind.
• Chapter 7 is more theoretically inclined; here we have tried to provethe ‘optimal preconditioning property’ observed for the midpoint rulein Chapter 5, for a class of scalar model problems (cf. Example 1).Here, only a partial theoretical explanation could be given.
• Finally, in Chapter 8 we report results obtained for collocation based onLobatto nodes (which are used in the standard MATLAB code bvp4c).
6
Our results show that this method cannot be successfully applied inthe presence of an essential singularity.
Our conclusion is that collocation methods as implemented in sbvp shouldbe seriously considered as candidates for numerically solving BVPs with anessential singularity. Concerning a posteriori error estimation and mesh se-lection, strategies based on the auxiliary midpoint rule and h-h/2 refinementshould be included as options, where the latter is the most expensive but, atthe same time, seems to be the most reliable alternative.
7
Chapter 2
Theoretical foundations
2.1 Problem class
Consider the following boundary value problem for a system of ordinarydifferential equations:
tαz′(t) = f(t, z(t)), 0 < t ≤ 1, (2.1a)
b(z(0), z(1)) = 0, (2.1b)
where α ≥ 1 and where z is required to be continuous, z ∈ C[0, 1]∩C1(0, 1].The functions f and b are continuous nonlinear mappings on appropriatedomains. When α = 1 the problem (2.1) is said to have a singularity ofthe first kind, while a singularity is of the second kind (or essential) for α > 1.
More generally, we discuss nonlinear systems of the form
T (t)z′(t) = f(t, z(t)), 0 < t ≤ 1, (2.2a)
z ∈ C[0, 1] ∩ C1(0, 1], (2.2b)
b(z(0), z(1)) = 0, (2.2c)
whereT (t) := diag(tα1I1, t
α2I2, . . . , tαrIr),
and the Ik are unit matrices with either αk ≥ 1 for 1 ≤ k ≤ r, or αk ≥ 1 for1 ≤ k ≤ r − 1 and αr = 0.
As a first step in the analysis of (2.2) we examine the solution structure oflinear systems
T (t)z′(t) = (M + A(t))z(t) + g(t), 0 < t ≤ 1, (2.3a)
8
z ∈ C[0, 1] ∩ C1(0, 1], (2.3b)
B0z(0) + B1z(1) = γ, (2.3c)
where the matrix M ∈ Rn×n is a block upper triangular matrix,
M :=
M11 M12 . . . M1r
0 M22 . . . M2r...
... . . ....
0 0 . . . Mrr
, (2.4)
and A, g ∈ C[0, 1]. Each matrix Mkk is a square matrix of the same size as Ik
which is assumed to be nonsingular when αk 6= 0 and has no eigenvalues thatare purely imaginary. When αr = 0 then Mrr = 0. Moreover, B0, B1 ∈ Rp×n
are constant matrices and γ ∈ Rp is a constant vector. In general, p ≤ n.
2.2 Analytical properties
The analytical properties of (2.2) and (2.3) have been discussed in full detailin [HW80b] and [HW79]. In these papers the Fredholm theory for linearsystems has been established and existence and smoothness results for non-linear problems have been provided. In this section we recapitulate the mostimportant of these fundamental results.
2.2.1 Linear systems
We first examine
tαz′(t) = Mz(t) + tα−ρg(t), 0 < t ≤ 1, α ≥ 1, (2.5)
where M is an n×n matrix whose eigenvalues λj satisfy Re λj 6= 0, j = 1 . . . n.The general solution of (2.5) reads
z(t) = Z(t)z(δ) + Z(t)
∫ t
δ
Z(s)−1s−ρg(s) ds, (2.6)
where
Z(t) =
{exp[M(δ1−α − t1−α)/(α− 1)], α 6= 1,(t/δ)M = exp [log(t/δ)M ], α = 1,
(2.7)
is the fundamental solution matrix satisfying
tαZ ′(t)−MZ(t) = 0, 0 < t ≤ 1, Z(δ) = I,
9
for 0 < δ ≤ 1. Let
Q :=1
2πi
∫
Γ−(λI −M)−1 dλ,
(2.8)
P :=1
2πi
∫
Γ+
(λI −M)−1 dλ,
where Γ− and Γ+ denote closed contours (oriented canonically) in the leftand right complex halfplane, respectively, such that each eigenvalue of Mis enclosed by either Γ− or Γ+. The matrices Q and P are projections ontothe invariant subspaces of M associated with the eigenvalues having negativeand positive real parts, respectively.Via the Jordan decomposition of M it is straightforward to obtain an explicitrepresentation of Z(t). This representation immediately yields
Lemma 2.2.1 Let α ≥ 1. Then for an arbitrary vector η, Zη ∈ C[0, 1] iffQη = 0.
For α > 1 we define1
(Bρg)(t) =
tρ−αZ(t)∫ t
0QZ−1(s)s−ρg(s) ds
+ tρ−αZ(t)∫ t
δPZ−1(s)s−ρg(s) ds, 0 < t ≤ 1,
−M−1g(0), t = 0.
(2.9)
Concerning continuous solutions of (2.5) we have
Lemma 2.2.2 Let α > 1, ρ ≤ α and g ∈ C[0, 1]. Then, every z ∈ C[0, 1] ∩C1(0, 1] which satisfies (2.5) has the form
z(t) = Z(t)Pz(δ) + tα−ρ(Bρg)(t). (2.10)
Clearly, this means that the rank[Q] linearly independent conditions given byQz(0) = 0 have to be satisfied for z to be continuous, z ∈ C[0, 1]. Moreover,we need to pose rank[P ] additional conditions at t = 1, Pz(1) = Pη, η ∈ Rn,for the uniqueness of z.
As a next step in the analysis we examine the system
T (t)z′(t) = Mz(t) + g(t), 0 < t ≤ 1, (2.11)
1The analysis for systems with a singularity of the first kind, α = 1, can be found in[HW76] and [HW78].
10
with matrices T (t) and M defined in §2.1. Let
M = D + U, D = diag(M11,M22, . . . , Mrr),
Z(t) = diag(Z1(t), . . . , Zr(t)),
P = diag(P1, . . . , Pr),
Q = diag(Q1, . . . , Qr)
and
R := I − P −Q =
{diag(0, . . . , 0, Ir), αr = 0,0, αr 6= 0,
where
Zk(t) =
{exp [Mkk(δ
1−αk − t1−αk)/(αk − 1)], αk 6= 1,exp [log(t/δ)Mkk], αk = 1,
(2.12)
and Pk, Qk are defined by (2.8) with M replaced by Mkk if αk 6= 0 andPk = Qk = 0 if αk = 0. In addition, define for 0 < δ ≤ 1
(B g)(t) = Z(t)
∫ t
0
QZ−1(s)T−1(s)g(s) ds
+ Z(t)
∫ t
δ
PZ−1(s)T−1(s)g(s) ds + R
∫ t
δ
g(s) ds. (2.13)
Lemma 2.2.3 Let g ∈ C[0, 1]. Then any continuous solution of (2.11) mustsatisfy
z(t) = Z(t)[Pz(δ) + Rz(δ)] + (B[Uz + g])(t)
= Z(t)(P + R)η + (B[Uz + g])(t), (2.14)
where η = (P + R)z(δ).
The most general linear equation that we shall consider is
T (t)z′(t) = (M +A(t))z(t)+g(t), 0 < t ≤ 1, z ∈ C[0, 1]∩C1(0, 1], (2.15)
where A, g ∈ C[0, 1] and(I −R)A(0) = 0. (2.16)
In order to show the existence and uniqueness of the solution of (2.15) acontraction argument is used. Consequently, the following result holds.Let p = rank[P + R], W be an n × p matrix which consists of linearlyindependent columns of (P + R), and define
X(t) := Y (t)W,
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where Y (t) is the unique solution of
T (t)Y ′(t) = (M + A(t))Y (t), 0 < t ≤ 1, Y (t) ∈ C[0, 1] ∩ C1(0, 1],
RY (δ) = R, PY (δ) = P.
In addition, let y be the unique particular solution of (2.15) subject to theboundary conditions
Ry(δ) = P y(δ) = 0.
Then we have
Theorem 2.2.1 Any solution of (2.15) has the form
z(t) = X(t)β + y(t) (2.17)
with a unique β ∈ Rp.
We now answer the question under which circumstances the solution of (2.15)given by (2.17) satisfies the boundary conditions
B0z(0) + B1z(1) = γ. (2.18)
Our aim is to establish conditions on B0 and B1 which lead to a Fredholmalternative for (2.15) and (2.18), cf. (2.3c). To do this it is convenient tointroduce the differential expression
l(z) = Tz′ − (M + A)z
and associate with it the operator defined by
Lz = l(z)
for z ∈ D = {z ∈ C[0, 1], T z′ ∈ C[0, 1], B0z(0) + B1z(1) = 0}. Then we have
Theorem 2.2.2 Ifrank [B0, B1] = k, (2.19)
then L is Fredholm with index p− k. Furthermore, if L−1 exists, it is bounded.
In applications we are primarily interested in the case when L is Fredholmwith index zero. We therefore assume that B0, B1 are p× n matrices, γ is ap-vector and that (2.19) holds with k = p. On substitution of (2.17) in (2.18)we find
Theorem 2.2.3 The problem (2.3) has a unique solution for all g ∈ C[0, 1]and γ ∈ Rp iff the p× p matrix [B0X(0) + B1X(1)] is nonsingular.
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Remark 1. The boundary conditions (2.3c) are equivalent to
B0Rz(0) + B1z(1) = γ (2.20)
whereγ = γ + B0(M + R)−1(I −R)g(0).
It turns out that the form (2.20) is advantageous for some numerical schemesapplied to the boundary value problem (2.3).
Remark 2. Moreover, it can be shown that the restriction that the solutionbe continuous at t = 0 is not useful when constructing numerical schemes.It turns out that the relation
Qz(0) = Q(M + R)−1((R− I)g(0) + Rz(0))) (2.21)
is more satisfactory.Equations (2.21) and (2.20) are the n linearly independent boundaryconditions which must be employed when the problem (2.3) is discretizedby a difference scheme. This and other related questions are discussed in[HW79].
2.2.2 Nonlinear case
We now formulate smoothness results for nonlinear problems (2.2),
T (t)z′(t) = f(t, z(t)), 0 < t ≤ 1,
z ∈ C[0, 1] ∩ C1(0, 1],
b(z(0), z(1)) = 0.
We first make a number of assumptions.
(N1) Problem (2.2) has a solution z(t).With this solution and some ρ > 0 we associate the spheres
sρ(z(t)) := {y ∈ Rn, |z(t)− y| ≤ ρ}
and the tube
Tρ := {(t, y), t ∈ [0, 1], y ∈ sρ(z(t))}.
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(N2) For some ρ > 0, f(t, z(t)) is continuously differentiable with respect toz, and ∂f
∂z(t, z) is continuous on Tρ.
(N3) For all y ∈ sρ(z(0)), the matrix
M(y) :=∂f
∂y(0, y)
has the fixed block upper triangular structure introduced in (2.4). Inaddition, the matrix
M := M(z(0))
satisfies all the conditions concerning the matrix M introduced in §2.1.
(N4) b(v, w) is a vector-valued function of dimension
p := rank[P + R]
which is continuously differentiable on sρ(z(0))× sρ(z(1)).
(N5) The solution z(t) is isolated. This means that the linearized problem
T (t)u′(t)−G(t)u(t) = 0,
u ∈ C[0, 1] ∩ C1(0, 1],
B0u(0) + B1u(1) = 0
with
G(t) =∂f(t, z(t))
∂y, B0 =
∂b(z(0), z(1))
∂v, B1 =
∂b(z(0), z(1))
∂w
has only the trivial solution.
The requirement that the solution of (2.2) be continuous at t = 0 obviouslyimposes the following restriction on the solution:
(I −R)f(0, z(0)) = 0. (2.23)
(N3) implies that (2.23) can be inverted locally for (I −R)z(0), i.e.
(I −R)z(0) = ϕ(Rz(0)), (2.24)
whencez(0) = ϕ(Rz(0)) + Rz(0) =: ψ(Rz(0)). (2.25)
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This result is of value since it can be used to modify the boundary conditions(2.2c). We substitute (2.25) into (2.2c) and obtain
b(ψ(Rz(0)), z(1)) = 0
, which involves Rz(0) rather than z(0). Hence, without loss of generality,we may consider boundary conditions of the form
b(Rz(0), z(1)) = 0.
In the sequel, use will be made of the derivative of ψ(Rz) with respect to z.Substitution of (2.24) into (2.23), differentiation with respect to Rz(0), anduse of (2.25) yield
∂ψ
∂z= (M(ψ(Rz)) + R)−1R.
Equation (2.23) immediately yields
Lemma 2.2.4 Let z satisfy (2.2a) and (2.2b). Then Tz′ ∈ C[0, 1] and
limt→0+
T (t)z′(t) = Rz′(0).
This lemma says that any component zi of z in a block associated with αk ≥ 1satisfies limt→0+ tαkz′i(t) = 0, while the components (if any) associated withαr = 0 are in C1[0, 1]. This smoothness result can be extended if furtherrestrictions are imposed on the problem.
Lemma 2.2.5 Assume that
(i) all αk are integers,
(ii) the real parts of the eigenvalues of the matrix Mkk are negative wheneverαk = 1,
(iii) f ∈ Cm(Tρ).
Then
(i) z ∈ Cm[0, 1] ∩ Cm+1(0, 1],
(ii) when αr 6= 0, (i.e. R = 0), and ∂kf∂tk
(0, z(0)) = 0, k = 0, . . . , m, then
zk(0) = 0, k = 0, . . . , m.
15
The situation m = ∞ is the one most frequently encountered in practice.
Remark. We now examine the structure of the solution when (2.2) is linear,i.e.
T (t)z′(t) = (M + A(t))z(t) + g(t), 0 < t ≤ 1,
z ∈ C[0, 1] ∩ C1(0, 1],
B0Rz(0) + B1z(1) = γ,
see §2.1.Let us denote by y the unique particular solution of
T (t)y ′(t) = (M + A(t))y(t) + g(t), 0 < t ≤ 1, y(t) ∈ C[0, 1] ∩ C1(0, 1],
subject to boundary conditions
Ry(0) = P y(δ) = 0,
and let Y (t) satisfy
T (t)Y ′(t) = (M + A(t))Y (t), 0 < t ≤ 1, Y (t) ∈ C[0, 1] ∩ C1(0, 1],
RY (0) = R, PY (δ) = P.
Here, Theorem 2.2.3 can be reformulated in the following way:
Theorem 2.2.4 The problem (2.3) has a unique solution for all g ∈ C[0, 1]and γ ∈ Rp iff the p × p matrix [B0R + B1Y (1)]W is nonsingular. Thesolution in this case reads
y(t) = y(t) + Y (t)W ([B0R + B1Y (1)]W )−1(γ −B1y(1)).
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2.3 Numerical approach
When solving singular problems with a singularity of the first kind, col-location methods work satisfactorily and provide a dependable high orderapproximation for the solution of the underlying analytical problem. Thisis not the case in general, when other standard high order methods areapplied to solve singular problems. Many of those methods suffer fromorder reductions and become inefficient, see [HW77] for the discussion ofmultistep methods or [HW85] for the analysis of explicit Runge-Kuttaschemes. Collocation methods show advantageous convergence properties,see [HW78] and [Wei86] for the investigation of first and the second ordersingular systems (with a singularity of the first kind), respectively.
The aim of this section is to recapitulate the convergence results forcollocation methods applied to regular problems and to a certain class ofsingular problems. We also present the implementation of collocation in theMATLAB code sbvp.
2.3.1 Collocation methods for regular problems
We consider the problem (2.1) and set α = 0,
z′(t) = f(t, z(t)), 0 < t ≤ 1, (2.26a)
b(z(0), z(1)) = 0. (2.26b)
For the numerical analysis we introduce the mesh
∆ = (τ0, τ1, . . . , τN),
with τ0 = 0, τN = 1, hi := τi+1 − τi and h := max0≤i≤N hi. We denote thecorresponding grid vectors by
u∆ = (u0, u1, . . . , uN) ∈ R(N+1)n.
The norm on the space of grid vectors is given by
‖u∆‖∆ = max0≤k≤N
|uk|.
For a continuous function x ∈ C[0, 1], we denote by R∆ the pointwise pro-jection onto the space of grid vectors,
R∆(x) = (x(τ0), x(τ1), . . . , x(τN)).
17
For collocation, m equidistantly spaced points are inserted in each subintervalJi = [τi, τi+1]. This yields the (fine) grid2
∆m := {ti,j : ti,j = τi + jδi, i = 0, . . . , N − 1, j = 0, . . . , m + 1}, (2.27)
where
δi :=hi
m + 1.
We seek to approximate z by a collocating function p(t) := pi(t), t ∈ Ji,where pi is a polynomial of degree ≤ m satisfying
p′i(ti,j) = f(ti,j, pi(ti,j)), i = 0, . . . , N − 1, j = 1, . . . , m, (2.28a)
pi(τi) = pi−1(τi), i = 1, . . . , N − 1, (2.28b)
b(p0(0), pN−1(1)) = 0. (2.28c)
The degree m of the polynomial function is sometimes referred to as the stageorder (Runge-Kutta terminology). For any collocation method with the stageorder m, the convergence order is at least m. For a suitable (non-equidistant!)choice of the collocation nodes ti,j, j = 1, . . . , m,, the convergence order ofthe error in the mesh points, τk, 0 ≤ k ≤ N , can be improved (up to 2m forGaussian nodes, cf. [BS73]). This effect is known as superconvergence.For the regular boundary value problem (2.26) the following convergenceresult holds (see [AMR88] and [AKW02]):
Theorem 2.3.1 Let z(t) be an isolated solution of (2.26) with sufficientlysmooth data. Then for any collocation scheme of the form (2.28), there existconstants ρ, h0 > 0 such that the following statements hold for all meshes ∆with h ≤ h0:
(i) There exists a unique solution p(t) of (2.28) in a tube of radius ρ aroundz(t).
(ii) This solution can be computed by Newton’s method which convergesquadratically provided that the initial guess p[0](t) is sufficiently closeto z(t).
(iii) The following error estimates hold:
‖R∆(p)−R∆(z)‖∆ = O(hm+ν), (2.29a)
‖p− z‖∞ = O(hm+ν), (2.29b)
‖p(l) − z(l)‖∞ = O(hm+1−l), l = 1, . . . ,m, (2.29c)
where ν = 0 if m is even and ν = 1 if m is odd.
2For convenience, we denote τi by ti,0 ≡ ti−1,m+1, i = 1, . . . , N .
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2.3.2 Collocation for α = 1
The convergence theory for the above collocation methods applied to a classof linear singular boundary value problems of the form
z′(t) =M(t)
tz(t) + g(t), 0 < t ≤ 1, (2.30a)
B0z(0) + B1z(1) = γ, (2.30b)
where g ∈ C[0, 1], was presented in [HW78].
According to the theoretical and numerical results presented in [AKW02]and [HW78] the situation in the singular case is more or less the same as inthe regular case. Theorem 2.3.1 also holds, but for a limited class of singularproblems. This means that certain restrictions have to be imposed, namely,the real parts of the eigenvalues of M(0) are assumed to be nonpositive3.Furthermore, in the case of singular systems the superconvergence behaviorcannot be expected in general. In applications, where M(0) is diagonalizable,the stage order hm can be uniformly improved by at least one power of h. Inthe general case however, the convergence order may be negatively affectedby logarithmic terms occurring in the estimates (2.28) when m is odd (see[HW78]).
Collocation methods were successfully implemented in standard codesdesigned to solve regular boundary value problems. The best known arethe FORTRAN 90 code COLNEW, see [ACR78], [ACR81], and the standardMATLAB code bvp4c. The applicability of these codes in the singular caseis somewhat limited, see [AKKW].
Recently, another MATLAB solver called sbvp4 has been developed at theDepartment of Applied Mathematics and Numerical Analysis, Vienna Uni-versity of Technology. This code is based on collocation, and its purpose isto approximate solutions of nonlinear singular problems,
z′(t) =M(t)
t− az(t) + g(t, z(t)), a < t ≤ b, (2.31a)
b(z(a), z(b)) = 0. (2.31b)
The grid selection strategy is based on an estimate for the global error of thecollocation solution. The original idea for this error estimate was presented
3The results for a general spectrum of M(0) in the context of second order systems canbe found in [Wei86].
4The package is freely available from http ://www.math.tuwien.ac.at/∼ewa.
19
in [Ste78], see also [Zad76], and is closely related to the principle of DefectCorrection. In the current version of sbvp, a modified defect definition (see[AKW02],[AKKW02]) is used. The advantage of this modification is to pro-vide an asymptotically correct a posteriori error estimate for all grid points,including collocation points5. The proof of the asymptotical correctness ofthe error estimate for the regular case was given in [AKW02], and this proofwas recently extended to the singular case where α = 1 in [AKW].
2.3.3 Collocation for α > 1
Analytical properties of singular systems with a singularity of the secondkind and convergence results for finite difference schemes in the context ofthese problems have been discussed in [HW80b] and [HW79], respectively.Theoretical results on the convergence properties for collocation schemesapplied to such problems are not yet available. This and the fact thatcollocation works satisfactorily for problems with a singularity of the firstkind was the motivation for the present work.
The main goal was to carry out comprehensive numerical investigations inorder to complete, at least experimentally, the picture of the properties ofcollocation methods when applied to problems with an essential singularity.Moreover, we wanted to find out why the error estimate implemented in sbvpfails in this case. A future goal is to develop a new module for our code whichwould be suitable for the treatment of nonlinear boundary value problemswith an essential singularity, including an asymptotically correct a posteriorierror estimate as the basis for mesh adaptation.
5The original version works satisfactory only in the mesh points.
20
Chapter 3
Test problems
In this section we list the model problems used in the numerical experiments.
• Example 1: We first consider the linear scalar problem (terminal valueproblem)
z′(t) =1
tαz(t) + et − et
tα, (3.1a)
z(1) = e, (3.1b)
with the exact solution z(t) = et. Here, M = 1, A(t) = 0 and T (t) = tα,which means that the only eigenvalue of M is positive. Consequently,one condition, (3.1b), has to be posed in order to obtain a uniquesolution of (3.1). This example can be found in [HW79].
• Example 2: Next we have a linear system
z′(t) =1
tβ+2
0 −1 0 00 0 −1 00 0 0 −1ρ 0 0 0
z(t) (3.2a)
+1
tβ+2
0 0 0 00 βtβ+1 0 00 0 2βtβ+1 0
d(1/t)− ρ 0 0 3βtβ+1
z(t),
(4 −2 0 1
−2 2 −1 0
)z(0) =
(00
), (3.2b)
(4 2 0 −12 2 1 0
)z(1) =
(11
), (3.2c)
21
cf. [HW80b]. We choose the function d(t) = 1 and the constant ρ = 1.In this case
M =
0 −1 0 00 0 −1 00 0 0 −11 0 0 0
,
r = 1 and T (t) = tβ+2I1, where I1 ∈ R× and β > 0. The eigenvaluesof M are
(λ1, λ2, λ3, λ4) =
√2
2(1 + i, 1− i,−1 + i,−1− i),
and this means that according to the theory two initial conditions,(3.2b), equivalent to Qz(0)=0 are necessary for the solution to be con-tinuous. The remaining two conditions, (3.2c), now yield the unique-ness of z. The exact solution of this problem is not known.
• Example 3: This problem is nonlinear,
z′(t) =
−1t(z1(t) + z2(t))
−1t(2z2(t) + z3(t))
12t3
z3(t) + 12t2
z3(t)z4(t) + 12t
z1(t)z3(t)− 3tz3(t)
0
, (3.3a)
(1 0 0 00 1 0 0
)z(0) =
(00
), (3.3b)
(1 0 0 10 1 0 0
)z(1) =
( −1−1
), (3.3c)
see [HW79]. Here, r = 3 and T (t) = diag(tI1, t3I2, I3), where I1 ∈ R×
and I2 = I3 = 1. In the case of a nonlinear problem we need toinvestigate the structure of fz(t, z(t)), where
f(t, z(t)) =
−(z1(t) + z2(t))−(2z2(t) + z3(t))
12z3(t) + t
2z3(t)z4(t) + t2
2z1(t)z3(t)− 3t2z3(t)
0
.
Consequently,
fz(t, z(t)) =
−1 −1 0 0
0 −2 −1 0
0 0 1/2 0
0 0 0 0
+
0 0 0 0
0 0 0 0
0 0 tz4(t)2
+ t2z1(t)2
− 3t2 tz3(t)2
0 0 0 0
.
22
The eigenvalues of M are
(λ1, λ2, λ3, λ4) = (−1,−2,1
2, 0)
and therefore two initial conditions are necessary for the solution to becontinuous. For this problem the exact solution is not known either.
• Example 4: We consider the nonlinear problem
z′(t) =
− ct2
z2(t)−( c
t2+ 2
c)z1(t) + c
t2z21(t) + 2
tz2(t)
−4tz3(t) + c(1− ctz4(t))z4(t)
cz3(t)
, (3.4a)
(1 1 0 00 0 1 0
)z(0) =
(00
), (3.4b)
(0 1 −c −11 0 0 −c
)z(1) =
(00
). (3.4c)
for a c ∈ R+. This problem was discussed in [HW79].For T (t) = diag(t2I1, tI2, I3), with I1 ∈ R× and I2, I3 ∈ R, thestructure of the right-hand side is
f(t, z(t)) =
−cz2(t)
−(c + 2t2
c)z1(t) + cz2
1(t) + 2tz2(t)−4z3(t) + ct(1− ctz4(t))z4(t)
cz3(t)
,
and therefore,
fz(t, z(t)) := M + A(t)
=
0 −c 0 0
−c 0 0 0
0 0 −4 0
0 0 0 0
+
0 0 0 0
−2t2
c+ 2cz1(t) 2t 0 0
0 0 0 ct− 2c2t2z4(t)
0 0 c 0
.
The eigenvalues of M read
(λ1, λ2, λ3, λ4) = (c,−c,−4, 0).
On noting that P + Q = diag(I4, 0), where I4 ∈ R×, it followsimmediately that the necessary condition for (3.4) to be well-posed,(I −R)A(0) = 0, is satisfied if z1(0) = 0, cf. (2.16).
The exact solution of (3.4) is again not known.
23
• Example 5: This problem is linear, cf. [HW80a],
z′(t) =1
t2
(0 −1−1 0
)z(t), (3.5a)
(1 −2
)z(0) = 0, (3.5b)
(1 0
)z(1) = 1. (3.5c)
The exact solution is given by
z(t) = (z1(t), z2(t))T = (e
t−1t ,−e
t−1t )T .
The matrix M ,
M =
(0 −1−1 0
)
has two nonzero eigenvalues
(λ1, λ2) = (−1, 1),
and T (t) = t2I1 where I1 ∈ R×.
• Example 6: The next problem,
z′(t)=1
tα
(0 −1−1 0
)(z(t)−
(2et
e−t
))+
(2et
−e−t
), (3.6a)
(1 −2
)z(0)= 0, (3.6b)
(1 0
)z(1) = 2e, (3.6c)
is a modification of Example 5 with the exact solution
z(t) = (z1(t), z2(t))T = (2et, e−t)T
and unchanged matrix M , and T (t) = tαI1, I1 ∈ R2×2.
• Example 7: The problem
z′(t) =1
t2
−z2(t)−z3(t)−z4(t)
1− e−z1(t)/2
, (3.7a)
(1 0 0 00 1 0 0
)z(0) =
(00
), (3.7b)
(0 0 1 00 0 0 1
)z(1) =
(01
). (3.7c)
24
can be found in [MR83]. This model is nonlinear, T (t) = t2I1, whereI1 ∈ R× and
M =
0 −1 0 0
0 0 −1 0
0 0 0 −1
1/2 0 0 0
.
The eigenvalues of M are
(λ1, λ2, λ3, λ4) =4√
2
2(1 + i, 1− i,−1 + i,−1− i).
The exact solution is unknown.
In order to use sbvp to solve the model problems numerically the boundaryconditions stated above in two sets need to be rewritten as a (formally)coupled system of dimension n.
25
Chapter 4
Numerical tests based on thecollocation solver sbvpcol
In this section we will present various numerical results which have beenobtained in order to check the hypothesis that the convergence behavior ofcollocation methods as stated in Theorem 2.3.1 is also valid for BVPs witha singularity of the second kind.
Our investigations were based on the code sbvpcol, the collocation solverunderlying sbvp. In order to be able to call this routine we have to choose afew options such as:
• ‘tau 0’ . . . given mesh
• ‘ColPts’ . . . specifies the distribution of collocation points, with twoavailable options which have been used in our investigations: equidistantand gauss.
• ‘Basis’ . . . choice of basis for the internal representation of collocationpolynomials. Default is RungeKutta.
• ‘Degree’ . . . highest degree of basis polynomials. In the sequel wedenote this by m.
Any other unspecified options that concern the solution process are set todefault.
4.1 Convergence results for the global error
In Tables 1.0–9.3, the convergence results for collocation methods are given.By err we denote the maximal global error, i.e. the difference between the
26
numerical solution obtained by sbvpcol and the exact solution, measuredin the discrete maximum norm, for those cases where the exact solution isexplicitly known (Examples 1, 5 and 6).In all other examples, the unknown exact solution is replaced by a properapproximation, namely a reference solution obtained by means of sbvpcolusing a very small step size.Furthermore, p and const represent the empirical convergence rate anderror constant, respectively.Note that for all problems tested, the convergence orders at the mesh points(mesh) and at the collocation plus mesh points (coll) were both observed.
Now, we give a comprehensive overview of the numerical experiments.
• Example 1 ( (3.1) on page 21 )
This is a simple one-dimensional problem with a smooth solution.We have tested three different types of singularity (α = 1, 2, 3, seeTables 1.1–1.3), and also a regular version (α = 0, see Table 1.0).For α = 1 this is a problem with a singularity of the first kind, forwhich the code sbvp has already proved its efficiency (see [AKKW02]).The other two choices, α = 2, 3, are crucial for our present study.As it can be seen they show a slightly less advantageous behavior ascompared to the case α = 1.In particular, superconvergence of Gauss methods at the mesh pointsis not observed in the singular case, and the uniform convergence orderseems to be slightly deteriorating with increasing α.For equidistant nodes, the uniform convergence order m (= stage order)is observed throughout7.
• Example 2 ( (3.2) on page 21 )
For this linear 4-dimensional problem we also consider three cases,namely β = 0, 1, 2, each of them corresponding to a singularity of thesecond kind. See Tables 2.1–2.3 and 2.7–2.9.
7 We have also performed some experiments with a (numerically) less smooth solution,g(t) = sin(t10) cos(15t) instead of g(t) = et, for the analogous problem
z′(t) =1tα
(z(t)− g(t)) + g′(t), z(1) = g(1),
see Tables 8.1–8.3. In this case, high orders are better visible because the derivatives ofg(t) are larger in size, hence the global error level is also larger, and rounding errors do notspoil the observable orders. Note that for this problems, the classical (super-) convergenceorders are observed for Gaussian points. This may possibly be attributed to the fact thatz(0) = z′(0) = . . . = z(10)(0) = 0.
27
Here, in all three cases considered, the superconvergence order of collo-cation at Gaussian points is approximately attained at the mesh points.On the equidistant grid, the stage order m is obtained.
• Example 3 ( (3.3) on page 22 )
Here, no order reductions are encountered; see Tables 3.1 and 3.3.
For this example, sbvpcol reports a message about the maximum num-ber of function evaluations being exceeded9. This is probably causedby convergence problems of the Newton iteration which is used to solvethe system of collocation equations.
• Example 4 ( (3.4) on page 23 )
According to [HW79], we can expect convergence only for a suitablechoice of the starting iterate. We have used the initial approximation
y0 =
[(g1 +
tg2
c
)e−
ct ;−
(g1 +
tg2(− tc+ 1)
c
)e−
ct ; 0 ; d
]
where (d, c) = (2, 1.5), and g1, g2 are chosen as g1 = cec, g2 =
− (d+1)c2ec
c+1.
Here, the results for the convergence order are not fully satisfactory,in particular for the case of Gaussian collocation with m = 2 (see Ta-bles 4.1 and 4.3).
Equidistant collocation performs satisfactorily for m = 2, 4, with a con-vergence order m. However, for m = 6 only an order 4 is observed (Ta-ble 4.3). The fact that no convergence order higher than 4 is observedfor any of the collocation methods applied to this problem suggests apossible explanation for the order reductions. If the exact solution ofthis test problem (which is unknown, unfortunately) is not sufficientlysmooth, we cannot expect an arbitrary convergence order for colloca-tion schemes. Thus we conjecture that the order reductions down toorder 4 for Example 4 are caused by the fact that the exact solutionz(t) of (3.4) is no more than five times continuously differentiable.
In any case, global errors smaller than ≈ 1e−8 are not observed.
This effect may also be caused by an inappropriate choice of the refer-ence solution (obtained with h = 1.5626e−03). However, it is not easyto improve this reference solution.
In this example, difficulties in the Newton solution process were alsoreported by sbvpcol (similarly as in Example 3).
9‘MaxFunEvals’= 80000,‘MaxIter’= 50000, ‘TolX’= 1e−6
28
• Example 5 ( (3.5) on page 23 )
For this two-dimensional example the exact solution is explicitly known.The test results are presented in Tables 5.1 and 5.5.
In all cases the full conventional orders are observed, including super-convergence at the meshpoints for Gaussian collocation.
• Example 6 ( (3.6) on page 24 )
According to our numerical results, see Tables 6.1–6.3 and 6.17–6.19for three versions (α = 1, 2, 3), we can affirm the same behavior as forExample 1. A convergence order higher than m + 1 is not observed,and for larger values of α an order reduction becomes visible.
• Example 7 ( (3.7) on page 24 )
For this example, the same conclusions about the convergence orderas for Example 5 may be drawn (see Tables 7.1 and 7.3). (Note thatthese results were obtained with the help of a reference solution withh = 3.125e−03.)
Summarizing these results, we state that the stage order is obtained in allcases. As in the case of a singularity of the first kind (α = 1), superconver-gence of Gauss methods cannot be observed except in some special cases.
29
4.2 Convergence results for first and second
derivatives
For collocation applied to regular boundary value problems it is well knownthat the first derivative is approximated (uniformly in t) with the sameasymptotic quality as the solution itself. For higher derivatives, the order ofthis approximation gradually decreases (cf. e.g. Theorem 2.3.1 or [AMR88]).For equidistant collocation nodes, for instance, we have
p(k)coll(t)− z(k)(t) = O(hm+1−k)
for k = 1, . . . ,m, where m is the stage order of the method.In this section, we investigate the approximation of the first and second
derivatives by the collocation method implemented in sbvp when applied toproblems with an essential singularity. This is motivated by the importantrole which the approximation quality of the derivatives plays in the theoryof error estimates (cf. e.g. [AKW02]).
For some of our examples this numerical investigation is difficult or im-possible. In particular, for the case where the exact solution is not known(Examples (3.2), (3.3), (3.4), (3.7)), it was not possible to generate a ref-erence solution sufficiently accurate for this purpose. Therefore these ex-periments were only performed for Examples 3.1, 3.5 and 3.6 (with exactsolution known). For these examples, the derivative of the global error wasobtained as the difference between the derivative of the polynomial (colloca-tion) approximation, obtained using the MATLAB function polyder, and thederivative of the exact solution.
• Example 1 ( (3.1) on page 21 )
Here, also for higher values of the parameter α, the asymptotic approx-imation quality of the first and second derivatives is nearly as good asin the regular case α = 0, namely O(hm) and O(hm−1), respectively;see Tables 1.4–1.11. Slight order reductions which might(?) occur de-pending on α > 1 are hard to diagnose numerically.
Note that at the mesh points the polynomial from the interval left tothis point has been used.
• Example 5 ((3.5) on page 23)
For this 2-dimensional example, the results are very similar as for Ex-ample 1, see Tables 5.2–5.3.
• Example 6 ((3.6) on page 24)
30
Here, for higher values of α, the observed convergence orders are signif-icantly smaller than m or m− 1, respectively, see Tables 6.4–6.11. Forα > 1, the amount of order reduction seems to be about O(1− 1/α).
The conclusion is: An order reduction may occur in the approximation ofthe derivatives which is, however, not severe for moderate values of α.
31
Chapter 5
A posteriori error estimate andconditioning
Up to now we only have investigated the performance of the collocation solversbvpcol. Concerning the performance of the full code sbvp, however, anotherimportant question is whether, in the case of an essential singularity, the aposteriori error estimate implemented in sbvp produces an asymptoticallycorrect global error estimate.
Note that for regular problems or problems with a singularity of thefirst kind, the asymptotic correctness of this estimate has been proven, see[AKW02], [AKW]. For problems with an essential singularity, however, nei-ther theoretical nor numerical results have been available so far.
For the algorithmic details of this estimate we refer to [AKW02]. Here weonly note that it is based on the idea of defect correction, and that a cheaplow order scheme (backward Euler) is applied twice to estimate the error ofsbvpcol.
As we shall see, this estimate does not work satisfactorily in the case of anessential singularity. There is strong evidence that this is (at least partially)caused by the ill-conditioning of the backward Euler equations in this case.Therefore, we also pay attention to the conditioning of the various systemmatrices involved, and we also shall try to apply preconditioning techniquesto improve conditioning where necessary.
32
5.1 A posteriori error estimate and condi-
tioning in sbvp
5.1.1 Conditioning of the collocation equations
For solving a system of collocation equations, say F (p) = 0, sbvpcol appliesNewton’s method, where the Jacobian matrix DF corresponds to a linearizedversion of the collocation equations. For numerical stability and for theperformance of Newton’s method, the condition number of DF ,
cond(DF ) = ‖DF‖ ‖DF−1‖,is a crucial parameter.
In the sequel we will investigate these condition numbers and their asymp-totical orders w.r.t. h, using the MATLAB functions cond and condest. Thebest possible behavior we might expect is cond(DF ) = O(h−1), which holdsfor a stable, correctly scaled discretization of a regular, first order boundaryvalue problem. (For a system of collocation equations, ‘correct scaling’ meansthat the coefficients in the continuity and boundary conditions are scaled toh−1, which is the natural scaling occurring in the collocation equations; see[AKKW02]).
Note that, by default, cond(A) returns the condition number of a matrix Aw.r.t. the 2-norm, while condest(A) computes a lower bound for the conditionnumber w.r.t. the 1-norm.
Our corresponding numerical results are discussed in the sequel.
• Example 1 ( (3.1) on page 21 )
The results given in Tables 1.12 and 1.13 show that the condition num-bers of the matrices DF from sbvpcol have the optimal asymptoticbehavior O(h−1) for α = 0 and α = 1, For α > 1, the essential sin-gularity causes them to decrease (approximately) like h−α, see Tables1.14 and 1.15.
• Example 2 ( (3.2) on page 21 )
Also in this case, the order of cond(DF ) significantly decreases withincreasing β and is observed to be near −(β + 2), see Tables 2.4–2.6.(Note that for all values β = 0, 1, 2, this is a problem with an essentialsingularity.)
• Example 3 ( (3.3) on page 22 )
For this nonlinear example we observe cond(DF ) = O(h−3), which maybe expected due the t−3-term present in the Jacobian, see Table 3.2.
33
• Example 4 ( (3.4) on page 23 )
We observe the same behavior as before: cond(DF ) ≈ h−α, whereα = 2 characterizes the degree of singularity; see Table 4.2.
• Example 5 ( (3.5) on page 23 )
Again we observe cond(DF ) ≈ h−α, with α = 2, see Table 5.4.
• Example 6 ( (3.6) on page 24 )
The results for this example can be found in Tables 6.12–6.15. Theinfluence of α on the condition number is again clearly visible.
These results can be interpreted in the following way: In all cases, the (nega-tive) power of h which was observed in cond(DF ) corresponds to the highestnegative power, say −α, of t occurring in the right hand side of the corre-sponding singular problem, which implies ‖DF‖ = O(h−α). But this meansthat ‖DF−1‖ = O(1), and in this sense the collocation methods applied tothese singular problems might be called stable, which is consistent with theresults from §4.1.
However, we have to be careful with this interpretation: Such a ‘stabilityestimate’, even if it can be proven theoretically, does not yet imply that thecollocation method is convergent (with stage order m), as observed in §4.1.
Moreover, as we shall see in §5.3 for the case of the implicit midpoint rule,there exist examples where the asymptotic conditioning can be improved bya clever choice of a preconditioner.
5.1.2 A posteriori error estimates
We now describe our results concerning the performance of the a posteriorierror estimate implemented in sbvp (cf. [AKW02]) for problems with anessential singularity.
An overview of these results can be seen from Figures 1.1, 2.1, . . . , 7.1and 9.1, where the exact global error (‘exact’) is plotted together with itsestimate produced by the function sbvperr called by sbvp (a plot of thesolution is also provided) for default tolerances tol = 1e−12. In particular,singularities of the first kind (where theory and numerical experience tellsus that sbvperr gives reliable results) are compared with essentially singularsituations.
• For Example 1, the corresponding results for α = 1, 2, 3 are displayedin Figure 1.1.
34
In contrast to α = 1, the quality of the error estimate deteriorates nearthe singular point for α = 2.
For α = 3, Figure 1.1 shows that the error estimate completely fails.
The results of the similar experiment done with a less smooth functiong(t) = cos(15t) are presented in Tables 9.1–9.3 and Figure 9.1.
• In Figure 2.1 we observe a similar effect for the essentially singularExample 2, with β = 0, 1, 2.
• In Figures 3.1, 4.1, 5.1 and 7.1, plots of the error estimate provided bysbvperr are presented for Examples 3, 4, 5 and 7. A similar, but lessdrastic, behavior as before is observed.
• Figure 6.1 demonstrates a behavior of the error estimate for Example 6similar to that presented in Figure 1.1.
These experimental results clearly show that the error estimate implementedin sbvperr is useless for problems with an essential singularity.
This is possibly caused by the inability of the auxiliary scheme – thebackward Euler method – to cope with such a problem type. Therefore, inthe next section we consider the Euler scheme and investigate its convergenceand conditioning properties.
5.2 The backward Euler method
The backward Euler discretization of a boundary value problem for a firstorder ODE system y′ = F (t, y) is given by the simple difference scheme
ηi − ηi−1
hi
= F (ti, ηi), i = 1, . . . , N (5.1)
on a grid {ti}, with step size hi = ti − ti−1, plus boundary conditions. Itsclassical convergence order is 1.
This can also be interpreted as a collocation method producing apiecewise linear approximation on each interval [ti−1, ti]. Therefore, a slightmodification of the routine sbvpcol enabled us to perform experiments forthe backward Euler scheme, with a constant step size hi ≡ h. This is notthe most efficient way to implement (5.1) but it was convenient for a first test.
For a more thorough investigation of (5.1) we have concentrated on Ex-ample 1, with different choices of the parameter α. Although this is a simplescalar terminal value problems, the above (negative!) results concerning errorestimation show that it is worthwhile to have a closer look at this example.
35
5.2.1 Backward Euler by means of sbvpcol
Tables 1.16 and 1.17 show numerical results for Example 1. For α = 0, 1, 2,the conventional convergence order 1 is observed, but for α = 3 the methodis rapidly divergent (Table 1.16). This effect may be related to the ill-conditioning of the Euler system for α > 1, which can be clearly seen fromTable 1.17: While for α = 2 the ill-conditioning is present but not fatallylarge, the condition number explodes for h → 0 in the case of α = 3.
Of course, a terminal value problem is a very special case of a boundaryvalue problem, and its numerical solution by a one-step scheme is a simpleiteration. Therefore the size of the condition number of the correspondingbidiagonal (Euler) matrix may be not so significant as in the general case.However, it is worthwhile to test preconditioning as a possible means toimprove the convergence properties.
5.2.2 An implementation of backward Euler with pre-conditioning
In sbvpcol, standard preconditioning is implemented for the continuity equa-tions, which is also optimal for the special case of backward Euler for regularsituations or problems with a singularity of the first kind. For the essen-tially singular case the condition numbers are very large, and the question iswhether this can be improved by a proper scaling (depending on the valueof α). To investigate this question we have developed a direct MATLAB im-plementation for the backward Euler equations for the case of a linear scalarboundary value problem (independent of sbvpcol), which enables us to testdifferent ways of preconditioning.
For Example 1, the backward Euler scheme yields the linear system
ηi − ηi−1
h=
1
tαiηi + g(ti), i = 1, . . . , N, (5.2a)
ηN = e, (5.2b)
with g(t) = et − et/tα. N is number of mesh points (N ∈{10, 20, 40, 80, 160}),
36
and h = 1/N . The matrix of this linear system reads (dimension N+1×N+1)
A =
− 1h
1h− 1
tα10 . . . 0 0 0
0 − 1h
1h− 1
tα2. . . 0 0 0
......
.... . .
... 0 00 0 0 . . . 1
h− 1
tαN−20 0
0 0 0 . . . − 1h
1h− 1
tαN−10
0 0 0 . . . 0 − 1h
1h− 1
tαN
0 0 0 . . . 0 0 1h
(5.3)
with a natural preconditioning for the boundary (end) condition alreadyincluded (last row). In Tables 1.18 and 1.20 the corresponding conditionnumbers are given (with respect to the Euclidean and maximum norms) andtheir size is in accordance with the results from Table 1.17.
In further series of experiments we have tested different versions of pre-conditioning this matrix, with the aim of improving its condition number andthe method’s asymptotic behavior for h → 0 for α > 1. A slight improvementwas obtained by diagonal (left)-preconditioning, i.e. premultiplication of Awith a diagonal matrix P with entries tα−1
i . The corresponding conditionnumbers are given in Tables 1.19 and 1.21. However, these numbers are stillvery unsatisfactory.
Our conclusion from these experiments is simply the following: For prob-lems with an essential singularity, the backward Euler scheme must be ex-pected to lead to ill-conditioned systems and to behave unstably, with noobvious way to improve this behavior by preconditioning.
For our simple Example 1, a natural explanation is the following: Since anend condition is specified at the right endpoint, the ODE is integrated fromright to left and the backward Euler scheme is de facto an explicit method.The corresponding iteration reads
ηi−1 := (1− h/tαi )ηi − hg(ti), i = N, N−1, . . . ,
starting with ηN = e, where the factor (1 − h/tαi ) becomes large for α > 1near t = 0. This hints at an unstable behavior.
Thus, a possible remedy could be to use the forward Euler scheme – butonly for this particular example, certainly not in general. However, instead weconsider another simple (symmetric!) difference scheme, namely the implicitmidpoint rule, which should behave more robustly and for which theoreticalconvergence results are available, cf. [HW79].
37
5.3 Midpoint rule
The implicit midpoint rule corresponds to the difference scheme
ηi − ηi−1
hi
= F
(ti−1 + ti
2,ηi−1 + ηi
2
), i = 1, . . . , N, (5.4)
and it is also equivalent to the collocation method of Gauss type with stageorder m = 1 (collocation at the midpoint of each collocation interval [ti−1, ti]).Its classical (super-) convergence order is 2.
Again we restrict our considerations to our scalar model problem, Exam-ple 1.
5.3.1 Midpoint rule by means of sbvpcol
In Tables 1.22 and 1.23 the numerical results for the implicit midpoint rule(realized using sbvpcol) are shown. Here, the full (super-) convergence order2 is observed, despite the fact that the condition numbers are large and growwith α. In general, however, we can only expect a convergence order 1 + γ,where 0 < γ = γ(α) < 1, cf. [HW79].
If we intend to consider the midpoint rule as an alternative to the Eulerscheme for the purpose of error estimation, an appropriate preconditioningmay play a crucial role. Therefore, we have tested different ways of precon-ditioning.
5.3.2 An implementation of the midpoint rule withpreconditioning
In Tables 1.24–1.27 we present the condition numbers obtained for Exam-ple 1, using a direct implementation of the midpoint rule, with and withoutpreconditioning.
For Example 1, the midpoint rule with constant step size h = 1/N yieldsthe linear system
ηi − ηi−1
h=
ηi−1 + ηi
2 ti−1/2α
+ g(ti−1/2), i = 1, . . . , N, (5.5a)
ηN = e, (5.5b)
where ti−1/2 := ti−h/2 and g(t) = et− et
tα. The corresponding system matrix
38
reads (end condition scaled to 1/h as before)
B =
− 1h− 1
2t 12
α1h− 1
2t 12
α . . . 0 0
0 − 1h− 1
2t 32
α . . . 0 0
......
. . ....
...0 0 . . . 0 00 0 . . . 1
h− 1
2t(N− 3
2 )α 0
0 0 . . . − 1h− 1
2t(N− 1
2 )α
1h− 1
2t(N− 1
2 )α
0 0 . . . 0 1h
. (5.6)
The results obtained for the observed asymptotical order of the conditionnumbers of B are presented in Tables 1.24 and 1.26, for the Euclidean andmaximum norm, respectively. We see that the condition numbers are large(however, not so extremely large as for the implicit Euler scheme), withnegative orders depending on α.
For α > 1, the asymptotical behavior of the size of the entries in the i-throw of the matrix B, for i → 0, i.e., for t → 0, is O(t−α). Therefore a naturalidea is to ‘cancel’ this effect by (left)-preconditioning with factors O(tα−1),such that all rows are scaled to O(1/h). If we multiply the i-th row by afactor tα−1
i (as above for the backward Euler scheme), we obtain the modifiedmatrix
B′ =
− tα−11
h− tα−1
1
2t 12
α
tα−11
h− 2tα−1
1
2t 12
α . . . 0 0
0 − tα−12
h− tα−1
2
2t 32
α . . . 0 0
......
. . ....
...0 0 . . . 0 0
0 0 . . .tα−1N−1
h− tα−1
N−1
2tN− 3
2
α 0
0 0 . . . − tα−1N
h− tα−1
N
2tN− 1
2
α
tα−1N
h− tα−1
N
2tN− 1
2
α
0 0 . . . 0 1h
,
(5.7)which evidently satisfies ‖B′‖∞ = O(h−1). Now the question is how the norminverse B′−1 behaves for h → 0. Such an estimate for ‖B′−1‖ is not easy toobtain analytically; some preliminary results are given in Chapter 7.
Our corresponding numerical results concerning the norm of the inverseB′−1 can be found in Table 1.25 (2-norm) and Table 1.27 (max-norm). Theseresults show that our preconditioner is optimal - for both norms we observe
39
‖B′−1‖ = O(1) for h → 0. Together with ‖B′‖ = O(h−1) this is the optimalestimate which may be expected, i.e., our preconditioner is optimal1.
Due to our numerical results obtained for the midpoint rule, this methodappears to be a promising candidate for the purpose of a posteriori error es-timation via defect correction, instead of the backward Euler scheme whichfails for α > 1. However, in the present version of sbvp, this way of errorestimation is not implemented. Therefore we content ourselves with a cor-responding test for Example 1, since this is a terminal value problem whichcan be solved by an initial value code available at our department.
This experiment is documented in Table 1.43. We see that – in contrast tothe backward Euler method – the quality of this error estimate is completelysatisfactory for α = 0 and α = 1, but begins to deteriorate for larger valuesof α. It seems not to be really robust w.r.t. an essential singularity.
1Obviously, our preconditioner does not provide a proper scaling for α = 0. Conse-quently, we can neglect the (unsatisfactory) results for this case.
40
Chapter 6
Error estimation based on ah-h/2 strategy
In Chapter 5 we have seen that the a posteriori error estimation procedureimplemented in sbvp is completely unreliable for problems with an essentialsingularity. The modified version based on the midpoint rule presented at theend of Chapter 5 performed significantly better, but with decreasing qualityfor larger values of α.
In this section we present our numerical results for a classical, but compu-tationally more expensive procedure based on grid halving (‘h-h/2 estimate’).The classical justification for the h-h/2 estimate is based on an asymptoticrepresentation for the global error,
zh = z + Chp + O(hp+1), zh/2 = z + C(h/2)p + O(hp+1), (6.1)
from which the estimate
zh − z =1
1− 2−p(zh − zh/2) + O(hp+1) (6.2)
can be immediately derived. (Here, z denotes the exact solution, and zh, zh/2
are the collocation solutions obtained with step size h and h/2, respectively.)For problems with an essential singularity, so far there exists no theoreticaljustification for (6.1). However, our satisfactory numerical results for collo-cation methods reported in Chapter 4 motivate us to test the performanceof the h-h/2 estimate.
Thus, we compute two approximations using sbvpcol, with step sizes hand h/2, respectively, and estimate the global error z − zh on the basis of(6.2). Thus we compute
eest :=1
1− 2−p(zh − zh/2),
41
compare this estimate with the true error etrue = zh − z, and observe theasymptotical order of the difference, which should hopefully be higher thanthe convergence order of the underlying method, i.e. the order of etrue. Inthose cases where the exact solution z is not known, we again use a referencesolution obtained on a very fine mesh to obtain etrue as precisely as possible.
All numerical results to be discussed in the sequel are based on equidistantgrids and an even number of collocation points (m = 2, 4, 6). In the corre-sponding Tables, ‘err’ denotes the norm of the error of the error estimate,i.e. err = ‖eest − etrue‖, and p′ denotes its observed order.
Here are our results for the seven test problems from Chapter 3:
• Example 1
The results are given in Tables 1.34–1.37, for α = 0, 1, 2, 3, respectively.These numbers are to be compared with Tables 1.30–1.33 showing theglobal errors ‖etrue‖ (with observed orders p = m).
We throughout observe uniform values p′ ≈ p + 1, or higher, and fur-thermore p′ ≈ p + 2 at the mesh points.
The asymptotic quality of this estimate is superior to the estimatebased on the midpoint rule discussed above, especially for higher valuesof α.
We also have tested this example using a reference solution instead ofthe exact solution, in order to test the effect of the resulting inaccuracyin etrue. Tables 1.38–1.41 show that, on finer grids, this effect spoilsthe orders which would be otherwise observed. The same has to beexpected for all examples where only a numerical reference solution isavailable (Examples 2, 3, 4 and 7).
• Example 2
The results for β = 0, 1, 2 are given in Tables 2.10–2.12, based on areference solution obtained numerically. These are to be comparedwith Tables 2.7–2.9 showing the corresponding global errors ‖etrue‖.Although these results are less ‘regular’ than for Example 1 concerningthe orders observed, the absolute values of the error estimate comparedto the errors show that the h-h/2 estimate yields reliable results.
• Example 3
Tables 3.3 and 3.4 show the global errors and the corresponding resultsfor the estimate, respectively (based on a numerical reference solution).Our conclusions are the same as for Example 2.
42
• Example 4
See Tables 4.3 and 4.4; similar remarks as for Examples 2 and 3 apply.Here, for m = 6 a significant order reduction is observed. Such aneffect is usually caused by a certain lack of smoothness of z, but theuse of reference solution may also play a role. We have not furthertried to analyze this point; anyway, the h-h/2 estimate again behavessatisfactorily.
• Example 5
See Tables 5.5 and 5.6 (exact solution known). The observed orders areacceptable, but their variation in the case of the estimate is somewhatirregular. However, the performance of the estimate is satisfactory.
• Example 6
See Tables 6.16–6.19 for the global errors, and Tables 6.20–6.23 for theerror of the corresponding h-h/2 estimate (α = 0, 1, 2, 3, exact solutionknown). We may draw similar conclusions as for Example 5.
• Example 7
See Tables 7.3 and 7.4, obtained using a reference solution. Again, theorders observed for the error estimate are irregular, but, throughout,the error of the estimate appears to be significantly smaller than theerror itself.
Summarizing, we may claim that the method of a posteriori error estima-tion by h-h/2 strategy should seriously be considered for implementation ina further version of sbvp. Our tests have shown that this error estimatebehaves robustly and reliably for all problems tested featuring an essentialsingularity.
43
Chapter 7
On the stability of thepreconditioned midpoint rule
This chapter is devoted to a theoretical explanation for the positive effect ofpreconditioning observed for the midpoint rule in Example 1.
Let us consider the case of an essential singularity with α ≥ 2. We use aslight but natural modification of our preconditioner: The rows of the matrixB (cf. 5.6) are not premultiplied by tα−1
i but by tα−1i−1/2. Numerically, this
makes hardly a difference: cf. Table 1.42, to be compared with Table 1.27,which indicate that ‖B′−1‖ = O(1) holds for both versions of B′. But thestructure of the modified matrix becomes simpler with the latter choice. Wethus consider
B′ =
t 12
α−1(− 1h− 1
2t 12
α ) t 12
α−1( 1h− 1
2t 12
α ) . . . 0
0 t 32
α−1(− 1h− 1
2t 32
α ) . . . 0
......
. . ....
0 0 . . . tα−1N− 1
2
( 1h− 1
2tN− 1
2
α )
0 0 . . . 1/h
.
In the sequel we try to derive a bound for the norm of the inverse B′−1, inorder to prove ‖B′−1‖ = O(1) for h → 0. Throughout, ‖ · ‖ denotes themaximum norm for matrices.
1. First, we apply the following simple estimate, which can easily be provento be correct:
If D is an invertible matrix sufficiently ‘close’ to B′, in the sense that
‖I −D−1B′‖ ≤ m < 1
44
holds, then B′ is also invertible and satisfies
‖B′−1‖ ≤ 1
1−m‖D−1‖. (7.1)
We now apply this estimate with D = diag(B′) (the diagonal of B′). Withthe denotation
sk :=tα−1k− 1
2
h+
1
2tk− 12
,
s′k :=tα−1k− 1
2
h− 1
2tk− 12
,
we have
D−1 =
− 1s1
0 0 . . . 0 0 0
0 − 1s2
0 . . . 0 0 0
0 0 − 1s3
. . . 0 0 0
......
.... . .
...... 0
0 0 0 . . . 0 − 1sN
0
0 0 0 . . . 0 0 h
(7.2)
and
I −D−1B′ =
0 − s′1s1
0 0 0 0 0
0 0 − s′2s2
0 0 0 0
0 0 0 . . . 0 0 0
......
.... . .
...... 0
0 0 0 . . . 0 − s′NsN
0
0 0 0 . . . 0 0 0
. (7.3)
Thus we have
‖D−1‖ = max{h, max1≤k≤N
∣∣∣ 1
sk
∣∣∣}with
max1≤k≤N
∣∣∣ 1
sk
∣∣∣ ≤(
min1≤k≤N
|sk|)−1
≤(
minh/2≤t≤1−h/2
|f(t; h, α)|)−1
,
where
f(t; h, α) =tα−1
h+
1
2t=
tα + h/2
th.
45
An elementary calculation shows that, in the interval [h/2, 1−h/2] this func-
tion attains its minimum at tmin = (h/(2(α−1)))1/α, with
(f(tmin))−1 = O(h1/α).
Thus we have‖D−1‖ = O(h1/α) for h → 0. (7.4)
Furthermore,
‖I −D−1B′‖ = max1≤k≤N
∣∣∣s′k
sk
∣∣∣ = max1≤k≤N
|g(tk−1/2; h, α)| ≤≤ max
h/2≤t≤1−h/2|g(t; h, α)|
where
g(t; h, α) =tα−1
h− 1
2ttα−1
h+ 1
2t
=tα − h
2
tα + h2
.
It is easy to verify that g(t; h, α) is monotonously increasing fort ∈ [h/2, 1− h/2], and for α ≥ 2, |g(t; h, α)| takes its maximum at t = h/2,with
|g(h/2; h, α)| = 1− (h/2)α−1
1 + (h/2)α−1 = 1−O(hα−1).
Thus we have
‖I −D−1B′‖ = m with m = 1−O(hα−1). (7.5)
Finally, (7.1) together with (7.4) and (7.5) yield
‖B′−1‖ = O(h1−α) ·O(h1/α) = O(h−1+1/α) for h → 0. (7.6)
This is not what we had hoped for, namely a uniform O(1) estimate. Forarbitrary α ≥ 2, we only obtain the uniform bound ‖B′−1‖ ≤ O(h−1). Theabove estimate is obviously too crude.
2. To obtain a better estimate, we explicitly computed the inverse B′−1. Tothis end, we used the computer algebra system MAPLE to invert the 5 × 5version of B′,
(− t 12
α−1
h− 1
2t 12
) (t 12
α−1
h− 1
2t 12
) 0 0 0
0 (− t 32
α−1
h− 1
2t 32
) (t 32
α−1
h− 1
2t 32
) 0 0
0 0 (− t 52
α−1
h− 1
2t 52
) (t 52
α−1
h− 1
2t 52
) 0
0 0 0 (tα−172
h− 1
2t 72
) (tα−172
h− 1
2t 72
)
0 0 0 0 1/h
.
46
Its inverse reads
− 2α−1hhα−1+2α−1 M
01 − 2α−13h
3αhα−1+2α−1 M11 − 2α−15h
5αhα−1+2α−1 M21 − 2α−17h
7αhα−1+2α−1 M31 hM4
1
0 − 2α−13h3αhα−1+2α−1 M
12 − 2α−15h
5αhα−1+2α−1 M22 − 2α−17h
7αhα−1+2α−1 M32 hM4
2
0 0 − 2α−15h5αhα−1+2α−1 M
23 − 2α−17h
7αhα−1+2α−1 M33 hM4
3
0 0 0 − 2α−17h7αhα−1+2α−1 M
34 hM4
4
0 0 0 0 h
,
where
M lk :=
{ ∏lµ=k
(2µ−1)αhα−1−2α−1
(2µ−1)αhα−1+2α−1 , k ≤ l
1, k > l.(7.7)
From this result it is easy to guess the structure of the inverse B′−1 for generaldimension, which can be verified to be correct. To obtain a sharp estimatefor ‖B′−1‖, it is now necessary to estimate terms of the form M l
k for arbitraryk ≤ l. We tried to derive such estimates using asymptotic properties of theGamma function. However, a complete result including the desired estimatefor ‖B′−1‖ could not be obtained so far.
47
Chapter 8
Lobatto distribution
We now briefly report a few numerical experiments which have been obtainedusing sbvp with collocation points of Lobatto type. In particular, we use the3-stage scheme, where the endpoints and the midpoint of each collocationinterval are used as collocation points. This choice is motivated by the factthat this is exactly the scheme which is used in the standard MATLAB codebvp4c.
Note that the application of such a scheme to a singular problem is notstraightforward: The right-hand side of the given ODE cannot be evaluatedat t = 0, which is a collocation point. For test examples where the exactsolution is known the remedy is simple, namely we provide the value of z′(0)instead. In the general case, Lemma 2.2.5 could provide this value.
In Tables 1.28 and 1.29 the respective results for Exampe 1 are given.For α = 0 and α = 1 we obtain the classical convergence order m + 1 = 4,but for α > 1 a slight order reduction is visible (Table 1.28). Moreover, thecondition number of the associated system matrix grows with α and becomesvery large (Table 1.29).
It is at least doubtful whether Lobatto methods are suitable candidatesfor the solution of BVPs with an essential singularity. However, more testruns will be necessary to get a clear picture.
48
Bibliography
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[ACR81] , Collocation software for boundary value ODEs, ACMTransactions on Mathematical Software 7 (1981), no. 2, 209–222.
[AKKW] W. Auzinger, G. Kneisl, O. Koch, and E. Weinmuller,A collocation code for boundary value problems in ordi-nary differential equations, To appear in Numer. Algo-rithms. Also available as ANUM Preprint Nr. 18/01 athttp://www.math.tuwien.ac.at/~inst115/preprints.htm.
[AKKW02] , A solution routine for singular boundaryvalue problems, Techn. Rep. ANUM Preprint Nr.1/02, Inst. for Appl. Math. and Numer. Anal., Vi-enna Univ. of Technology, Austria, 2002, Available athttp://www.math.tuwien.ac.at/~inst115/preprints.htm.
[AKW] W. Auzinger, O. Koch, and E. Weinmuller, Analysis of a new er-ror estimate for collocation methods applied to singular boundaryvalue problems., Submitted to SIAM J. Numer. Anal.
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[BS73] C. de Boor and B. Swartz, Collocation at Gaussian points, SIAMJ. Numer. Anal. 10 (1973), 582–606.
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[HW76] F.R. de Hoog and R. Weiss, Difference methods for boundaryvalue problems with a singularity of the first kind, SIAM J. Nu-mer. Anal. 13 (1976), 775–813.
[HW77] , The application of linear multistep methods to singularinitial value problems, Math. Comp. 32 (1977), 676–690.
[HW78] , Collocation methods for singular boundary value prob-lems, SIAM J. Numer. Anal. 15 (1978), 198–217.
[HW79] , The numerical solution of boundary value problems withan essential singularity, SIAM J. Numer. Anal. 16 (1979), 637–669.
[HW80a] , An approximation theory for boundary value problemson infinite intervals, Computing 24 (1980), 227–239.
[HW80b] , On the boundary value problem for systems of ordinarydifferential equations with a singularity of the second kind, SIAMJ. Math. Anal. 11 (1980), 41–60.
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50
51
Figure 1.1 : plot of solution and error using sbvp for Example 1
1. a) Solution, α =1 b) Error, α =1
2. a) Solution, α =2 b) Error, α = 2
3. a) Solution, α =3 b) Error, α = 3
52
TABLE 1.0 : global error for Example 1 -α=0-
1. sbvpcol, equidistant, m=4
h err mesh
err coll
p mesh
p coll
const mesh
const coll
1.00e-01 2.11e-08 2.18e-08 5.00e-02 1.32e-09 1.34e-09 4.00 4.02 2.12e-04 2.28e-04 2.50e-02 8.25e-11 8.31e-11 4.00 4.01 2.11e-04 2.22e-04 1.25e-02 5.15e-12 5.18e-12 4.00 4.01 2.11e-04 2.17e-04 6.25e-03 3.22e-13 3.23e-13 4.00 4.00 2.11e-04 2.14e-04
2. sbvpcol, Gauss, m=2
h err mesh
err coll
p mesh
p coll
const mesh
const coll
1.00e-01 1.39e-07 2.10e-05 5.00e-02 8.68e-09 2.68e-06 4.00 2.97 1.39e-03 1.97e-02 2.50e-02 5.42e-10 3.38e-07 4.00 2.99 1.39e-03 2.06e-02 1.25e-02 3.39e-11 4.24e-08 4.00 2.99 1.39e-03 8.64e-01 6.25e-03 2.12e-12 5.31e-09 4.00 3.00 1.37e-03 2.14e-02
3. sbvpcol, Gauss, m=3
h err mesh
err coll
p mesh
p coll
const mesh
const coll
5.000e-01 1.57e-07 6.87e-05 2.500e-01 2.43e-09 4.88e-06 6.01 3.82 1.01e-05 9.68e-04 1.250e-01 3.79e-11 3.25e-07 6.00 3.91 9.98e-06 1.10e-03 6.250e-02 5.92e-13 2.09e-08 6.00 3.95 9.93e-06 1.20e-03 3.125e-02 9.33e-15 1.33e-09 5.99 3.98 9.57e-06 1.29e-03
4. sbvpcol, Gauss, m=4
h err mesh
err coll
p mesh
p coll
const mesh
const coll
5.000e-01 1.55e-10 1.64e-06 2.500e-01 6.02e-13 5.77e-08 +8.01 4.83 3.99e-08 4.65e-05 1.250e-01 2.44e-15 1.91e-09 +7.95 4.91 3.66e-08 5.24e-05 6.250e-02 4.44e-16 6.17e-11 +2.46 4.96 4.06e-13 5.73e-05 3.125e-02 8.88e-16 1.96e-12 –1.00 4.98 2.78e-17 6.09e-05
53
TABLE 1.1 : global error for Example 1 -α=1-
1. sbvpcol, equidistant, m=4
h err mesh
err coll
p mesh
p coll
const mesh
const coll
1.00e-01 1.53e-08 1.63e-08 5.00e-02 9.57e-10 9.88e-10 4.00 4.05 1.51e-04 1.83e-04 2.50e-02 5.98e-11 6.08e-11 4.00 4.02 1.54e-04 1.69e-04 1.25e-02 3.73e-12 3.77e-12 4.00 4.01 1.53e-04 1.63e-04 6.25e-03 2.35e-13 2.41e-13 3.99 3.97 1.45e-04 1.34e-04
2. sbvpcol, Gauss, m=2
h err mesh
err coll
p mesh
p coll
const mesh
const coll
1.00e-01 9.61e-06 2.01e-05 5.00e-02 1.18e-06 2.67e-06 3.03 2.91 1.03e-02 1.97e-02 2.50e-02 1.46e-07 3.37e-07 3.01 2.99 9.73e-03 2.07e-02 1.25e-02 1.82e-08 4.24e-08 3.00 2.99 9.34e-03 2.08e-02 6.25e-03 2.27e-09 5.31e-09 3.00 3.00 9.32e-03 2.08e-02
3. sbvpcol, Gauss, m=3
h err mesh
err coll
p mesh
p coll
const mesh
const coll
1.00e-01 5.42e-08 1.34e-07 5.00e-02 3.32e-09 8.63e-09 4.03 3.96 5.79e-04 1.22e-03 2.50e-02 2.05e-10 5.46e-10 4.01 3.98 5.55e-04 1.30e-03 1.25e-02 1.28e-11 3.44e-11 4.01 3.99 5.40e-04 1.35e-03 6.25e-03 7.97e-13 2.16e-12 4.00 4.00 5.32e-04 1.45e-03
4. sbvpcol, Gauss, m=4
h err mesh
err coll
p mesh
p coll
const mesh
const coll
1.00e-01 2.48e-10 6.35e-10 5.00e-02 7.60e-12 2.03e-11 5.03 4.97 2.65e-05 5.89e-05 2.50e-02 2.35e-13 6.43e-13 5.01 4.98 2.53e-05 6.14e-05 1.25e-02 7.33e-15 2.04e-14 5.00 4.98 2.45e-05 6.01e-05 6.25e-03 1.33e-15 2.22e-15 2.46 3.20 3.51e-10 2.50e-08
54
TABLE 1.2 : global error for Example 1 -α=2-
1. sbvpcol, equidistant, m=4
h err mesh
err coll
p mesh
p coll
const mesh
const coll
1.00e-01 1.29e-08 1.40e-08 5.00e-02 8.00e-10 8.37e-10 4.01 4.06 1.31e-04 1.60e-04 2.50e-02 5.00e-11 5.11e-11 4.00 4.03 1.28e-04 1.48e-04 1.25e-02 3.12e-12 3.16e-12 4.00 4.02 1.28e-04 1.39e-04 6.25e-03 1.95e-13 1.96e-13 4.00 4.01 1.28e-04 1.35e-04
2. sbvpcol, Gauss, m=2
h err mesh
err coll
p mesh
p coll
const mesh
const coll
1.00e-01 3.63e-05 3.63e-05 5.00e-02 5.55e-06 5.55e-06 2.71 2.71 1.86e-02 1.86e-02 2.50e-02 8.59e-07 8.59e-07 2.69 2.69 1.75e-02 1.75e-02 1.25e-02 1.34e-07 1.34e-07 2.68 2.68 1.69e-02 1.69e-02 6.25e-03 2.10e-08 2.10e-08 2.68 2.68 1.69e-02 1.69e-02
3. sbvpcol, Gauss, m=3
h err mesh
err coll
p mesh
p coll
const mesh
const coll
1.00e-01 1.69e-07 1.69e-07 5.00e-02 8.56e-09 8.62e-09 4.30 4.29 3.37e-03 3.30e-03 2.50e-02 4.84e-10 5.46e-10 4.14 3.98 2.08e-03 1.30e-03 1.25e-02 2.92e-11 3.43e-11 4.05 3.99 1.49e-03 1.35e-03 6.25e-03 1.70e-12 2.15e-12 4.10 4.00 1.85e-03 1.40e-03
4. sbvpcol, Gauss, m=4
h err mesh
err coll
p mesh
p coll
const mesh
const coll
1.00e-01 1.10e-09 1.10e-09 5.00e-02 4.03e-11 4.03e-11 4.77 4.77 6.51e-05 6.51e-05 2.50e-02 1.55e-12 1.55e-12 4.70 4.70 5.25e-05 5.25e-05 1.25e-02 6.02e-14 6.02e-14 4.69 4.69 4.97e-05 4.97e-05 6.25e-03 2.00e-15 2.22e-15 4.91 4.76 1.34e-04 6.90e-05
55
TABLE 1.3 : global error for Example 1 -α=3-
1. sbvpcol, equidistant, m=4
h err mesh
err coll
p mesh
p coll
const mesh
const coll
1.00e-01 1.11e-08 1.24e-08 5.00e-02 7.02e-10 7.39e-10 3.98 4.07 1.06e-04 1.45e-04 2.50e-02 4.38e-11 4.50e-11 4.00 4.04 1.13e-04 1.32e-04 1.25e-02 2.73e-12 2.77e-12 4.00 4.02 1.13e-04 1.24e-04 6.25e-03 1.72e-13 1.73e-13 3.99 4.00 1.07e-04 1.15e-04
2. sbvpcol, Gauss, m=2
h err mesh
err coll
p mesh
p coll
const mesh
const coll
1.00e-01 6.83e-05 6.83e-05 5.00e-02 1.16e-05 1.16e-05 2.55 2.55 2.42e-02 2.42e-02 2.50e-02 2.01e-06 2.01e-06 2.54 2.54 2.34e-02 2.34e-02 1.25e-02 3.49e-07 3.49e-07 2.53 2.53 2.27e-02 2.27e-02 6.25e-03 6.09e-08 6.09e-08 2.52 2.52 2.18e-02 2.18e-02
3. sbvpcol, Gauss, m=3
h err mesh
err coll
p mesh
p coll
const mesh
const coll
1.00e-01 1.56e-07 1.56e-07 5.00e-02 8.32e-09 8.62e-09 4.23 4.18 2.65e-03 2.36e-03 2.50e-02 4.60e-10 5.46e-10 4.18 3.98 2.28e-03 1.30e-03 1.25e-02 2.75e-11 3.44e-11 4.07 3.99 1.52e-03 1.35e-03 6.25e-03 1.67e-12 2.15e-12 4.04 4.00 1.34e-03 1.41e-03
4. sbvpcol, Gauss, m=4
h err mesh
err coll
p mesh
p coll
const mesh
const coll
1.00e-01 2.12e-09 2.12e-09 5.00e-02 9.07e-11 9.07e-11 4.55 4.55 7.46e-05 7.46e-05 2.50e-02 3.92e-12 3.92e-12 4.53 4.53 7.11e-05 7.11e-05 1.25e-02 1.71e-13 1.71e-13 4.52 4.52 6.77e-05 6.77e-05 6.25e-03 5.99e-15 5.99e-15 4.84 4.84 2.76e-04 2.76e-04
56
TABLE 1.4 : convergence of first derivative for Example 1 -α=0-
1. sbvpcol, equidistant, m=4
h err mesh *
err coll
p mesh *
p coll
const mesh *
const coll
1.00e-01 4.14e-07 4.14e-07 5.00e-02 2.65e-08 2.65e-08 3.96 3.96 3.80e-03 3.80e-03 2.50e-02 1.68e-09 1.68e-09 3.98 3.98 4.00e-03 4.00e-03 1.25e-02 1.08e-10 1.08e-10 3.96 3.96 3.70e-03 3.70e-03 6.25e-03 2.60e-11 2.60e-11 2.05 2.05 8.66e-07 8.66e-07
2. sbvpcol, Gauss, m=2
h err mesh *
err coll
p mesh *
p coll
const mesh *
const coll
1.00e-01 2.15e-03 2.15e-03 5.00e-02 5.52e-04 5.52e-04 1.99 1.99 2.10e-01 2.10e-01 2.50e-02 1.40e-04 1.40e-04 1.98 1.98 2.08e-01 2.08e-01 1.25e-02 3.52e-05 3.52e-05 1.99 1.99 2.16e-01 2.16e-01 6.25e-03 8.82e-06 8.82e-06 2.00 2.00 2.25e-01 2.25e-01
3. sbvpcol, Gauss, m=3
h err mesh *
err coll
p mesh *
p coll
const mesh *
const coll
1.00e-01 2.15e-05 2.15e-05 5.00e-02 2.76e-06 2.76e-06 2.96 2.96 1.96e-02 1.96e-02 2.50e-02 3.50e-07 3.50e-07 2.98 2.98 2.08e-02 2.08e-02 1.25e-02 4.40e-08 4.40e-08 2.99 2.99 2.16e-02 2.16e-02 6.25e-03 5.51e-09 5.51e-09 3.00 3.00 2.25e-02 2.25e-02
4. sbvpcol, Gauss, m=4
h err mesh *
err coll
p mesh *
p coll
const mesh *
const coll
1.00e-01 1.54e-07 1.54e-07 5.00e-02 9.86e-09 9.86e-09 3.96 3.96 1.40e-03 1.40e-03 2.50e-02 6.25e-10 6.25e-10 3.98 3.98 1.49e-03 1.49e-03 1.25e-02 3.95e-11 3.99e-11 3.98 3.97 1.49e-03 1.49e-03 6.25e-03 8.22e-12 7.38e-12 2.26 2.43 7.90e-07 1.68e-06
* solution computed at mesh points by left polynomial
57
TABLE 1.5 : convergence of first derivative for Example 1 -α=1-
1. sbvpcol, equidistant, m=4
h err mesh *
err coll
p mesh *
p coll
const mesh *
const coll
1.00e-01 4.14e-07 4.14e-07 5.00e-02 2.65e-08 2.65e-08 3.96 3.96 3.80e-03 3.80e-03 2.50e-02 1.67e-09 1.67e-09 3.98 3.98 4.00e-03 4.00e-03 1.25e-02 1.08e-10 1.08e-10 3.96 3.96 3.70e-03 3.70e-03 6.25e-03 2.60e-11 9.57e-12 2.05 3.50 8.66e-07 4.80e-04
2. sbvpcol, Gauss, m=2
h err mesh *
err coll
p mesh *
p coll
const mesh *
const coll
1.00e-01 2.15e-03 2.16e-03 5.00e-02 5.52e-04 5.52e-04 1.96 1.97 1.96e-01 2.02e-01 2.50e-02 1.40e-04 1.40e-04 1.98 1.98 2.08e-01 2.08e-01 1.25e-02 3.52e-05 3.52e-05 1.99 1.99 2.16e-01 2.16e-01 6.25e-03 8.82e-06 8.82e-06 2.00 2.00 2.25e-01 2.25e-01
3. sbvpcol, Gauss, m=3
h err mesh *
err coll
p mesh *
p coll
const mesh *
const coll
1.00e-01 2.15e-05 2.16e-05 5.00e-02 2.76e-06 2.76e-06 2.96 2.97 1.96e-02 2.02e-02 2.50e-02 3.49e-07 3.50e-07 2.98 2.98 2.08e-02 2.08e-02 1.25e-02 4.40e-08 4.40e-08 2.99 2.99 2.15e-02 2.15e-02 6.25e-03 5.51e-09 5.51e-09 3.00 3.00 2.25e-02 2.25e-02
4. sbvpcol, Gauss, m=4
h err mesh *
err coll
p mesh *
p coll
const mesh *
const coll
1.00e-01 1.54e-07 1.54e-07 5.00e-02 9.86e-09 9.86e-09 3.96 3.96 1.40e-03 1.40e-03 2.50e-02 6.25e-10 6.25e-10 3.98 3.98 1.49e-03 1.49e-03 1.25e-02 3.95e-11 3.99e-11 3.98 3.97 1.49e-03 1.49e-03 6.25e-03 8.22e-12 7.38e-12 2.26 2.43 7.90e-07 1.68e-06
* solution computed at mesh points by left polynomial
58
TABLE 1.6 : convergence of first derivative for Example 1 -α=2-
1. sbvpcol, equidistant, m=4
h err mesh *
err coll
p mesh *
p coll
const mesh *
const coll
1.00e-01 4.14e-07 4.14e-07 5.00e-02 2.93e-08 2.93e-08 3.82 3.82 2.74e-03 2.74e-03 2.50e-02 2.24e-09 2.24e-09 3.71 3.71 1.97e-03 1.97e-03 1.25e-02 1.75e-10 1.75e-10 3.67 3.67 1.70e-03 1.70e-03 6.25e-03 2.60e-11 1.24e-11 2.75 3.81 3.00e-05 3.12e-05
2. sbvpcol, Gauss, m=2
h err mesh *
err coll
p mesh *
p coll
const mesh *
const coll
1.00e-01 2.51e-03 2.51e-03 5.00e-02 7.44e-04 7.44e-04 1.75 1.75 1.41e-01 1.41e-01 2.50e-02 2.25e-04 2.25e-04 1.73 1.73 1.33e-01 1.33e-01 1.25e-02 6.88e-05 6.88e-05 1.71 1.71 1.24e-01 1.24e-01 6.25e-03 2.12e-05 2.12e-05 1.70 1.70 1.18e-01 1.18e-01
3. sbvpcol, Gauss, m=3
h err mesh *
err coll
p mesh *
p coll
const mesh *
const coll
1.00e-01 2.26e-05 2.26e-05 5.00e-02 2.76e-06 2.76e-06 3.03 3.03 2.42e-02 2.42e-02 2.50e-02 3.50e-07 3.50e-07 2.98 2.98 2.08e-02 2.08e-02 1.25e-02 4.40e-08 4.40e-08 2.99 2.99 2.16e-02 2.16e-02 6.25e-03 5.51e-09 5.51e-09 3.00 3.00 2.25e-02 2.25e-02
4. sbvpcol, Gauss, m=4
h err mesh *
err coll
p mesh *
p coll
const mesh *
const coll
1.00e-01 2.35e-07 2.35e-07 5.00e-02 1.69e-08 1.69e-08 3.79 3.79 1.45e-03 1.45e-03 2.50e-02 1.29e-09 1.29e-09 3.72 3.72 1.17e-03 1.17e-03 1.25e-02 9.89e-11 9.89e-11 3.70 3.70 1.09e-03 1.09e-03 6.25e-03 8.22e-12 7.38e-12 3.58 3.74 6.43e-04 1.30e-03
* solution computed at mesh points by left polynomial
59
TABLE 1.7 : convergence of first derivative for Example 1 -α=3-
1. sbvpcol, equidistant, m=4
h err mesh *
err coll
p mesh *
p coll
const mesh *
const coll
1.00e-01 7.27e-07 7.27e-07 5.00e-02 6.17e-08 6.17e-08 3.56 3.56 2.64e-03 2.64e-03 2.50e-02 5.31e-09 5.31e-09 3.54 3.54 2.49e-03 2.49e-03 1.25e-02 4.59e-10 4.59e-10 3.53 3.53 2.40e-03 2.40e-03 6.25e-03 2.56e-11 2.56e-11 4.17 4.17 3.96e-02 3.96e-02
2. sbvpcol, Gauss, m=2
h err mesh *
err coll
p mesh *
p coll
const mesh *
const coll
1.00e-01 4.44e-03 4.44e-03 5.00e-02 1.47e-03 1.47e-03 1.59 1.59 1.73e-01 1.73e-01 2.50e-02 4.99e-04 4.99e-04 1.56 1.56 1.57e-01 1.57e-01 1.25e-02 1.72e-04 1.72e-04 1.54 1.54 1.46e-01 1.46e-01 6.25e-03 5.95e-05 5.95e-05 1.53 1.53 1.40e-01 1.40e-01
3. sbvpcol, Gauss, m=3
h err mesh *
err coll
p mesh *
p coll
const mesh *
const coll
1.00e-01 2.15e-05 2.26e-05 5.00e-02 2.76e-06 2.76e-06 2.96 3.03 1.96e-02 2.42e-02 2.50e-02 3.49e-07 3.50e-07 2.98 2.98 2.08e-02 2.08e-02 1.25e-02 4.40e-08 4.40e-08 2.99 2.99 2.15e-02 2.16e-02 6.25e-03 5.51e-09 5.51e-09 3.00 3.00 2.25e-02 2.25e-02
4. sbvpcol, Gauss, m=4
h err mesh *
err coll
p mesh *
p coll
const mesh *
const coll
1.00e-01 4.37e-07 4.37e-07 5.00e-02 3.70e-08 3.70e-08 3.56 3.56 1.59e-03 1.59e-03 2.50e-02 3.19e-09 3.19e-09 3.54 3.54 1.49e-03 1.49e-03 1.25e-02 2.76e-10 2.76e-10 3.53 3.53 1.44e-03 1.44e-03 6.25e-03 2.02e-11 2.02e-11 3.77 3.77 4.13e-03 4.13e-03
* solution computed at mesh points by left polynomial
60
TABLE 1.8 : convergence of second derivative for Example 1 -α=0-
1. sbvpcol, equidistant, m=4
h err mesh *
err coll
p mesh *
p coll
const mesh *
const coll
1.00e-01 4.35e-05 4.35e-05 5.00e-02 5.55e-06 5.55e-06 2.97 2.97 4.06e-02 4.06e-02 2.50e-02 7.00e-07 7.00e-07 2.99 2.99 4.31e-02 4.31e-02 1.25e-02 8.94e-08 8.94e-08 2.97 2.97 4.01e-02 4.01e-02 6.25e-03 3.08e-08 1.69e-08 1.54 2.41 7.62e-05 3.45e-03
2. sbvpcol, Gauss, m=2
h err mesh *
err coll
p mesh *
p coll
const mesh *
const coll
1.00e-01 1.31e-01 1.31e-01 5.00e-02 6.68e-02 6.68e-02 0.98 0.98 1.25e+00 1.25e+00 2.50e-02 3.37e-02 3.37e-02 0.99 0.99 1.28e+00 1.28e+00 1.25e-02 1.70e-02 1.70e-02 1.00 1.00 1.33e+00 1.33e+00 6.25e-03 8.48e-03 8.48e-03 0.99 0.99 1.30e+00 1.30e+00
3. sbvpcol, Gauss, m=3
h err mesh *
err coll
p mesh *
p coll
const mesh *
const coll
1.00e-01 2.61e-03 2.61e-03 5.00e-02 6.66e-04 6.66e-04 1.97 1.97 2.44e-01 2.44e-01 2.50e-02 1.68e-04 1.68e-04 1.99 1.99 2.59e-01 2.59e-01 1.25e-02 4.22e-05 4.22e-05 1.99 1.99 2.59e-01 2.59e-01 6.25e-03 1.06e-05 1.06e-05 2.00 2.00 2.70e-01 2.70e-01
4. sbvpcol, Gauss, m=4
h err mesh *
err coll
p mesh *
p coll
const mesh *
const coll
1.00e-01 3.09e-05 3.09e-05 5.00e-02 3.95e-06 3.95e-06 2.97 2.97 2.87e-02 2.87e-02 2.50e-02 5.00e-07 5.00e-07 2.98 2.98 3.01e-02 3.01e-02 1.25e-02 6.35e-08 6.35e-08 2.98 3.03 2.94e-02 3.55e-02 6.25e-03 1.65e-08 1.65e-08 1.95 2.12 3.24e-04 6.50e-04
* solution computed at mesh points by left polynomial
61
TABLE 1.9 : convergence of second derivative for Example 1 -α=1-
1. sbvpcol, equidistant, m=4
h err mesh *
err coll
p mesh *
p coll
const mesh *
const coll
1.00e-01 4.35e-05 4.35e-05 5.00e-02 5.55e-06 5.55e-06 2.97 2.97 4.07e-02 4.07e-02 2.50e-02 7.00e-07 7.00e-07 2.99 2.99 4.27e-02 4.27e-02 1.25e-02 8.94e-08 8.94e-08 2.97 2.97 4.01e-02 4.01e-02 6.25e-03 3.08e-08 1.67e-08 1.54 2.42 7.49e-05 3.60e-03
2. sbvpcol, Gauss, m=2
h err mesh *
err coll
p mesh *
p coll
const mesh *
const coll
1.00e-01 1.31e-01 1.31e-01 5.00e-02 6.68e-02 6.68e-02 0.98 0.98 1.25e+00 1.25e+00 2.50e-02 3.37e-02 3.37e-02 0.99 0.99 1.29e+00 1.29e+00 1.25e-02 1.69e-02 1.69e-02 1.00 1.00 1.33e+00 1.33e+00 6.25e-03 8.48e-03 8.48e-03 1.00 1.00 1.33e+00 1.33e+00
3. sbvpcol, Gauss, m=3
h err mesh *
err coll
p mesh *
p coll
const mesh *
const coll
1.00e-01 2.61e-03 2.61e-03 5.00e-02 6.66e-04 6.66e-04 1.97 1.97 2.44e-01 2.44e-01 2.50e-02 1.68e-04 1.68e-04 1.99 1.99 2.59e-01 2.59e-01 1.25e-02 4.22e-05 4.22e-05 1.99 1.99 2.59e-01 2.59e-01 6.25e-03 1.06e-05 1.06e-05 2.00 2.00 2.70e-01 2.70e-01
4. sbvpcol, Gauss, m=4
h err mesh *
err coll
p mesh *
p coll
const mesh *
const coll
1.00e-01 3.09e-05 3.09e-05 5.00e-02 3.95e-06 3.95e-06 2.97 2.97 2.87e-03 2.87e-03 2.50e-02 5.00e-07 5.00e-07 2.98 2.98 3.00e-03 3.00e-03 1.25e-02 6.13e-08 6.35e-08 3.03 2.98 3.54e-02 2.95e-03 6.25e-03 1.42e-08 1.65e-08 2.11 1.95 6.50e-04 3.24e-04
* solution computed at mesh points by left polynomial
62
TABLE 1.10 : convergence of second derivative for Example 1 -α=2-
1. sbvpcol, equidistant, m=4
h err mesh *
err coll
p mesh *
p coll
const mesh *
const coll
1.00e-01 4.35e-05 4.35e-05 5.00e-02 5.55e-06 5.55e-06 2.97 2.97 4.06e-02 4.06e-02 2.50e-02 7.01e-07 7.01e-07 2.98 2.98 4.18e-02 4.18e-02 1.25e-02 1.07e-07 1.07e-07 2.71 2.71 1.54e-02 1.54e-02 6.25e-03 3.08e-08 1.67e-08 1.79 2.68 2.73e-04 1.35e-02
2. sbvpcol, Gauss, m=2
h err mesh *
err coll
p mesh *
p coll
const mesh *
const coll
1.00e-01 1.31e-01 1.31e-01 5.00e-02 6.68e-02 6.68e-02 0.98 0.98 1.24e+00 1.24e+00 2.50e-02 3.37e-02 3.37e-02 0.99 0.99 1.28e+00 1.28e+00 1.25e-02 1.69e-02 1.69e-02 1.00 1.00 1.33e+00 1.33e+00 6.25e-03 8.60e-02 8.60e-02 0.97 0.97 1.21e+00 1.21e+00
3. sbvpcol, Gauss, m=3
h err mesh *
err coll
p mesh *
p coll
const mesh *
const coll
1.00e-01 2.61e-03 2.61e-03 5.00e-02 6.66e-04 6.66e-04 1.97 1.97 2.44e-01 2.44e-01 2.50e-02 1.68e-04 1.68e-04 1.98 1.98 2.59e-01 2.59e-01 1.25e-02 4.22e-05 4.22e-05 1.99 1.99 2.59e-01 2.59e-01 6.25e-03 1.06e-05 1.06e-05 2.00 2.00 2.70e-01 2.70e-01
4. sbvpcol, Gauss, m=4
h err mesh *
err coll
p mesh *
p coll
const mesh *
const coll
1.00e-01 3.09e-05 3.09e-05 5.00e-02 3.95e-06 3.95e-06 2.97 2.97 2.87e-02 2.87e-02 2.50e-02 5.25e-07 5.25e-07 2.91 2.91 2.44e-02 2.44e-02 1.25e-02 7.88e-08 7.88e-08 2.73 2.73 1.27e-02 1.27e-02 6.25e-03 1.42e-08 1.65e-08 2.48 2.26 4.08e-03 1.57e-03
* solution computed at mesh points by left polynomial
63
TABLE 1.11 : convergence of second derivative for Example 1 -α=3-
1. sbvpcol, equidistant, m=4
h err mesh *
err coll
p mesh *
p coll
const mesh *
const coll
1.00e-01 5.56e-05 5.56e-05 5.00e-02 9.21e-06 9.21e-06 2.59 2.59 2.16e-02 2.16e-02 2.50e-02 1.56e-06 1.56e-06 2.56 2.56 1.97e-02 1.97e-02 1.25e-02 2.66e-07 2.66e-07 2.55 2.55 1.90e-02 1.90e-02 6.25e-03 3.02e-08 3.02e-08 3.13 3.13 2.41e-01 2.41e-01
2. sbvpcol, Gauss, m=2
h err mesh *
err coll
p mesh *
p coll
const mesh *
const coll
1.00e-01 1.31e-01 1.31e-01 5.00e-02 7.28e-02 7.28e-02 0.85 0.85 9.27e-01 9.27e-01 2.50e-02 4.69e-02 4.69e-02 0.63 0.63 4.81e-01 4.81e-01 1.25e-02 3.09e-02 3.09e-02 0.60 0.60 4.29e-01 4.29e-01 6.25e-03 2.08e-02 2.08e-02 0.57 0.57 3.76e-01 3.76e-01
3. sbvpcol, Gauss, m=3
h err mesh *
err coll
p mesh *
p coll
const mesh *
const coll
1.00e-01 2.60e-03 2.60e-03 5.00e-02 6.65e-04 6.65e-04 1.97 1.97 2.43e-01 2.43e-01 2.50e-02 1.68e-04 1.68e-04 1.98 1.98 2.51e-01 2.51e-01 1.25e-02 4.22e-05 4.22e-05 1.99 1.99 2.59e-01 2.59e-01 6.25e-03 1.06e-05 1.06e-05 2.00 2.00 2.70e-01 2.70e-01
4. sbvpcol, Gauss, m=4
h err mesh *
err coll
p mesh *
p coll
const mesh *
const coll
1.00e-01 4.32e-05 4.32e-05 5.00e-02 7.14e-06 7.14e-06 2.60 2.60 1.72e-02 1.72e-02 2.50e-02 1.20e-06 1.20e-06 2.57 2.57 1.58e-02 1.58e-02 1.25e-02 2.06e-07 2.06e-07 2.55 2.55 1.46e-02 1.46e-02 6.25e-03 3.04e-08 3.04e-08 2.76 2.76 3.68e-02 3.68e-02
* solution computed at mesh points by left polynomial
64
TABLE 1.12 : matrix condition estimates for Example 1 -α=0-
1. sbvpcol, equidistant, m=4
h condestDF cond ord. condest
ord. cond
const condest
const cond
1.00e-01 2.05e+01 1.12e+01 5.00e-02 3.27e+01 2.24e+01 -0.67 -1.00 4.34+00 1.13e+00 2.50e-02 5.77e+01 4.48e+01 -0.82 -1.00 2.79e+00 1.11e+00 1.25e-02 1.08e+02 8.98e+01 -0.91 -1.00 2.04e+00 1.11e+00 6.25e-03 2.09e+02 1.80e+02 -0.95 -1.00 1.67e+00 1.12e+00
2. sbvpcol, Gauss, m=2
h condestDF cond ord. condest
ord. cond
const condest
const cond
1.00e-01 1.60e+01 1.15e+01 5.00e-02 2.86e+01 2.32e+01 -0.83 -1.01 2.34e+00 1.12e+00 2.50e-02 5.38e+01 4.65e+01 -0.91 -1.01 1.85e+00 1.14e+00 1.25e-02 1.04e+02 9.31e+01 -0.96 -1.00 1.58e+00 1.15e+00 6.25e-03 2.05e+02 1.86e+02 -0.98 -1.00 2.83e+00 1.16e+00
3. sbvpcol, Gauss, m=3
h condestDF cond ord. condest
ord. cond
const condest
const cond
1.00e-01 1.82e+01 2.15e+01 5.00e-02 3.06e+01 1.07e+01 -0.75 -1.01 3.24e+00 1.05e+00 2.50e-02 5.58e+01 4.32e+01 -0.87 -1.01 2.28e+00 1.06e+00 1.25e-02 1.06e+02 8.66e+01 -0.93 -1.00 1.80e+00 1.07e+00 6.25e-03 2.07e+02 1.73e+02 -0.96 -1.00 1.55e+00 1.09e+00
4. sbvpcol, Gauss, m=4
h condestDF cond ord. condest
ord. cond
const condest
const cond
1.00e-01 2.05e+01 1.03e+01 5.00e-02 3.27e+01 2.06e+01 -0.67 -1.00 4.34e+00 1.02e+00 2.50e-02 5.78e+01 4.14e+01 -0.82 -1.01 2.79e+00 1.01e+00 1.25e-02 1.08e+02 8.31e+01 -0.91 -1.01 2.04e+00 1.02e+00 6.25e-03 2.09e+02 1.66e+02 -0.95 -1.00 1.67e+00 1.05e+00
65
TABLE 1.13 : matrix condition estimates for Example 1 -α=1-
1. sbvpcol, equidistant, m=4
h condestDF cond ord. condest
ord. cond
const condest
const cond
1.00e-01 9.70e+01 3.18e+01 5.00e-02 1.54e+02 5.62e+01 -0.67 -0.82 2.09e+01 4.80e+00 2.50e-02 2.68e+02 1.10e+02 -0.80 -0.97 1.40e+01 3.06e+00 1.25e-02 4.97e+02 2.20e+02 -0.89 -1.00 1.01e+01 2.78e+00 6.25e-03 9.53e+02 4.40e+02 -0.94 -1.00 8.05e+00 2.73e+00
2. sbvpcol, Gauss, m=2
h condestDF cond ord. condest
ord. cond
const condest
const cond
1.00e-01 4.55e+01 2.02e+01 5.00e-02 8.05e+01 3.90e+01 -0.82 -0.95 6.84e+00 2.27e+00 2.50e-02 1.50e+02 7.79e+01 -0.90 -1.00 5.39e+00 1.97e+00 1.25e-02 2.90e+02 1.56e+02 -0.95 -1.00 4.55e+00 1.92e+00 6.25e-03 5.70e+02 3.13e+02 -0.97 -1.00 4.08e+00 1.93e+00
3. sbvpcol, Gauss, m=3
h condestDF cond ord. condest
ord. cond
const condest
const cond
1.00e-01 9.75e+01 3.67e+01 5.00e-02 1.62e+02 6.85e+01 -0.74 -0.90 1.79e+02 4.61e+00 2.50e-02 2.92e+02 1.36e+02 -0.85 -0.99 1.28e+02 3.56e+00 1.25e-02 5.52e+02 2.72e+02 -0.92 -1.00 9.91e+01 3.38e+00 6.25e-03 1.07e+03 5.45e+02 -0.96 -1.00 8.34e+01 3.36e+00
4. sbvpcol, Gauss, m=4
h condestDF cond ord. condest
ord. cond
const condest
const cond
1.00e-01 1.78e+02 6.02e+01 5.00e-02 2.83e+02 1.08e+02 -0.67 -0.85 3.84e+01 8.59e+00 2.50e-02 4.93e+02 2.13e+02 -0.80 -0.98 2.58e+01 5.81e+00 1.25e-02 9.13e+02 4.26e+02 -0.89 -1.00 1.86e+01 5.33e+00 6.25e-03 1.75e+03 8.52e+02 -0.94 -1.00 1.48e+01 5.26e+00
66
TABLE 1.14 : matrix condition estimates for Example 1 -α=2-
1. sbvpcol, equidistant, m=4
h condestDF cond ord. condest
ord. cond
const condest
const cond
1.00e-01 2.69e+03 1.16e+03 5.00e-02 8.25e+03 3.95e+03 -1.62 -1.76 6.53e+01 2.01e+01 2.50e-02 2.80e+04 1.53e+04 -1.76 -1.95 4.21e+01 1.13e+01 1.25e-02 1.02e+05 6.09e+04 -1.86 -1.99 2.88e+01 9.78e+00 6.25e-03 3.88e+05 2.44e+05 -1.93 -2.00 2.19e+01 9.47e+00
2. sbvpcol, Gauss, m=2
h condestDF cond ord. condest
ord. cond
const condest
const cond
1.00e-01 1.41e+03 8.93e+02 5.00e-02 4.61e+03 2.82e+03 -1.71 -1.66 2.76e+01 1.97e+01 2.50e-02 1.70e+04 1.12e+04 -1.88 -1.99 1.65e+01 7.25e+00 1.25e-02 6.49e+04 4.48e+04 -1.94 -2.00 1.34e+01 6.92e+00 6.25e-03 2.54e+05 1.80e+05 -1.97 -2.00 1.17e+01 6.91e+00
3. sbvpcol, Gauss, m=3
h condestDF cond ord. condest
ord. cond
const condest
const cond
1.00e-01 6.74e+03 4.05e+03 5.00e-02 1.89e+04 1.06e+04 -1.49 -1.38 2.19e+02 1.68e+02 2.50e-02 6.27e+04 3.66e+04 -1.73 -1.79 1.07e+02 4.95e+01 1.25e-02 2.34e+05 1.46e+05 -1.90 -2.00 5.67e+01 2.29e+01 6.25e-03 9.02e+05 5.87e+05 -1.95 -2.00 4.61e+02 2.26e+01
4. sbvpcol, Gauss, m=4
h condestDF cond ord. condest
ord. cond
const condest
const cond
1.00e-01 1.80e+04 1.25e+04 5.00e-02 6.47e+04 3.57e+04 -1.84 -1.52 2.59e+02 3.80e+02 2.50e-02 1.75e+05 9.33e+04 -1.44 -1.39 8.69e+02 5.60e+02 1.25e-02 6.30e+05 3.72e+05 -1.85 -2.00 1.93e+02 5.91e+01 6.25e-03 2.40e+06 1.49e+06 -1.93 -2.00 1.35e+02 5.76e+01
· condestDF:=condest(DF,1), cond:=cond(DF,2)
67
TABLE 1.15 : matrix condition estimates for Example 1 -α=3-
1. sbvpcol, equidistant, m=4
h condestDF cond ord. condest
ord. cond
const condest
const cond
1.00e-01 1.02e+05 5.27e+04 5.00e-02 6.12e+05 3.30e+05 -2.58 -2.65 2.69e+02 1.18e+02 2.50e-02 4.07e+06 2.51e+06 -2.73 -2.93 1.70e+02 5.15e+01 1.25e-02 2.92e+07 1.99e+07 -2.85 -2.99 1.12e+02 4.11e+01 6.25e-03 2.21e+08 1.59e+08 -2.92 -3.00 8.25e+01 3.89e+01
2. sbvpcol, Gauss, m=2
h condestDF cond ord. condest
ord. cond
const condest
const cond
1.00e-01 1.16e+05 6.95e+04 5.00e-02 6.34e+05 3.85e+05 -2.44 -2.47 4.18e+02 2.36e+02 2.50e-02 3.74e+06 2.16e+06 -2.56 -2.49 2.95e+02 2.25e+02 1.25e-02 2.17e+07 1.42e+07 -2.54 -2.72 3.25e+02 9.47e+01 6.25e-03 1.56e+08 1.14e+08 -2.85 -3.00 8.09e+01 2.74e+01
3. sbvpcol, Gauss, m=3
h condestDF cond ord. condest
ord. cond
const condest
const cond
1.00e-01 1.06e+06 6.68e+05 5.00e-02 7.36e+06 3.75e+06 -2.79 -2.49 1.73e+03 2.16e+03 2.50e-02 4.23e+07 2.11e+07 -2.52 -2.49 3.85e+03 2.15e+03 1.25e-02 2.38e+08 1.19e+08 -2.50 -2.50 4.26e+03 2.12e+03 6.25e-03 1.36e+09 6.98e+08 -2.52 -2.55 3.89e+03 1.67e+03
4. sbvpcol, Gauss, m=4
h condestDF cond ord. condest
ord. cond
const condest
const cond
1.00e-01 7.35e+06 3.98e+06 5.00e-02 4.62e+07 2.22e+07 -2.65 -2.48 1.64e+04 1.32e+04 2.50e-02 2.60e+08 1.25e+08 -2.49 -2.49 2.65e+04 1.27e+04 1.25e-02 1.52e+09 7.06e+08 -2.55 -2.50 2.12e+04 1.24e+04 6.25e-03 8.66e+09 3.99e+09 -2.51 -2.50 2.59e+04 1.24e+04
68
TABLE 1.16 : global error for Example 1 – Backward Euler method-
1. sbvpcol, α=0, backward Euler, number of collocation points m= 1
h err mesh
err coll
p mesh
p coll
const mesh
const coll
1.00e-01 5.22e-02 5.22e-02 5.00e-02 2.55e-02 2.55e-02 1.03 1.03 5.64e.01 5.64e.01 2.50e-02 1.26e-02 1.26e-02 1.02 1.02 5.37e-01 5.37e-01 1.25e-02 6.28e-03 6.28e-03 1.00 1.00 5.13e-01 5.13e-01 6.25e-03 3.13e-03 3.13e-03 1.00 1.00 5.10e-01 5.10e-01
2. sbvpcol, α=1, backward Euler
h err mesh
err coll
p mesh
p coll
const mesh
const coll
1.00e-01 3.95e-02 3.95e-02 5.00e-02 1.90e-02 1.90e-02 1.06 1.06 4.49e-01 4.49e-01 2.50e-02 9.27e-03 9.27e-03 1.04 1.04 4.23e-01 4.23e-01 1.25e-02 4.58e-03 4.58e-03 1.02 1.02 3.92e-01 3.92e-01 6.25e-03 2.28e-03 2.28e-03 1.01 1.01 3.79e-01 3.79e-01
3. sbvpcol, α=2, backward Euler
h err mesh
err coll
p mesh
p coll
const mesh
const coll
1.00e-01 3.40e-02 3.40e-02 5.00e-02 1.60e-02 1.60e-02 1.09 1.09 4.16e-01 4.16e-01 2.50e-02 7.81e-03 7.81e-03 1.03 1.03 3.55e-01 3.55e-01 1.25e-02 3.84e-03 3.84e-03 1.02 1.02 3.42e-01 3.42e-01 6.25e-03 1.91e-03 1.91e-03 1.01 1.01 3.18e-01 3.18e-01
4. sbvpcol, α=3, backward Euler
h err mesh
err coll
p mesh
p coll
const mesh
const coll
1.00e-01 7.20e+00 7.20e+00 5.00e-02 5.93e+01 5.93e+01 -3.04 -3.04 6.54e-03 6.54e-03 2.50e-02 1.73e+05 1.73e+05 -11.5 -11.5 6.27e-14 6.27e-14 1.25e-02 8.70e+09 8.70e+09 -15.6 -15.6 1.64e-20 1.64e-20 6.25e-03 5.99e+17 5.99e+17 -26.0 -26.0 2.45e-40 2.45e-40
69
TABLE 1.17 : matrix condition estimates for Example 1 -Backward Euler method -
1. sbvpcol, α=0, backward Euler, number of the collocation points m= 1
h condestDF cond ord. condest
ord. cond
const condest
const cond
1.00e-01 1.71e+01 1.43e+01 5.00e-02 2.96e+01 2.86e+01 -0.80 -1.00 2.73e+00 1.43e+00 2.50e-02 5.49e+01 5.73e+01 -0.89 -1.00 2.07e+00 1.44e+00 1.25e-02 1.05e+02 1.15e+02 -0.94 -1.00 1.70e+00 1.43e+00 6.25e-03 2.07e+02 2.29e+02 -0.97 -1.00 1.50e+00 1.43e+00
2. sbvpcol, α=1, backward Euler
h condestDF cond ord. condest
ord. cond
const condest
const cond
1.00e-01 1.75e+01 1.05e+01 5.00e-02 3.00e+01 2.06e+01 -0.78 -0.97 2.92e+00 1.12e+00 2.50e-02 5.50e+01 4.11e+01 -0.87 -1.00 1.46e+00 1.04e+00 1.25e-02 1.05e+02 8.22e+01 -0.93 -1.00 1.76e+00 1.03e+00 6.25e-03 2.05e+02 1.65e+02 -0.97 -1.00 1.53e+00 1.02e+00
3. sbvpcol, α=2, backward Euler
h condestDF cond ord. condest
ord. cond
const condest
const cond
1.00e-01 3.80e+02 4.70e+02 5.00e-02 4.35e+03 7.04e+03 -3.51 -3.90 1.15e-01 5.86e-02 2.50e-02 1.55e+05 2.79e+05 -5.16 -5.31 8.45e-04 8.69e-04 1.25e-02 2.10e+07 4.32e+07 -7.09 -7.27 7.01e-07 6.23e-07 6.25e-03 1.92e+10 4.54e+10 -9.84 -10.0 4.02e-12 3.41e-12
4. sbvpcol, α=3, backward Euler
h condestDF cond ord. condest
ord. cond
const condest
const cond
1.00e-01 6.32e+05 1.06e+06 5.00e-02 2.51e+09 4.94e+09 -12.0 -12.2 7.00e-07 6.82e-07 2.50e-02 1.00e+15 2.12e+15 -18.6 -18.7 1.55e-15 2.24e-15 1.25e-02 4.92e+23 1.43e+24 -28.9 -29.3 5.61e-32 2.20e-32 6.25e-03 2.08e+37 2.68e+28 -45.3 -14.1 3.50e-63 1.38e-03
· condestDF:=condest(DF,1), cond:=cond(DF,2)
70
TABLE 1.18 : matrix condition estimates for Example 1 - Backward Euler method -
-preconditioning 1/h -
1. α=0, backward Euler, number of the collocation points m= 1
h condest(A) cond(A) ord. condest
ord. cond
const condest
const cond
norm A
norm inv(A)
1.00e-01 1.24e+01 9.44e+00 1.88e+01 5.02e-01 5.00e-02 2.50e+01 1.83e+01 -1.02 -0.96 1.19e+00 1.04e+00 3.89e+01 4.72e-01 2.50e-02 5.03e+01 3.61e+01 -1.01 -0.98 1.22e+00 9.89e–01 7.89e+01 4.57e-01 1.25e-02 1.01e+02 7.15e+01 -1.00 -0.99 1.24e+00 9.49e–01 1.59e+02 4.50e-01 6.25e-03 2.03e+02 1.42e+02 -1.01 -0.99 1.22e+00 9.22e–01 3.19e+02 4.46e-01
2. α=1, backward Euler
h condest(A) cond(A) ord. condest
ord. cond
const condest
const cond
norm A
norm inv(A)
1.00e-01 1.05e+01 6.87e+00 1.83e+01 3.75e-01 5.00e-02 2.05e+01 1.33e+01 -0.97 -0.95 1.12e+00 7.65e-01 3.84e+01 3.46e-01 2.50e-02 3.95e+01 2.61e+01 -0.95 -0.97 1.20e+00 7.23e-01 7.85e+01 3.32e-01 1.25e-02 7.95e+01 5.16e+01 -1.01 -0.98 9.55e–01 6.91e-01 1.59e+02 3.25e-01 6.25e-03 1.60e+02 1.03e+02 -1.00 -0.99 9.74e–01 6.70e-01 3.19e+02 3.22e-01
3. α=2, backward Euler
h condest(A) cond(A) ord. condest
ord. cond
const condest
const cond
norm A
norm inv(A)
1.00e-01 1.60e+02 1.50e+02 9.11e+01 1.65e+01 5.00e-02 2.00e+03 2.36e+03 -3.64 -3.98 3.63e-02 1.58e-02 3.81e+02 6.20e+00 2.50e-02 7.47e+04 9.62e+04 -5.22 -5.35 3.20e-04 2.61e-04 1.56e+03 6.16e+01 1.25e-02 1.03e+07 1.51e+07 -7.11 -7.29 3.03e-07 2.01e-07 6.32e+03 2.38e+03 6.25e-03 9.53e+09 1.59e+10 -9.85 -10.1 1.86e-12 1.14e-12 2.54e+04 6.26e+05
4. α=3, backward Euler
h condest(A) cond(A) ord. condest
ord. cond
const condest
const cond
norm A
norm inv(A)
1.00e-01 3.11e+05 3.70e+05 9.90e+02 3.74e+02 5.00e-02 1.25e+09 1.74e+09 -12.0 -12.2 3.32e-07 2.33e-07 7.98e+03 2.18e+05 2.50e-02 5.01e+14 7.49e+14 -18.6 -18.7 7.64e-16 7.84e-16 6.40e+04 1.17e+10 1.25e-02 2.46e+23 4.13e+23 -28.9 -29.0 2.79e-32 2.26e-32 5.12e+05 8.07e+17 6.25e-03 1.04e+37 1.95e+37 -45.3 -45.4 1.75e-63 1.50e-63 4.10e+06 4.75e+30
. condest:=condest(A,1), cond:=cond(A,2), norm(A) = norm(A,2)
71
TABLE 1.19 : matrix condition estimates for Example 1 - Backward Euler method - -preconditioning tˆ(α-1) -
1. α=0, backward Euler, number of the collocation points m= 1
h condest(A) cond(A) ord. condest
ord. cond
const condest
const cond
norm A
norm inv(A)
1.00e-01 9.12e+01 5.62e+01 1.40e+02 4.02e-01 5.00e-02 3.72e+02 2.07e+02 -2.03 -1.88 8.54e-01 7.39e-01 5.72e+02 3.61e-01 2.50e-02 1.50e+03 7.87e+02 -2.01 -1.93 8.92e-01 6.37e-01 2.32e+03 3.39e-01 1.25e-02 6.04e+03 3.07e+03 -2.01 -1.96 9.16e-01 5.66e-01 9.33e+03 3.29e-01 6.25e-03 2.42e+04 1.21e+04 -2.00 -1.98 9.30e-01 5.23e-01 3.74e+04 3.23e-01
2. α=1, backward Euler
h condest(A) cond(A) ord. condest
ord. cond
const condest
const cond
norm A
norm inv(A)
1.00e-01 9.50e+00 6.87e+00 1.83e+01 3.75e-01 5.00e-02 1.95e+01 1.33e+01 -1.04 -0.95 8.72e-01 7.65e–01 3.84e+01 3.46e-01 2.50e-02 3.95e+01 2.61e+01 -1.02 -0.97 9.23e-01 1.06e+00 7.85e+01 3.32e-01 1.25e-02 7.95e+01 5.16e+01 -1.01 -0.98 9.55e-01 6.94e–01 1.59e+02 3.25e-01 6.25e-03 1.60e+02 1.03e+02 -1.01 -1.00 9.55e-01 6.53e-01 3.19e+02 3.22e-01
3. α=2, backward Euler
h condest(A) cond(A) ord. condest
ord. cond
const condest
const cond
norm A
norm inv(A)
1.00e-01 1.01e+02 1.09e+02 1.67e+01 6.50e+00 5.00e-02 1.05e+03 1.28e+03 -3.37 -3.55 4.23e-02 3.0e-02 3.55e+01 3.60e+01 2.50e-02 2.95e+04 3.85e+04 -4.81 -4.91 5.76e-04 5.23e-04 7.40e+01 5.20e+02 1.25e-02 2.93e+06 4.40e+06 -6.63 -6.84 6.95e-07 4.30e-07 1.52e+02 2.89e+04 6.25e-03 2.11e+09 3.37e+09 -9.49 -9.58 2.53e-12 2.57e-12 3.09e+02 1.09e+07
4. α=3, backward Euler
h condest(A) cond(A) ord. condest
ord. cond
const condest
const cond
norm A
norm inv(A)
1.00e-01 3.70e+04 3.94e+04 1.63e+01 2.42e+03 5.00e-02 6.78e+07 8.17e+07 -10.8 -11.0 5.35e-07 3.78e-07 3.45e+01 2.37e+06 2.50e-02 1.02e+13 1.50e+13 -17.2 -17.5 2.85e-15 1.45e-15 7.22e+01 2.08e+11 1.25e-02 2.17e+21 3.44e+21 -27.7 -27.8 4.88e-32 4.81e-32 1.49e+02 2.31e+19 6.25e-03 3.61e+34 6.66e+34 -43.9 -44.1 5.67e-63 3.45e-63 3.05e+02 2.18e+32
. condest:=condest(A,1), cond:=cond(A,2), norm(A) = norm(A,2)
72
TABLE 1.20 : matrix condition estimates based on max. norm for Example 1 - Backward Euler method -
- preconditioning 1/h - 1. α=1, backward Euler, number of collocation points m= 1
h cond(A) ord. cond
const condest
norm (A)
norm inv(A)
1.00e-01 8.33e+00 1.90e+01 4.38e-01 5.00e-02 1.57e+01 -0.91 1.01e+00 3.90e+01 4.02e-01 2.50e-02 3.04e+01 -0.95 9.03e–01 7.90e+01 3.85e-01 1.25e-02 5.99e+01 -0.98 8.23e–01 1.59e+02 3.76e-01 6.25e-03 1.19e+02 -0.99 7.81e–01 3.19e+02 3.72e-01
2. α=2, backward Euler
h cond(A) ord. cond
const condest
norm (A)
norm inv(A)
1.00e-01 2.68e+02 1.00e+02 2.68e+00 5.00e-02 4.49e+03 -4.07 2.30e-03 4.00e+02 1.12e+01 2.50e-02 1.87e+05 -5.38 4.49e-04 1.60e+03 1.17e+02 1.25e-02 3.15e+07 -7.40 2.65e-07 6.40e+03 4.92e+03 6.25e-03 3.63e+10 -10.2 1.39e-12 2.56e+04 1.42e+06
3. α=3, backward Euler
h cond(A) ord. cond
const condest
norm (A)
norm inv(A)
1.00e-01 6.99e+05 1.00e+03 6.99e+02 5.00e-02 3.28e+09 -12.2 4.45e-07 8.00e+03 4.09e+05 2.50e-02 1.55e+15 -18.9 9.80e-16 6.40e+04 2.42e+10 1.25e-02 9.60e+23 -29.2 2.51e-32 5.12e+05 1.88e+18 6.25e-03 5.10e+37 -45.6 1.63e-63 4.10e+06 1.25e+31
73
TABLE 1.21 : matrix condition estimates based on max. norm for Example 1 - Backward Euler method -
-preconditioning tˆ(α-1) - 1. α=1, backward Euler, number of collocation points m = 1
h cond(A) ord. cond
const condest
norm (A)
norm inv(A)
1.00e-01 8.33e+00 1.90e+01 4.38e-01 5.00e-02 1.57e+01 -0.91 1.01e+00 3.90e+01 4.02e-01 2.50e-02 3.04e+01 -0.95 9.03e–01 7.90e+01 3.85e-01 1.25e-02 5.99e+01 -0.98 8.22e–01 1.59e+02 3.76e-01 6.25e-03 1.19e+02 -0.99 7.81e–01 3.19e+02 3.72e-01
2. α=2, backward Euler
h cond(A) ord. cond
const condest
norm (A)
norm inv(A)
1.00e-01 2.02e+02 1.90e+01 1.06e+01 5.00e-02 2.55e+03 -3.66 4.44e-02 3.90e+01 6.53e+01 2.50e-02 7.95e+04 -4.96 8.92e-04 7.90e+01 1.01e+03 1.25e-02 9.68e+06 -6.93 6.33e-07 1.59e+02 6.09e+04 6.25e-03 7.98e+09 -9.69 3.55e-12 3.19e+02 2.50e+07
3. α=3, backward Euler
h cond(A) ord. cond
const condest
norm (A)
norm inv(A)
1.00e-01 8.22e+04 1.90e+01 4.32e+03 5.00e-02 1.76e+08 -11.1 7.09e-07 3.90e+01 4.52e+06 2.50e-02 3.47e+13 -17.6 2.30e-15 7.90e+01 4.39e+11 1.25e-02 8.69e+21 -27.9 6.97e-32 1.59e+02 5.46e+19 6.25e-03 3.61e+34 -41.9 1.47e-58 3.19e+02 5.79e+32
74
TABLE 1.22 : global error for Example 1 - Midpoint rule -
1. sbvpcol, α=0, Midpoint rule, number of collocation points m=1
h err mesh
Err Coll
p mesh
p coll
const mesh
const coll
1.00e-01 8.34e-04 3.13e-03 5.00e-02 2.08e-04 8.15e-04 2.00 1.94 8.37e-02 2.73e-01 2.50e-02 5.21e-05 2.08e-04 2.00 1.97 8.34e-02 2.98e-01 1.25e-02 1.30e-05 5.25e-05 2.00 1.99 8.34e-02 3.15e-01 6.25e-03 3.26e-06 1.32e-05 2.00 1.99 8.33e-02 3.25e-01
2. sbvpcol, α=1, Midpoint rule
h err mesh
Err Coll
p mesh
p coll
const mesh
const coll
1.00e-01 1.34e-03 1.34e-03 5.00e-02 3.34e-04 8.15e-04 2.00 0.72 1.35e-01 6.99e-03 2.50e-02 8.35e-05 2.08e-04 2.00 1.97 1.34e-01 2.98e-01 1.25e-02 2.09e-05 5.25e-05 2.00 1.98 1.34e-01 3.15e-01 6.25e-03 5.21e-06 1.32e-05 2.00 1.99 1.34e-01 3.25e-01
3. sbvpcol, α=2, Midpoint rule
h err mesh
Err Coll
p mesh
p coll
const mesh
const coll
1.00e-01 1.62e-03 3.11e-03 5.00e-02 4.02e-04 8.14e-04 2.01 1.93 1.66e-01 2.67e-01 2.50e-02 1.01e-04 2.08e-04 2.00 1.97 1.61e-01 2.97e-01 1.25e-02 2.51e-05 5.25e-05 2.00 1.98 1.61e-01 3.15e-01 6.25e-03 6.29e-06 1.32e-05 2.00 1.99 1.61e-01 3.25e-01
4. sbvpcol, α=3, Midpoint rule
h err mesh
Err Coll
p mesh
p coll
const mesh
const coll
1.00e-01 1.83e-03 3.11e-03 5.00e-02 4.52e-04 8.13e-04 2.02 1.93 1.90e-01 2.68e-01 2.50e-02 1.13e-04 2.08e-04 2.00 1.97 1.83e-01 2.96e-01 1.25e-02 2.82e-05 5.25e-05 2.00 1.98 1.80e-01 3.14e-01 6.25e-03 7.05e-06 1.32e-05 2.00 1.99 1.81e-01 3.25e-01
75
TABLE 1.23 : matrix condition estimates for Example 1 -Midpoint rule - 1. sbvpcol, α=0, Midpoint rule, number of collocation points m=1
h condestDF cond ord. condest
ord. cond
const condest
const cond
1.00e-01 1.65e+01 1.42e+01 5.00e-02 2.92e+01 2.85e+01 -0.82 -1.01 2.47e+00 1.39e+00 2.50e-02 5.44e+01 5.72e+01 -0.90 -1.00 1.96e+00 1.43e+00 1.25e-02 1.05e+02 1.14e+02 -0.95 -1.00 1.64e+00 1.43e+00 6.25e-03 2.06e+02 2.29e+02 -0.97 -1.00 1.48e+00 1.43e+00
2. sbvpcol, α=1, Midpoint rule
h condestDF cond ord. condest
ord. cond
const condest
Const cond
1.00e-01 2.00e+01 1.09e+01 5.00e-02 3.51e+01 2.14e+01 -0.81 -0.98 3.09e+00 1.14e+00 2.50e-02 6.51e+01 4.30e+01 -0.89 -1.00 2.42e+00 1.08e+00 1.25e-02 1.25e+02 8.62e+01 -0.94 -1.00 2.04e+00 1.06e+00 6.25e-03 2.45e+02 1.73e+02 -0.97 -1.00 3.48e+00 1.07e+00
3. sbvpcol, α=2, Midpoint rule
h condestDF cond ord. condest
ord. cond
const condest
const cond
1.00e-01 2.32e+02 1.59e+02 5.00e-02 7.87e+02 6.21e+02 -1.76 -1.97 3.99e+00 1.71e+00 2.50e-02 2.87e+03 2.49e+03 -1.86 -2.00 2.96e+00 1.55e+00 1.25e-02 1.09e+04 9.99e+03 -1.93 -2.01 2.35e+00 1.53e+00 6.25e-03 4.25e+04 4.01e+04 -1.96 -2.00 2.02e+00 1.54e+00
4. sbvpcol, α=3, Midpoint rule
h condestDF cond ord. condest
ord. cond
const condest
const cond
1.00e-01 4.29e+03 3.46e+03 5.00e-02 2.72e+04 2.11e+04 -2.67 -2.61 9.27e+00 8.58e+00 2.50e-02 1.97e+05 1.68e+05 -2.86 -2.99 5.22e+00 2.69e+00 1.25e-02 1.50e+06 1.34e+06 -2.92 -3.00 4.12e+00 2.60e+00 6.25e-03 1.16e+07 1.08e+07 -2.96 -3.00 3.50e+00 2.59e+00
76
TABLE 1.24 : matrix condition estimates for Example 1 - Midpoint rule -
- preconditioning 1/h - 1. α=0, Midpoint rule, number of collocation points m=1
h condest(B) cond(B) ord. condest
ord. cond
const condest
const cond
norm B
norm inv(B)
1.00e-01 1.26e+01 9.68e+00 1.98e+01 4.89e-01 5.00e-02 2.53e+01 1.85e+01 -1.00 -0.93 1.24e+00 1.13e+00 3.99e+01 4.65e-01 2.50e-02 5.06e+01 3.62e+01 -1.00 -0.97 1.27e+00 1.02e+00 7.99e+01 4.53e-01 1.25e-02 1.01e+02 7.16e+01 -1.00 -0.98 1.28e+00 9.60e–01 1.60e+02 4.48e-01 6.25e-03 2.02e+02 1.42e+02 -1.00 -0.99 1.26e+00 9.44e–01 3.20e+02 4.45e-01
2. α=1, Midpoint rule
h condest(B) cond(B) ord. condest
ord. cond
const condest
const cond
norm B
norm inv(B)
1.00e-01 1.10e+01 7.11e+00 2.00e+01 3.56e-01 5.00e-02 1.95e+01 1.34e+01 -0.83 -0.91 1.64e+00 8.66e-01 4.00e+01 3.36e-01 2.50e-02 3.95e+01 2.61e+01 -1.02 -0.96 9.23e–01 7.48e-01 8.00e+01 3.27e-01 1.25e-02 7.95e+01 5.16e+01 -1.01 -0.98 9.55e–01 7.00e-01 1.60e+02 3.22e-01 6.25e-03 1.60e+01 1.03e+02 -1.00 -0.99 9.74e–01 6.71e-01 3.20e+02 3.20e-01
3. α=2, Midpoint rule
h condest(B) cond(B) ord. condest
ord. cond
const condest
const cond
norm B
norm inv(B)
1.00e-01 8.33e+01 8.25e+01 2.84e+02 2.91e-01 5.00e-02 3.81e+02 3.04e+02 -2.19 -1.88 5.34e–01 1.09e+00 1.13e+03 2.68e-01 2.50e-02 1.41e+03 1.17e+03 -1.89 -1.94 1.34e+00 9.00e–01 4.53e+03 2.58e-01 1.25e-02 5.69e+03 4.58e+03 -2.01 -1.97 8.40e–01 8.07e–01 1.81e+04 2.53e-01 6.25e-03 2.29e+04 1.82e+04 -2.01 -1.99 8.63e–01 7.60e–01 7.25e+04 2.51e-01
4. α=3, Midpoint rule
h condest(B) cond(B) ord. condest
ord. cond
const condest
const cond
norm B
norm inv(B)
1.00e-01 1.30e+03 1.44e+03 5.66e+03 2.54e-01 5.00e-02 1.09e+04 1.04e+04 -3.07 -2.85 1.12e+00 2.02e+00 4.53e+04 2.29e-01 2.50e-02 8.92e+04 7.91e+04 -3.04 -2.93 1.22e+00 1.61e+00 3.62e+05 2.18e-01 1.25e-02 7.45e+05 6.18e+05 -3.06 -2.97 1.12e+00 1.40e+00 2.90e+06 2.13e-01 6.25e-03 5.82e+06 4.88e+06 -2.97 -2.98 1.69e+00 1.30e+00 2.32e+07 2.11e-01
. condest:=condest(B,1), cond:=cond(B,2), norm(B) = norm(B,2)
77
TABLE 1.25 : matrix condition estimates for Example 1 - Midpoint rule -
-preconditioning tˆ(α-1) - 1. α=0, Midpoint rule, number of collocation points m=1
h condest(B’) cond(B’) ord. condest
ord. cond
const condest
const cond
norm B’
norm inv(B’)
1.00e-01 1.03e+02 5.80e+01 1.47e+02 3.95e-01 5.00e-02 3.76e+02 2.09e+02 -1.87 -1.85 1.40e+00 8.20e-01 5.87e+02 3.56e-01 2.50e-02 1.55e+03 7.91e+02 -2.04 -1.92 8.25e–01 6.64e-01 2.35e+03 3.37e-01 1.25e-02 6.06e+03 3.07e+03 -1.97 -1.96 1.09e+00 5.80e-01 9.39e+03 3.27e-01 6.25e-03 2.42e+04 1.21e+04 -2.00 -1.98 9.57e–01 5.27e-01 3.76e+04 3.22e-01
2. α=1, Midpoint rule
h condest(B’) cond(B’) ord. condest
ord. cond
const condest
const cond
norm B’
norm inv(B’)
1.00e-01 1.10e+01 7.11e+00 2.00e+01 3.56e-01 5.00e-02 1.95e+01 1.34e+01 -0.83 -0.91 1.64e+00 8.66e-01 4.00e+01 3.36e-01 2.50e-02 3.95e+01 2.61e+01 -1.02 -0.96 9.23e–01 7.51e-01 8.00e+01 3.27e-01 1.25e-02 7.95e+01 5.16e+01 -1.01 -0.98 9.55e–01 6.94e-01 1.60e+02 3.22e-01 6.25e-03 1.60e+02 1.03e+02 -1.01 -1.00 9.55e–01 6.53e-01 3.20e+02 3.20e-01
3. α=2, Midpoint rule
h condest(B’) cond(B’) ord. condest
ord. cond
const condest
const cond
norm B’
norm inv(B’)
1.00e-01 9.54e+00 9.64e+00 2.87e+01 3.36e-01 5.00e-02 1.94e+01 1.89e+01 -1.02 -0.97 9.03e–01 1.03e+00 5.71e+01 3.32e-01 2.50e-02 4.11e+01 3.78e+01 -1.08 -1.00 7.56e–01 9.45e–01 1.14e+02 3.31e-01 1.25e-02 7.86e+01 7.56e+01 -0.94 -1.00 1.30e+00 9.45e–01 2.28e+02 3.32e-01 6.25e-03 1.58e+02 1.58e+02 -1.01 -1.00 9.51e–01 9.53e–01 4.56e+02 3.32e-01
4. α=3, Midpoint rule
h condest(B’) cond(B’) ord. condest
ord. cond
const condest
const cond
norm B’
norm inv(B’)
1.00e-01 3.35e+01 3.06e+01 5.67e+01 5.67e+01 5.00e-02 7.53e+01 6.57e+01 -1.17 -1.10 2.27e+00 2.42e+00 1.13e+02 5.79e–01 2.50e-02 1.60e+02 1.39e+02 -1.09 -1.08 2.90e+00 2.58e+00 2.27e+02 6.11e–01 1.25e-02 3.31e+02 2.89e+02 -1.05 -1.06 3.34e+00 2.83e+00 4.54e+02 6.37e–01 6.25e-03 6.81e+02 5.95e+02 -1.04 -1.04 3.46e+00 3.01e+00 9.08e+02 6.56e–01
. condest:=condest(B’,1), cond:=cond(B’,2), norm(B’) = norm(B’,2)
78
TABLE 1.26 : matrix condition estimates based on max. norm for Example 1 –Midpoint rule -
- preconditioning 1/h - 1. α=1, Midpoint rule, number of collocation points m=1
h cond(B) ord. cond
const cond
norm (B)
norm inv(B)
1.00e-01 8.17e+00 2.00e+01 4.08e-01 5.00e-02 1.55e+01 -0.92 9.74e-01 4.00e+01 3.87e-01 2.50e-02 3.02e+01 -0.96 8.68e-01 8.00e+01 3.77e-01 1.25e-02 5.96e+01 -0.98 8.11e-01 1.60e+02 3.73e-01 6.25e-03 1.18e+02 -0.99 7.94e-01 3.20e+02 3.70e-01
2. α=2, Midpoint rule
h Cond(B) ord. cond
const cond
norm (B)
norm inv(B)
1.00e-01 1.32e+02 4.00e+02 3.29e-01 5.00e-02 4.90e+02 -2.05 1.04e+00 1.60e+03 3.06e-01 2.50e-02 1.89e+03 -1.95 1.43e+00 6.40e+03 2.95e-01 1.25e-02 7.40e+03 -1.97 1.32e+00 2.56e+04 2.89e-01 6.25e-03 2.93e+04 -1.99 1.23e+00 1.02e+05 2.87e-01
3. α=3, Midpoint rule
h Cond(B) ord. cond
const cond
norm (B)
norm inv(B)
1.00e-01 2.28e+03 8.00e+03 2.85e-01 5.00e-02 1.68e+04 -2.88 3.00e+00 6.40e+04 2.62e-01 2.50e-02 1.28e+05 -2.93 2.59e+00 5.12e+05 2.50e-01 1.25e-02 9.98e+05 -2.96 2.29e+00 4.10e+06 2.44e-01 6.25e-03 7.89e+06 -2.98 2.10e+00 3.28e+07 2.41e-01
79
TABLE 1.27 : matrix condition estimates based on max. norm for Example 1 –Midpoint rule -
-preconditioning tˆ(α-1) - 1. α=1, Midpoint rule, number of collocation points m=1
h Cond(B’) ord. cond
const cond
norm (B’)
norm inv(B’)
1.00e-01 8.17e+00 2.00e+01 4.08e-01 5.00e-02 1.55e+01 -0.92 9.73e-01 4.00e+01 3.87e-01 2.50e-02 3.02e+01 -0.96 8.68e-01 8.00e+01 3.77e-01 1.25e-02 5.96e+01 -0.98 8.11e-01 1.60e+02 3.73e-01 6.25e-03 1.18e+02 -0.99 7.94e-01 3.20e+02 3.70e-01
2. α=2, Midpoint rule
h Cond(B’) ord. cond
const cond
norm (B’)
norm inv(B’)
1.00e-01 1.70e+01 4.00e+01 4.24e-01 5.00e-02 3.40e+01 -1.00 1.70e+00 8.00e+01 4.25e-01 2.50e-02 6.80e+01 -1.00 1.70e+00 1.60e+02 4.25e-01 1.25e-02 1.36e+02 -1.00 1.70e+00 3.20e+02 4.26e-01 6.25e-03 2.73e+02 -1.01 1.66e+00 6.40e+02 4.27e-01
3. α=3, Midpoint rule
h Cond(B’) ord. cond
const cond
norm (B’)
norm inv(B’)
1.00e-01 4.46e+01 8.00e+01 5.57e-01 5.00e-02 9.96e+01 -1.16 3.09e+00 1.60e+02 6.23e-01 2.50e-02 2.18e+02 -1.13 3.37e+00 3.20e+02 6.81e-01 1.25e-02 4.67e+02 -1.10 3.78e+00 6.40e+02 7.30e-01 6.25e-03 9.84e+02 -1.08 4.20e+00 1.28e+03 7.69e-01
80
TABLE 1.28 : global error for Example 1 - Lobatto points - 1. sbvpcol, α=0, Lobatto points, number of collocation points m=3
h err mesh
err coll
p mesh
p coll
const mesh
const coll
1.00e-01 1.41e-07 6.56e-07 5.00e-02 8.75e-09 4.26e-08 4.01 3.95 1.46e-03 5.78e-03 2.50e-02 5.45e-10 2.71e-09 4.01 3.97 1.43e-03 6.27e-03 1.25e-02 3.40e-11 1.71e-10 4.00 3.99 1.41e-03 6.60e-03 6.25e-03 2.12e-12 1.08e-11 4.00 3.99 1.39e-03 6.80e-03
2. sbvpcol, α=1, Lobatto points
h err mesh
err coll
p mesh
p coll
const mesh
const coll
1.00e-01 2.54e-07 6.55e-07 5.00e-02 1.55e-08 4.26e-08 4.03 3.94 2.73e-03 5.75e-03 2.50e-02 9.59e-10 2.71e-09 4.02 3.97 2.63e-03 6.26e-03 1.25e-02 5.95e-11 1.71e-10 4.01 3.99 2.55e-03 6.60e-03 6.25e-03 3.70e-12 1.08e-11 4.01 3.99 2.50e-03 6.80e-03
3. sbvpcol, α=2, Lobatto points
h err mesh
err coll
p mesh
p coll
const mesh
const coll
1.00e-01 6.63e-07 6.63e-07 5.00e-02 5.14e-08 5.14e-08 3.69 3.69 3.25e-03 3.25e-03 2.50e-02 4.00e-09 4.00e-09 3.68 3.68 3.18e-03 3.18e-03 1.25e-02 3.13e-10 3.13e-10 3.68 3.68 3.12e-03 3.12e-03 6.25e-03 2.45e-11 2.45e-11 3.67 3.67 3.07e-03 3.07e-03
4. sbvpcol, α=3, Lobatto points
h err mesh
err coll
p mesh
p coll
const mesh
const coll
1.00e-01 1.28e-06 1.28e-06 5.00e-02 1.09e-07 1.09e-07 3.55 3.55 4.54e-03 4.54e-03 2.50e-02 9.41e-09 9.41e-09 3.54 3.54 4.35e-03 4.35e-03 1.25e-02 8.17e-10 8.17e-10 3.52 3.52 4.18e-03 4.18e-03 6.25e-03 7.06e-11 7.06e-11 3.53 3.53 4.30e-03 4.30e-03
81
TABLE 1.29 : matrix condition estimates for Example 1 - Lobatto points -
1. sbvpcol, α=0, Lobatto points, number of collocation points m=3
h condest(DF) cond(DF) ord. condest
ord. cond
Const condest
const cond
norm DF
norm inv(DF)
1.00e-01 1.81e+01 1.16e+01 2.11e+01 5.48e-01 5.00e-02 3.06e+01 2.32e+01 -0.75 -1.00 3.19e+00 1.16e+00 4.27e+01 5.43e-01 2.50e-02 5.57e+01 4.65e+01 -0.87 -1.00 2.27e+00 1.15e+00 8.58e+01 5.42e-01 1.25e-02 1.06e+02 9.31e+01 -0.93 -1.00 1.80e+00 1.16e+00 1.72e+02 5.42e-01 6.25e-03 2.07e+02 1.86e+02 -0.96 -1.00 1.55e+00 1.17e+00 3.44e+02 5.42e-01
2. sbvpcol, α=1, Lobatto points
h condest(DF) cond(DF) ord. condest
ord. cond
Const condest
const cond
norm DF
norm inv(DF)
1.00e-01 3.09e+01 1.09e+01 2.63e+01 4.16e-01 5.00e-02 5.19e+01 2.06e+01 -0.75 -0.91 5.52e+00 1.34e+00 5.25e+01 3.92e-01 2.50e-02 9.37e+01 4.08e+01 -0.85 -0.99 4.04e+00 1.07e+00 1.05e+02 3.88e-01 1.25e-02 1.77e+02 8.15e+01 -0.92 -1.00 3.17e+00 1.02e+00 2.10e+02 3.88e-01 6.25e-03 3.44e+02 1.63e+02 -0.96 -1.00 2.67e+00 1.01e+00 4.20e+02 3.89e-01
3. sbvpcol, α=2, Lobatto points
h condest(DF) cond(DF) ord. condest
ord. cond
const condest
const cond
norm DF
norm inv(DF)
1.00e-01 7.33e+02 4.72e+02 4.43e+02 1.07e+01 5.00e-02 3.33e+03 1.99e+03 -2.18 -2.08 4.82e+01 3.96e+01 1.78e+03 1.12e+01 2.50e-02 1.41e+04 8.30e+03 -2.09 -2.06 6.40e+01 4.16e+01 7.12e+03 1.17e+01 1.25e-02 5.83e+04 3.42e+04 -2.04 -2.04 7.54e+01 4.44e+01 2.85e+04 1.20e+01 6.25e-03 2.36e+05 1.39e+05 -2.02 -2.03 8.29e+01 4.73e+01 1.14e+05 1.22e+01
4. sbvpcol, α=3, Lobatto points
h condest(DF) cond(DF) ord. condest
ord. cond
const condest
const cond
norm DF
norm inv(DF)
1.00e-01 1.65e+05 1.03e+05 8.68e+03 1.19e+01 5.00e-02 2.69e+06 1.68e+06 -4.02 -4.02 15.6e+01 9.81e+01 6.94e+04 2.42e+01 2.50e-02 4.31e+07 2.70e+07 -4.01 -4.01 1.65e+01 1.03e+01 5.56e+05 4.86e+01 1.25e-02 6.91e+08 4.32e+08 -4.00 -4.00 1.68e+01 1.05e+01 4.44e+06 9.72e+01 6.25e-03 1.11e+10 6.92e+09 -4.00 -4.00 1.68e+01 1.05e+01 3.56e+07 1.95e+02
. condest:=condest(DF,1), cond:=cond(DF,2), norm(DF) = norm(DF,2)
82
TABLE 1.30 : global error for Example 1 -α=0-
1. a) sbvpcol, equidistant, m=2, Error
h err mesh
err coll
p mesh
p coll
const mesh
const coll
1.00e-01 2.78e-04 2.83e-04 5.00e-02 6.95e-05 7.02e-05 2.00 2.01 2.79e-02 2.92e-02 2.50e-02 1.74e-05 1.75e-05 2.00 2.01 2.78e-02 2.87e-02 1.25e-02 4.34e-06 4.35e-06 2.00 2.00 2.78e-02 2.83e-02 6.25e-03 1.09e-06 1.09e-06 2.00 2.00 2.78e-02 2.81e-02
2. a) sbvpcol, equidistant, m=4, Error
h err mesh
err coll
p mesh
p coll
const mesh
const coll
1.00e-01 2.11e-08 2.18e-08 5.00e-02 1.32e-09 1.34e-09 4.00 4.02 2.12e-04 2.28e-04 2.50e-02 8.25e-11 8.31e-11 4.00 4.01 2.11e-04 2.22e-04 1.25e-02 5.15e-12 5.18e-12 4.00 4.01 2.11e-04 2.17e-04 6.25e-03 3.22e-13 3.23e-13 4.00 4.00 2.11e-04 2.14e-04
3. a) sbvpcol, equidistant, m=6, Error
h err mesh
err coll
p mesh
p coll
const mesh
const coll
5.000e-01 1.19e-08 1.28e-08 2.500e-01 1.82e-10 1.95e-10 6.03 6.04 7.77e-07 8.38e-07 1.250e-01 2.82e-12 2.94e-12 6.01 6.05 7.53e-07 8.53e-07 6.250e-02 4.42e-14 4.51e-14 6.00 6.03 7.36e-07 8.17e-07 3.125e-02 1.33e-15 1.55e-15 5.05 4.86 5.35e-08 3.19e-08
83
TABLE 1.31 : global error for Example 1 -α=1-
1. a) sbvpcol, equidistant, m=2, Error
h err mesh
err coll
p mesh
p coll
const mesh
const coll
1.00e-01 2.01e-04 2.10e-04 5.00e-02 5.04e-05 5.14e-05 2.00 2.03 1.99e-02 2.26e-02 2.50e-02 1.26e-05 1.27e-05 2.00 2.01 2.02e-02 2.15e-02 1.25e-02 3.15e-06 3.16e-06 2.00 2.01 2.01e-02 2.10e-02 6.25e-03 7.86e-07 7.89e-07 2.00 2.00 2.01e-02 2.06e-02
2. a) sbvpcol, equidistant, m=4, Error
h err mesh
err coll
p mesh
p coll
const mesh
const coll
1.00e-01 1.53e-08 1.63e-08 5.00e-02 9.57e-10 9.88e-10 4.00 4.05 1.51e-04 1.83e-04 2.50e-02 5.98e-11 6.08e-11 4.00 4.02 1.54e-04 1.69e-04 1.25e-02 3.73e-12 3.77e-12 4.00 4.01 1.53e-04 1.63e-04 6.25e-03 2.35e-13 2.41e-13 3.99 3.97 1.45e-04 1.34e-04
3. a) sbvpcol, equidistant, m=6, Error
h err mesh
err coll
p mesh
p coll
const mesh
const coll
5.000e-01 9.18e-09 1.11e-08 2.500e-01 1.33e-10 1.55e-10 6.10 6.16 6.32e-07 7.92e-07 1.250e-01 2.04e-12 2.24e-12 6.03 6.11 5.68e-07 7.46e-07 6.250e-02 3.24e-14 3.40e-14 5.98 6.05 5.12e-07 6.46e-07 3.125e-02 8.88e-16 1.33e-15 5.19 4.67 5.75e-08 1.44e-08
84
TABLE 1.32 : global error for Example 1 -α=2-
1. a) sbvpcol, equidistant, m=2, Error
h err mesh
err coll
p mesh
p coll
const mesh
const coll
1.00e-01 1.69e-04 1.78e-04 5.00e-02 4.21e-05 4.34e-05 2.01 2.03 1.73e-02 1.92e-02 2.50e-02 1.05e-05 1.07e-05 2.00 2.02 1.68e-02 1.85e-02 1.25e-02 2.63e-06 2.65e-06 2.00 2.01 1.69e-02 1.78e-02 6.25e-03 6.58e-07 6.60e-07 2.00 2.01 1.68e-02 1.74e-02
2. a) sbvpcol, equidistant, m=4, Error
h err mesh
err coll
p mesh
p coll
const mesh
const coll
1.00e-01 1.29e-08 1.40e-08 5.00e-02 8.00e-10 8.37e-10 4.01 4.06 1.31e-04 1.60e-04 2.50e-02 5.00e-11 5.11e-11 4.00 4.03 1.28e-04 1.48e-04 1.25e-02 3.12e-12 3.16e-12 4.00 4.02 1.28e-04 1.39e-04 6.25e-03 1.95e-13 1.96e-13 4.00 4.01 1.28e-04 1.35e-04
3. a) sbvpcol, equidistant, m=6, Error
h err mesh
err coll
p mesh
p coll
const mesh
const coll
5.000e-01 8.45e-09 2.500e-01 1.12e-10 1.28e-10 6.24 6.04 6.38e-07 5.58e-07 1.250e-01 1.71e-12 1.93e-12 6.03 6.05 4.80e-07 5.58e-07 6.250e-02 2.64e-14 2.80e-14 6.02 6.11 4.65e-07 6.38e-07 3.125e-02 2.22e-15 2.22e-15 3.57 3.66 5.28e-10 7.09e-10
85
TABLE 1.33 : global error for Example 1 -α=3-
1. a) sbvpcol, equidistant, m=2, Error
h err mesh
err coll
p mesh
p coll
const mesh
const coll
1.00e-01 1.46e-04 1.57e-04 5.00e-02 3.69e-05 3.81e-05 1.98 2.05 1.39e-02 1.76e-02 2.50e-02 9.21e-06 9.37e-06 2.00 2.02 1.49e-02 1.62e-02 1.25e-02 2.30e-06 2.32e-06 2.00 2.01 1.48e-02 1.57e-02 6.25e-03 5.75e-07 5.78e-07 2.00 2.01 1.47e-02 1.53e-02
2. a) sbvpcol, equidistant, m=4, Error
h err mesh
err coll
p mesh
p coll
const mesh
const coll
1.00e-01 1.11e-08 1.24e-08 5.00e-02 7.02e-10 7.39e-10 3.98 4.07 1.06e-04 1.45e-04 2.50e-02 4.38e-11 4.50e-11 4.00 4.04 1.13e-04 1.32e-04 1.25e-02 2.73e-12 2.77e-12 4.00 4.02 1.13e-04 1.24e-04 6.25e-03 1.72e-13 1.73e-13 3.99 4.00 1.07e-04 1.15e-04
3. a) sbvpcol, equidistant, m=6, Error
h err mesh
err coll
p mesh
p coll
const mesh
const coll
5.000e-01 7.70e-09 7.70e-09 2.500e-01 9.13e-11 1.22e-10 6.40 6.40 6.50e-07 6.49e-07 1.250e+01 1.52e-12 1.70e-12 5.91 6.17 3.31e-07 6.38e-07 6.250e-02 2.31e-14 2.46e-14 6.04 6.11 4.29e-07 5.55e-07 3.125e-02 5.11e-15 5.11e-15 2.18 2.27 9.65e-12 1.32e-11
86
TABLE 1.34 : Error of error estimate based on h-h/2 for Example 1 -α=0-
1. b) sbvpcol, equidistant, m=2, Error of error
h err mesh
err coll
p` mesh
p` coll
const mesh
const coll
1.00e-01 8.29e-08 3.22e-06 5.00e-02 5.18e-09 3.94e-07 4.00 3.03 8.29e-04 3.45e-03 2.50e-02 3.24e-10 4.88e-08 4.00 3.01 8.30e-04 3.30e-03 1.25e-02 2.02e-11 6.06e-09 4.00 3.01 8.30e-04 3.21e-03 6.25e-03 1.27e-12 7.56e-10 4.00 3.00 8.25e-04 3.16e-03
2. b) sbvpcol, equidistant, m=4, Error of error
h err mesh
err coll
p` mesh
p` coll
const mesh
const coll
5.000e-01 1.87e-08 1.49e-07 2.500e-01 2.91e-10 4.34e-09 6.01 5.11 1.21e-06 5.14e-06 1.250e-01 4.54e-12 1.29e-10 6.00 5.07 1.20e-06 4.90e-06 6.250e-02 7.07e-14 3.93e-12 6.01 5.04 1.20e-06 4.59e-06 3.125e-02 2.37e-16 1.20e-13 8.22 5.03 5.61e-04 4.52e-06
3. b) sbvpcol, equidistant, m=6, Error of error
h err mesh
err coll
p` mesh
p` coll
const mesh
const coll
5.000e-01 4.02e-12 3.01e-11 2.500e-01 1.65e-14 2.18e-13 +7.93 +7.11 9.80e-10 4.14e-09 1.250e-01 8.46e-17 1.79e-15 +7.60 +6.93 6.26e-10 3.22e-09 6.250e-02 2.01e-16 4.12e-16 –1.25 +2.12 6.31e-18 1.48e-13 3.125e-02 1.83e-15 1.83e-15 –3.18 –2.15 2.95e-20 1.07e-18
87
TABLE 1.35 : Error of error estimate based on h-h/2 for Example 1 -α=1-
1. b) sbvpcol, equidistant, m=2, Error of error
h err mesh
err coll
p` mesh
p` coll
const mesh
const coll
1.00e-01 2.12e-07 4.71e-06 5.00e-02 1.51e-08 6.12e-07 3.81 2.94 1.38e-03 4.14e+00 2.50e-02 9.42e-10 7.61e-08 4.00 3.01 2.40e-03 5.03e+00 1.25e-02 5.70e-11 9.48e-09 4.05 3.00 2.85e-03 4.95e+00 6.25e-03 3.62e-12 1.19e-09 3.98 2.99 2.11e-03 4.71e+00
2. b) sbvpcol, equidistant, m=4, Error of error
h err mesh
err coll
p` mesh
p` coll
const mesh
const coll
5.000e-01 4.89e-08 1.50e-07 2.500e-01 7.82e-10 5.56e-09 5.97 4.75 3.06e-06 4.03e-06 1.250e-01 1.23e-11 1.87e-10 5.99 4.89 3.15e-06 4.90e-06 6.250e-02 2.25e-13 6.07e-12 5.78 4.95 2.03e-06 5.50e-06 3.125e-02 3.94e-15 1.88e-13 5.84 5.01 2.39e-06 6.60e-06
3. b) sbvpcol, equidistant, m=6, Error of error
h err mesh
err coll
p` mesh
p` coll
const mesh
const coll
5.000e-01 1.02e-11 3.18e-11 2.500e-01 4.15e-14 2.85e-13 +7.94 +6.81 2.52e-09 3.56e-09 1.250e-01 2.75e-16 1.54e-15 +7.24 +7.53 9.44e-10 9.68e-09 6.250e-02 6.34e-17 8.81e-17 +2.12 +4.13 2.24e-14 8.30e-12 3.125e-02 1.82e-15 1.83e-15 –4.84 –4.37 9.39e-23 4.78e-22
88
TABLE 1.36 : Error of error estimate based on h-h/2 for Example 1 -α=2-
1. b) sbvpcol, equidistant, m=2, Error of error
h err mesh
err coll
p` mesh
p` coll
const mesh
const coll
1.00e-01 3.28e-07 5.13e-06 5.00e-02 2.06e-08 6.73e-07 3.99 2.93 3.23e-03 4.37e-03 2.50e-02 1.41e-09 8.60e-08 3.87 2.97 2.20e-03 4.89e-03 1.25e-02 8.44e-11 1.07e-08 4.07 3.00 4.61e-03 5.56e-03 6.25e-03 5.39e-12 1.35e-09 3.97 2.99 3.00e-03 5.35e-03
2. b) sbvpcol, equidistant, m=4, Error of error
h err mesh
err coll
p` mesh
p` coll
const mesh
const coll
5.000e-01 9.82e-08 9.82e-08 2.500e-01 1.70e-09 6.88e-09 5.86 3.84 5.69e-06 1.40e-06 1.250e-01 1.73e-11 2.11e-10 6.61 5.03 1.63e-05 7.30e-06 6.250e-02 3.45e-13 6.86e-12 5.65 4.94 2.17e-06 6.14e-06 3.125e-02 3.88e-15 2.11e-13 6.48 5.02 2.18e-05 7.63e-06
3. b) sbvpcol, equidistant, m=6, Error of error
h err mesh
err coll
p` mesh
p` coll
const mesh
const coll
5.000e-01 2.05e-11 2.500e-01 8.77e-14 3.49e-13 +7.87 +5.88 4.77e-09 1.20e-09 1.250e-01 4.76e-16 3.17e-15 +7.53 +6.79 2.97e-09 4.26e-09 6.250e-02 4.83e-16 7.05e-18 –0.02 +8.81 4.55e-16 2.86e-07 3.125e-02 1.84e-15 1.84e-15 –1.93 –8.03 2.29e-18 1.52e-27
89
TABLE 1.37 : Error of error estimate based on h-h/2 for Example 1 -α=3-
1. b) sbvpcol, equidistant, m=2, Error of error
h err mesh
err coll
p` mesh
p` coll
const mesh
const coll
1.00e-01 5.74e-07 5.62e-06 5.00e-02 2.89e-08 7.02e-07 4.31 3.00 1.18e-02 5.62e-03 2.50e-02 1.81e-09 9.02e-08 4.00 2.96 4.57e-03 5.01e-03 1.25e-02 1.13e-10 1.15e-08 4.00 2.98 4.62e-03 5.27e-03 6.25e-03 7.07e-12 1.44e-09 4.00 2.99 4.62e-03 5.60e-03
2. b) sbvpcol, equidistant, m=4, Error of error
h err mesh
err coll
p` mesh
p` coll
const mesh
const coll
5.000e-01 1.61e-07 1.61e-07 2.500e-01 1.16e-08 6.81e-09 3.80 4.56 2.25e-06 3.81e-06 1.250e+01 3.00e-11 2.11e-10 8.59 5.01 1.72e-03 7.08e-06 6.250e-02 4.76e-13 7.30e-12 5.98 4.86 7.48e-06 5.12e-06 3.125e-02 6.50e-15 2.26e-13 6.19 5.01 1.37e-05 7.93e-06
3. b) sbvpcol, equidistant, m=6, Error of error
h err mesh
err coll
p` mesh
p` coll
const mesh
const coll
5.000e-01 3.38e-11 3.38e-11 2.500e-01 5.05e-13 3.52e-13 +6.07 +6.59 2.26e-09 3.25e-09 1.250e+01 6.03e-16 3.26e-15 +9.70 +6.75 3.54e-07 4.10e-09 6.250e-02 1.41e-16 1.66e-16 +2.10 +4.30 4.71e-14 2.50e-11 3.125e-02 3.01e-15 3.01e-15 –4.42 –4.18 6.75e-22 1.54e-21
90
TABLE 1.38 : Error of error estimate based on h-h/2 using reference solution for Example 1
-α=0- 1. c) sbvpcol, equidistant, m=2, Error of error
h err mesh
err coll
p` mesh
p` coll
const mesh
const coll
5.000e-01 5.35e-05 4.60e-04 2.500e-01 4.94e-06 5.49e-05 3.44 3.07 5.80e-04 3.85e-03 1.250e-01 1.90e-06 8.04e-06 1.38 2.77 3.34e-05 2.56e-03 6.250e-02 1.71e-06 2.46e-06 0.15 1.71 2.60e-06 2.79e-04 3.125e-02 1.70e-06 1.79e-06 0.01 0.46 1.76e-06 8.79e-06
2. c) sbvpcol, equidistant, m=4, Error of error
h err mesh
err coll
p` mesh
p` coll
const mesh
const coll
5.000e-01 1.87e-08 1.49e-07 2.500e-01 2.92e-10 4.34e-09 6.00 5.11 1.20e-06 5.14e-06 1.250e-01 5.32e-12 1.30e-10 5.78 5.06 8.75e-07 4.84e-06 6.250e-02 8.57e-13 4.71e-12 2.64 4.78 1.28e-09 2.72e-06 3.125e-02 7.87e-13 9.04e-13 0.12 2.38 1.21e-12 3.47e-09
3. c) sbvpcol, equidistant, m=6, Error of error
h err mesh
err coll
p` mesh
p` coll
const mesh
const coll
5.000e-01 4.02e-12 3.01e-11 2.500e-01 1.33e-14 2.16e-13 +8.24 +7.12 1.22e-09 4.19e-09 1.250e-01 3.30e-15 1.31e-15 +2.00 +7.36 2.14e-13 5.80e-09 6.250e-02 3.42e-15 3.30e-15 –0.05 –1.33 2.98e-15 8.32e-17 3.125e-02 1.28e-15 1.28e-15 +1.41 +1.37 1.73e-13 1.46e-13
· Exact solution is the reference solution for step size h = 1.5625e-03
91
TABLE 1.39 : Error of error estimate based on h-h/2 using reference solution for Example 1
-α=1- 1. c) sbvpcol, equidistant, m=2, Error of error
h err mesh
err coll
p` mesh
p` coll
const mesh
const coll
5.000e-01 1.28e-04 2.26e-04 2.500e-01 9.39e-06 6.63e-05 3.77 1.78 1.74e-03 7.70e-04 1.250e-01 1.74e-06 1.03e-05 2.43 2.69 2.74e-04 2.77e-03 6.250e-02 1.26e-06 2.41e-06 0.46 2.09 4.51e-06 7.94e-04 3.125e-02 1.23e-06 1.38e-06 0.04 0.80 1.41e-06 2.22e-05
2. c) sbvpcol, equidistant, m=4, Error of error
h err mesh
err coll
p` mesh
p` coll
const mesh
const coll
5.000e-01 4.89e-08 1.50e-07 2.500e-01 7.83e-10 5.56e-09 5.97 4.75 3.06e-06 4.03e-06 1.250e-01 1.29e-11 1.88e-10 5.92 4.89 2.88e-06 4.88e-06 6.250e-02 7.95e-13 6.63e-12 4.02 4.82 5.50e-08 4.26e-06 3.125e-02 5.74e-13 7.56e-13 0.47 3.13 2.92e-12 3.94e-08
3. c) sbvpcol, equidistant, m=6, Error of error
h err mesh
err coll
p` mesh
p` coll
const mesh
const coll
5.000e-01 1.02e-11 3.18e-11 2.500e-01 4.01e-14 2.83e-13 +7.99 +6.81 2.60e-09 3.58e-09 1.250e-01 1.61e-15 2.11e-16 +4.64 +10.4 2.50e-11 5.09e-07 6.250e-02 1.27e-15 8.00e-16 +0.34 –1.92 3.27e-15 3.90e-18 3.125e-02 4.86e-16 4.93e-16 +1.38 +0.70 5.88e-14 5.53e-15
· Exact solution is the reference solution for step size h = 1.5625e-03
92
TABLE 1.40 : Error of error estimate based on h-h/2 using reference solution for Example 1
-α=2- 1. c) sbvpcol, equidistant, m=2, Error of error
h err mesh
err coll
p` mesh
p` coll
const mesh
const coll
5.000e-01 2.57e-04 2.57e-04 2.500e-01 1.84e-05 8.21e-05 3.80 1.65 3.58e-03 8.04e-04 1.250e-01 1.74e-06 1.11e-05 3.40 2.89 2.06e-03 4.51e-03 6.250e-02 1.08e-06 2.35e-06 0.69 2.23 7.24e-06 1.15e-03 3.125e-02 1.03e-06 1.20e-06 0.07 0.97 1.32e-06 3.49e-05
2. c) sbvpcol, equidistant, m=4, Error of error
h err mesh
err coll
p` mesh
p` coll
const mesh
const coll
5.000e-01 9.82e-08 9.82e-08 2.500e-01 1.70e-09 6.88e-09 5.86 3.84 5.69e-06 1.40e-06 1.250e-01 1.78e-11 2.12e-10 6.58 5.02 1.54e-05 7.27e-06 6.250e-02 8.21e-13 7.34e-12 4.44 4.85 1.80e-07 5.07e-06 3.125e-02 4.80e-13 6.84e-13 0.77 3.42 7.03e-12 9.70e-08
3. c) sbvpcol, equidistant, m=6, Error of error
h err mesh
err coll
p` mesh
p` coll
const mesh
const coll
5.000e-01 2.05e-11 2.500e-01 8.63e-14 3.49e-13 +7.89 +5.87 4.85e-09 1.20e-09 1.250e-01 4.76e-16 2.94e-15 +7.50 +6.89 2.84e-09 4.93e-09 6.250e-02 9.27e-16 6.73e-16 –0.96 +2.13 6.44e-17 2.46e-13 3.125e-02 4.76e-17 4.76e-17 +4.28 +3.82 1.34e-10 2.70e-11
· Exact solution is the reference solution for step size h = 1.5625e-03
93
TABLE 1.41 : Error of error estimate based on h-h/2 using reference solution for Example 1
-α=3- 1. c) sbvpcol, equidistant, m=2, Error of error
h err mesh
err coll
p` mesh
p` coll
const mesh
const coll
5.000e-01 1.27e-03 1.27e-03 2.500e-01 1.14e-04 1.14e-04 3.48 3.48 1.41e-02 1.41e-02 1.250e-01 2.14e-06 1.22e-05 5.74 3.23 3.24e-01 1.00e-02 6.250e-02 9.72e-07 2.30e-06 1.14 2.40 2.28e-05 1.81e-03 3.125e-02 9.03e-07 1.08e-06 0.11 1.09 1.30e-06 4.74e-05
2. c) sbvpcol, equidistant, m=4, Error of error
h err mesh
err coll
p` mesh
p` coll
const mesh
const coll
5.000e-01 1.61e-07 1.61e-07 2.500e-01 1.16e-08 6.81e-09 3.80 4.56 2.25e-06 3.81e-06 1.250e-01 3.04e-11 2.12e-10 8.57 5.01 1.67e-03 7.05e-06 6.250e-02 8.90e-13 7.71e-12 5.09 4.78 1.21e-06 4.38e-06 3.125e-02 4.23e-13 6.39e-13 1.07 3.59 1.74e-11 1.64e-07
3. c) sbvpcol, equidistant, m=6, Error of error
h err mesh
err coll
p` mesh
p` coll
const mesh
const coll
5.000e-01 3.38e-11 3.38e-11 2.500e-01 5.00e-13 3.53e-13 +6.08 +6.58 2.28e-09 3.24e-09 1.250e-01 8.25e-16 3.26e-15 +9.24 +6.76 1.84e-07 4.12e-09 6.250e-02 8.11e-17 5.64e-17 +3.35 +5.85 8.69e-13 6.33e-10 3.125e-02 4.31e-15 4.31e-15 –5.73 –6.26 1.01e-23 1.65e-24
· Exact solution is the reference solution for step size h = 1.5625e-03
94
TABLE 1.42 : matrix condition estimates based on max. norm for Example 1 –Midpoint rule -
-preconditioning (t-h/2)ˆ(α-1) - 1. α=1, Midpoint rule, number of the collocation points m=1
h cond(B’) ord. cond
const cond
norm (B’)
norm inv(B’)
1.00e-01 8.17e+00 2.00e+01 4.08e-01 5.00e-02 1.55e+01 -0.92 9.73e-01 4.00e+01 3.87e-01 2.50e-02 3.02e+01 -0.96 8.68e-01 8.00e+01 3.77e-01 1.25e-02 5.96e+01 -0.98 8.11e-01 1.60e+02 3.73e-01 6.25e-03 1.18e+02 -0.99 7.94e-01 3.20e+02 3.70e-01
2. α=2, Midpoint rule
h Cond(B’) ord. cond
const cond
norm (B’)
norm inv(B’)
1.00e-01 9.07e+00 2.00e+01 4.54e-01 5.00e-02 1.77e+01 -0.96 9.84e-01 4.00e+01 4.41e-01 2.50e-02 3.47e+01 -0.97 9.65e-01 8.00e+01 4.34e-01 1.25e-02 6.89e+01 -0.99 9.02e-01 1.60e+02 4.30e-01 6.25e-03 1.37e+02 -0.99 8.94e-01 3.20e+02 4.29e-01
3. α=3, Midpoint rule
h Cond(B’) ord. cond
const cond
norm (B’)
norm inv(B’)
1.00e-01 1.80e+01 2.00e+01 8.98e-01 5.00e-02 3.56e+01 -0.98 1.87e+00 4.00e+01 8.90e-01 2.50e-02 7.10e+01 -1.00 1.80e+00 8.00e+01 8.88e-01 1.25e-02 1.42e+02 -1.00 1.78e+00 1.60e+02 8.87e-01 6.25e-03 2.84e+02 -1.00 1.78e+00 3.20e+02 8.87e-01
95
TABLE 1.43 : Error estimate based on Midpoint rule for Example 1 1. α=0, Midpoint rule, number of the collocation points m=1
h err exact p exact err estimate p estimate errexc-errest p
2.00e-01 3.39e-07 4.00 3.39e-07 4.00 2.44e-10 +6.00 1.00e-01 2.11e-08 4.00 2.11e-08 4.00 3.80e-12 +6.12 5.00e-02 1.32e-09 4.00 1.32e-09 4.00 5.48e-14 +3.56 2.50e-02 8.25e-11 4.00 8.25e-11 4.00 4.66e-15 −0.82
1. α=1, Midpoint rule
h err exact p exact err estimate p estimate errexc-errest p 2.00e-01 2.47e-07 4.02 2.43e-07 4.00 2.51e-08 5.04 1.00e-01 1.53e-08 4.00 1.52e-08 3.99 7.61e-10 5.02 5.00e-02 9.57e-10 4.00 9.55e-10 4.00 2.35e-11 5.01 2.50e-02 5.98e-11 4.00 5.97e-11 4.00 7.27e-13 5.02
3. α=2, Midpoint rule
h err exact p exact err estimate p estimate errexc-errest p 2.00e-01 2.11e-07 4.03 2.06e-07 4.02 7.17e-08 4.46 1.00e-01 1.29e-07 4.01 1.28e-08 4.00 3.26e-09 4.72 5.00e-02 8.00e-10 4.00 7.96e-10 4.00 1.24e-10 4.80 2.50e-02 5.00e-11 4.00 4.99e-11 4.00 4.45e-12 4.80
4. α=3, Midpoint rule
h err exact p exact err estimate p estimate errexc-errest p 2.00e-01 1.85e-07 4.07 1.78e-07 4.02 1.50e-07 4.81 1.00e-01 1.11e-08 3.98 1.09e-08 3.97 5.35e-09 4.56 5.00e-02 7.02e-10 4.00 6.98e-10 4.00 2.26e-10 4.47 2.50e-02 4.38e-11 4.00 4.36e-11 4.00 1.02e-11 4.55
4. α=4, Midpoint rule
h err exact p exact err estimate p estimate errexc-errest p 2.00e-01 2.22e-07 4.44 1.44e-07 3.84 1.80e-07 4.45 1.00e-01 1.02e-08 4.02 1.01e-08 4.01 8.26e-09 4.42 5.00e-02 6.30e-10 4.00 6.26e-10 4.00 3.85e-10 4.44 2.50e-02 3.93e-11 4.00 3.91e-11 4.00 1.77e-11 4.43
4. α=4, Midpoint rule
h err exact p exact err estimate p estimate errexc-errest p 2.00e-01 2.86e-07 4.41 1.38e-07 3.93 2.22e-07 4.23 1.00e-01 1.34e-08 4.38 9.01e-09 3.99 1.18e-08 4.43 5.00e-02 6.42e-10 4.16 5.68e-10 3.99 5.48e-10 4.39 2.50e-02 3.58e-11 4.00 3.57e-11 4.00 2.62e-11 4.38
96
Figure 2.1 : plot of solution and error using sbvp for Example 2
1. a) Solution, β=0 b) Error, β=0
2. a) Solution, β=1 b) Error, β=1
3. a) Solution, β=2 b) Error, β=2
97
TABLE 2.1 : global error for Example 2 -β=0*-
1. sbvpcol, equidistant, m=4
h err mesh
err coll
p mesh
p coll
const mesh
const coll
1.00e-01 3.02e-02 3.02e-02 5.00e-02 1.26e-02 1.26e-02 1.26 1.26 5.50e–01 5.50e–01 2.50e-02 7.01e-05 7.01e-05 7.49 7.49 7.00e+07 7.00e+07 1.25e-02 2.19e-06 2.24e-06 5.00 4.97 7.23e+03 6.37e+03 6.25e-03 1.17e-07 1.34e-07 4.23 4.06 2.42e+02 1.19e+02
2. sbvpcol, Gauss, m =2
h err mesh
err coll
p mesh
p coll
const mesh
const coll
1.00e-01 3.12e-01 3.12e-01 5.00e-02 3.63e-02 3.63e-02 3.10 3.10 3.97e+02 3.97e+02 2.50e-02 3.44e-04 5.26e-04 6.72 6.11 2.01e+07 3.22e+06 1.25e-02 2.68e-05 7.16e-05 3.68 2.88 2.76e+02 2.13e+01 6.25e-03 1.67e-06 8.53e-06 4.00 3.07 1.09e+03 4.97e+01
3. sbvpcol, Gauss, m =3
h err mesh
err coll
p mesh
p coll
const mesh
const coll
1.00e-01 4.38e-02 4.38e-02 5.00e-02 3.50e-03 3.50e-03 3.65 3.64 1.93e+02 1.93e+02 2.50e-02 5.83e-05 7.11e-05 5.90 5.62 1.69e+05 7.19e+04 1.25e-02 9.21e-07 4.36e-06 5.98 4.03 2.26e+05 2.01e+02 6.25e-03 1.84e-08 2.78e-07 5.65 3.97 5.13e+04 1.58e+02
4. sbvpcol, Gauss, m =4
h err mesh
err coll
p mesh
p coll
const mesh
const coll
1.00e-01 2.03e-02 2.03e-02 5.00e-02 1.90e-03 1.90e-03 3.41 3.41 5.31e+01 5.31e+01 2.50e-02 5.89e-06 9.35e-06 8.33 7.67 1.32e+08 1.79e+07 1.25e-02 5.65e-08 3.58e-07 6.70 4.71 3.21e+05 3.23e+02 6.25e-03 3.91e-10 1.21e-08 7.17 4.89 2.57e+06 7.28e+01
(*) Exact solution is a reference solution for step size h = 3.125e-003
98
TABLE 2.2 : global error for Example 2 -β=1*-
1. sbvpcol, equidistant, m=4
h err mesh
err coll
p mesh
p coll
const mesh
const coll
1.00e-01 5.27e-03 5.27e-03 5.00e-02 1.23e-04 1.23e-04 5.42 5.42 1.42e+03 1.42e+03 2.50e-02 2.44e-06 2.55e-06 5.66 5.59 2.83e+03 2.31e+03 1.25e-02 1.42e-07 1.65e-07 4.10 3.95 9.06e+00 5.50e+00 6.25e-03 8.45e-09 9.19e-09 4.07 4.16 7.97e+00 1.39e+01
2. sbvpcol, Gauss, m=2
h err mesh
err coll
p mesh
p coll
const mesh
const coll
1.00e-01 3.61e-02 3.61e-02 5.00e-02 9.51e-04 9.51e-04 5.25 5.25 6.36e+03 6.36e+03 2.50e-02 3.96e-05 1.04e-04 4.59 3.12 8.80e+02 1.38e+01 1.25e-02 2.58e-06 1.28e-05 3.94 3.01 8.18e+01 6.99e+00 6.25e-03 1.53e-07 1.51e-06 4.07 3.08 1.43e+02 9.34e+00
3. sbvpcol, Gauss, m=3
h err mesh
err coll
p mesh
p coll
const mesh
const coll
1.00e-01 2.70e-03 2.70e-03 5.00e-02 4.68e-05 8.08e-05 5.85 5.06 1.91e+03 3.11e+02 2.50e-02 1.08e-06 5.71e-06 5.44 3.82 5.62e+02 7.60e+00 1.25e-02 1.96e-08 3.55e-07 5.78 4.01 1.95e+03 1.50e+01 6.25e-03 3.12e-10 2.34e-08 5.97 3.92 4.55e+03 1.03e+01
4. sbvpcol, Gauss, m =4
h err mesh
err coll
p mesh
p coll
const mesh
const coll
1.00e-01 6.58e-04 6.58e-04 5.00e-02 7.93e-06 1.28e-05 6.37 5.69 1.56e+03 3.21e+02 2.50e-02 4.34e-08 3.99e-07 7.51 5.00 4.71e+04 4.05e+01 1.25e-02 2.26e-10 1.33e-08 7.58 4.91 6.18e+04 2.93e+01 6.25e-03 9.40e-13 4.29e-10 7.91 4.95 2.55e+05 3.55e+01
(*) Exact solution is a reference solution for step size h = 3.125e-003
99
TABLE 2.3 : global error for Example 2 -β=2*-
1. sbvpcol, equidistant, m=4
h err mesh
err coll
p mesh
p coll
const mesh
const coll
1.00e-01 2.15e-03 2.15e-03 5.00e-02 1.40e-05 1.40e-05 7.22 7.22 3.53e+04 3.53e+04 2.50e-02 6.81e-07 7.60e-07 4.36 4.20 6.71e+00 4.11e+00 1.25e-02 4.21e-08 4.94e-08 4.01 3.94 1.83e+00 1.58e+00 6.25e-03 2.50e-09 2.76e-09 4.07 4.16 2.39e+00 4.15e+00
2. sbvpcol, Gauss, m =2
h err mesh
err coll
p mesh
p coll
const mesh
const coll
1.00e-01 3.10e-03 3.10e-03 5.00e-02 1.79e-04 3.09e-04 4.11 3.32 4.37e+01 6.56e+00 2.50e-02 1.32e-05 4.16e-05 3.76 2.89 1.14e+01 1.80e+00 1.25e-02 8.60e-07 5.22e-06 3.94 2.99 2.69e+01 2.61e+00 6.25e-03 5.22e-08 6.09e-07 4.04 3.10 4.21e+01 4.10e+00
3. sbvpcol, Gauss, m =3
h err mesh
err coll
p mesh
p coll
const mesh
const coll
1.00e-01 1.90e-03 1.90e-03 5.00e-02 1.10e-05 2.97e-05 7.44 6.00 5.21e+04 1.89e+03 2.50e-02 2.56e-07 1.84e-06 5.42 4.02 1.23e+02 5.02e+00 1.25e-02 4.23e-09 1.19e-07 5.92 3.95 7.78e+02 3.94e+00 6.25e-03 7.17e-11 7.56e-09 5.88 3.97 6.58e+02 4.30e+00
4. sbvpcol, Gauss, m =4
h err mesh
err coll
p mesh
p coll
const mesh
const coll
1.00e-01 8.12e-05 8.12e-05 5.00e-02 9.98e-07 3.09e-06 6.34 4.72 1.80e+02 4.23e+00 2.50e-02 6.88e-09 1.11e-07 7.18 4.79 2.20e+03 5.34e+00 1.25e-02 3.47e-11 3.38e-09 7.62 5.03 1.14e+04 1.32e+01 6.25e-03 1.48e-13 1.13e-10 7.87 4.91 3.36e+04 7.33e+00
(*) Exact solution is a reference solution for step size h = 3.125e-03
100
TABLE 2.4 : matrix condition estimates for Example 2 - β=0- 1. sbvpcol, equidistant, m=4
h condestDF cond ord. condest
ord. cond
const condest
const cond
1.00e-01 6.21e+04 1.95e+04 5.00e-02 1.88e+05 7.30e+04 -1.60 -1.90 1.58e+03 2.43e+02 2.50e-02 6.23e+05 2.90e+05 -1.73 -1.99 1.04e+03 1.88e+02 1.25e-02 2.24e+06 1.17e+06 -1.84 -2.01 6.95e+02 1.78e+02 6.25e-03 8.42e+06 4.68e+06 -1.91 -2.00 5.17e+02 1.83e+02
2. sbvpcol, Gauss, m=2
h condestDF cond ord. condest
ord. cond
const condest
const cond
1.00e-01 4.26e+04 1.79e+04 5.00e-02 1.08e+05 5.28e+04 -1.34 -1.56 1.95e+03 4.92e+02 2.50e-02 3.71e+05 2.13e+05 -1.79 -2.01 5.13e+02 1.28e+02 1.25e-02 1.41e+06 8.58e+05 -1.92 -2.01 3.07e+02 1.28e+02 6.25e-03 5.48e+06 3.45e+06 -1.96 -2.01 2.62e+02 5.23e+02
3. sbvpcol, Gauss, m=3
h condestDF cond ord. condest
ord. cond
const condest
const cond
1.00e-01 2.17e+05 7.62e+04 5.00e-02 5.38e+05 2.00e+05 -1.31 -1.39 1.07e+04 3.08e+03 2.50e-02 1.38e+06 6.96e+05 -1.35 -1.80 9.16e+03 9.18e+02 1.25e-02 5.09e+06 2.80e+06 -1.88 -2.01 1.32e+03 4.20e+02 6.25e-03 1.95e+07 1.13e+07 -1.94 -2.01 1.04e+03 4.26e+02
4. sbvpcol, Gauss, m=4
h condestDF cond ord. condest
ord. cond
const condest
const cond
1.00e-01 8.20e+05 2.48e+05 5.00e-02 2.01e+06 6.57e+05 -1.29 -1.41 4.19e+04 9.67e+03 2.50e-02 5.00e+06 1.77e+06 -1.32 -1.43 3.89e+04 9.04e+03 1.25e-02 1.36e+07 7.12e+06 -1.45 -2.01 2.41e+04 1.08e+03 6.25e-03 5.20e+07 2.86e+07 -1.94 -2.01 2.82e+03 1.08e+03
· condestDF:=condest(DF,1), cond:=cond(DF,2)
101
TABLE 2.5 : matrix condition estimates for Example 2 - β=1-
1. sbvpcol, equidistant, m=4
h condestDF cond ord. condest
ord. cond
const condest
const cond
1.00e-01 3.31e+06 1.60e+06 5.00e-02 1.90e+07 1.14e+07 -2.52 -2.83 9.94e+03 2.34e+03 2.50e-02 1.22e+08 8.98e+07 -2.68 -2.97 6.18e+03 1.52e+03 1.25e-02 7.36e+08 7.22e+08 -2.59 -3.01 8.57e+03 1.37e+03 6.25e-03 6.37e+09 5.81e+09 -3.11 -3.01 8.80e+02 1.36e+03
2. sbvpcol, Gauss, m=2
h condestDF cond ord. condest
ord. cond
const condest
const cond
1.00e-01 3.54e+06 1.26e+06 5.00e-02 1.96e+07 7.86e+06 -2.47 -2.64 1.19e+04 2.89e+03 2.50e-02 1.16e+08 6.36e+07 -2.57 -3.02 9.03e+03 9.36e+02 1.25e-02 6.74e+08 5.15e+08 -2.54 -3.02 1.01e+04 9.35e+02 6.25e-03 4.47e+09 4.15e+09 -2.73 -3.01 4.31e+03 9.61e+02
3. sbvpcol, Gauss, m=3
h condestDF cond ord. condest
ord. cond
const condest
const cond
1.00e-01 3.46e+07 1.24e+07 5.00e-02 2.19e+08 7.66e+07 -2.66 -2.63 7.52e+04 2.92e+04 2.50e-02 1.30e+09 4.55e+08 -2.57 -2.57 9.98e+04 3.49e+04 1.25e-02 7.51e+09 3.16e+09 -2.53 -2.80 1.13e+05 1.51e+04 6.25e-03 4.30e+10 2.54e+10 -2.52 -3.01 1.23+05 5.92e+03
4. sbvpcol, Gauss, m=4
h condestDF cond ord. condest
ord. cond
const condest
const cond
1.00e-01 2.13e+08 7.41e+07 5.00e-02 1.39e+09 4.55e+08 -2.71 -2.62 1.49e+03 1.78e+05 2.50e-02 8.31e+09 2.70e+09 -2.58 -2.57 6.19e+05 2.08e+05 1.25e-02 4.81e+10 1.57e+10 -2.53 -2.54 7.27e+05 2.32e+05 6.25e-03 2.75e+11 1.05e+11 -2.52 -2.74 7.83e+05 9.40e+04
· condestDF:=condest(DF,1), cond:=cond(DF,2)
102
TABLE 2.6 : matrix condition estimates for Example 2 - β=2- 1. sbvpcol, equidistant, m=4
h condestDF cond ord. condest
ord. cond
const condest
const cond
1.00e-01 1.67e+08 7.18e+07 5.00e-02 1.53e+09 9.37e+08 -3.20 -3.71 1.06e+05 1.41e+04 2.50e-02 2.35e+10 1.45e+10 -3.94 -3.95 1.14e+04 6.81e+03 1.25e-02 2.87e+11 2.33e+11 -3.61 -4.01 3.84e+04 5.50e+03 6.25e-03 3.70e+12 3.76e+12 -3.69 -4.01 2.74e+04 5.39e+03
2. sbvpcol, Gauss, m=2
h condestDF cond ord. condest
ord. cond
const condest
const cond
1.00e-01 2.18e+08 8.13e+07 5.00e-02 2.70e+09 1.12e+09 -3.63 -3.78 5.08e+04 1.34e+04 2.50e-02 3.46e+10 1.45e+10 -3.68 -3.69 4.42e+04 1.76e+04 1.25e-02 4.31e+11 1.81e+11 -3.64 -3.64 5.15e+04 2.11e+04 6.25e-03 5.29e+12 2.55e+12 -3.62 -3.82 5.64e+04 9.77e+03
3. sbvpcol, Gauss, m=3
h condestDF cond ord. condest
ord. cond
const condest
const cond
1.00e-01 4.50e+09 1.63e+09 5.00e-02 5.99e+10 2.20e+10 -3.73 -3.75 8.28e+05 2.91e+05 2.50e-02 7.59e+11 2.79e+11 -3.66 -3.67 1.03e+06 3.69e+05 1.25e-02 9.38e+12 3.47e+12 -3.63 -3.63 1.45e+07 5.24e+06 6.25e-03 1.15e+14 4.25e+13 -3.61 -3.62 1.25e+06 4.55e+05
4. sbvpcol, Gauss, m=4
h condestDF cond ord. condest
ord. cond
const condest
const cond
1.00e-01 4.95e+10 1.67e+10 5.00e-02 6.49e+11 2.19e+11 -3.71 -3.72 9.62e+06 3.18e+06 2.50e-02 8.15e+12 2.78e+12 -3.65 -3.66 1.16e+07 3.78e+06 1.25e-02 1.00e+14 3.44e+13 -3.62 -3.63 1.28e+07 4.25e+06 6.25e-03 1.23e+15 4.21e+14 -3.61 -3.61 1.35e+07 4.54e+06
· condestDF:=condest(DF,1), cond:=cond(DF,2)
103
TABLE 2.7 : global error for Example 2 - β=0- 1. a) sbvpcol, equidistant, m=2, Error
h err mesh
err coll
p mesh
p coll
const mesh
const coll
1.00e-01 2.25e-01 2.25e-01 5.00e-02 3.35e-02 3.35e-02 2.75 2.75 1.25e+02 1.25e+02 2.50e-02 1.84e-03 1.89e-03 4.19 4.15 9.37e+03 8.31e+03 1.25e-02 4.34e-04 4.50e-04 2.08 2.07 4.00e+00 3.93e+00 6.25e-03 8.65e-05 9.15e-05 2.33 2.30 1.17e+01 1.06e+01
2. a) sbvpcol, equidistant, m=4, Error
h err mesh
err coll
p mesh
p coll
const mesh
const coll
1.00e-01 3.02e-02 3.02e-02 5.00e-02 1.26e-02 1.26e-02 1.26 1.26 5.50e–01 5.50e–01 2.50e-02 7.01e-05 7.01e-05 7.49 7.49 7.00e+07 7.00e+07 1.25e-02 2.19e-06 2.24e-06 5.00 4.97 7.23e+03 6.37e+03 6.25e-03 1.17e-07 1.34e-07 4.23 4.06 2.42e+02 1.19e+02
3. a) sbvpcol, equidistant, m=6, Error
h err mesh
err coll
p mesh
p coll
const mesh
const coll
1.00e-01 2.73e-02 2.73e-02 5.00e-02 3.51e-03 3.51e-03 2.96 2.96 2.49e+01 2.49e+01 2.50e-02 1.84e-05 1.84e-05 7.58 7.58 2.53e+07 2.53e+07 1.25e-02 4.92e-08 4.92e-08 8.55 8.55 9.02e+08 9.02e+08 6.25e-03 5.25e-10 5.45e-10 6.55 6.50 1.43e+05 1.26e+03
· Exact solution is a reference solution for step size h = 3.125e-03
104
TABLE 2.8 : global error for Example 2 - β=1- 1. a) sbvpcol, equidistant, m=2, Error
H err mesh
err coll
p mesh
p coll
const mesh
const coll
1.00e-01 2.51e-02 2.51e-02 5.00e-02 7.21e-03 7.21e-03 1.80 1.80 1.58e+00 1.58e+00 2.50e-02 1.85e-03 1.85e-03 1.96 1.96 2.58e+00 2.58e+00 1.25e-02 4.44e-04 4.44e-04 2.06 2.06 3.68e+00 3.68e+00 6.25e-03 8.90e-05 8.90e-05 2.32 2.32 1.15e+01 1.15e+01
2. a) sbvpcol, equidistant, m=4, Error
H err mesh
err coll
p mesh
p coll
const mesh
const coll
1.00e-01 5.27e-03 5.27e-03 5.00e-02 1.23e-04 1.23e-04 5.42 5.42 1.42e+03 1.42e+03 2.50e-02 2.44e-06 2.55e-06 5.66 5.59 2.83e+03 2.31e+03 1.25e-02 1.42e-07 1.65e-07 4.10 3.95 9.06e+00 5.50e+00 6.25e-03 8.45e-09 9.19e-09 4.07 4.16 7.97e+00 1.39e+01
3. a) sbvpcol, equidistant, m=6, Error
H err mesh
err coll
p mesh
p coll
const mesh
const coll
1.00e-01 1.65e-03 1.65e-03 5.00e-02 8.24e-06 8.24e-06 7.65 7.65 7.30e+04 7.30e+04 2.50e-02 3.76e-08 3.76e-08 7.78 7.78 1.08e+05 1.08e+05 1.25e-02 4.40e-10 4.88e-10 6.42 6.27 7.22e+02 4.14e+02 6.25e-03 6.52e-12 7.84e-12 6.08 5.96 1.61e+02 1.07e+02
· Exact solution is a reference solution for step size h = 3.125e-03
105
TABLE 2.9 : global error for Example 2 - β=2- 1. a) sbvpcol, equidistant, m=2, Error
H err mesh
err coll
p mesh
p coll
const mesh
const coll
1.00e-01 5.05e-02 5.05e-02 5.00e-02 1.23e-02 1.23e-02 2.04 2.04 5.51e+00 5.51e+00 2.50e-02 3.03e-03 3.03e-03 2.02 2.02 5.24e+00 5.24e+00 1.25e-02 7.21e-04 7.21e-04 2.07 2.07 6.32e+00 6.32e+00 6.25e-03 1.44e-04 1.44e-04 2.32 2.32 1.89e+01 1.89e+01
2. a) sbvpcol, equidistant, m=4, Error
h err mesh
err coll
P mesh
p coll
const mesh
const coll
1.00e-01 2.15e-03 2.15e-03 5.00e-02 1.40e-05 1.40e-05 7.22 7.22 3.53e+04 3.53e+04 2.50e-02 6.81e-07 7.60e-07 4.36 4.20 6.71e+00 4.11e+00 1.25e-02 4.21e-08 4.94e-08 4.01 3.94 1.83e+00 1.58e+00 6.25e-03 2.50e-09 2.76e-09 4.07 4.16 2.39e+00 4.15e+00
3. a) sbvpcol, equidistant, m=6, Error
h err mesh
err coll
p mesh
p coll
const mesh
const coll
1.00e-01 1.21e-04 1.21e-04 5.00e-02 7.01e-07 7.01e-07 7.43 7.43 3.21e+03 3.21e+03 2.50e-02 7.33e-09 7.33e-09 6.58 6.58 2.56e+02 2.56e+02 1.25e-02 9.05e-11 1.07e-10 6.34 6.10 1.05e+02 4.37e+01 6.25e-03 1.41e-12 1.57e-12 6.00 6.09 2.40e+01 4.06e+01
· Exact solution is a reference solution for step size h = 3.125e-03
106
TABLE 2.10 : Error of error estimate based on h-h/2 for Example 2 - β=0- 1. b) sbvpcol, equidistant, m=2, Error of error
h err mesh
err coll
p` mesh
p` coll
const mesh
const coll
1.00e-01 3.73e-02 3.73e-02 5.00e-02 1.31e-02 1.31e-02 1.51 1.51 1.21e+00 1.21e+00 2.50e-02 1.76e-05 1.17e-04 9.54 6.81 3.36e+10 9.39e+06 1.25e-02 1.09e-06 1.75e-05 4.02 2.74 4.84e+01 2.89e+00 6.25e-03 4.93e-07 3.07e-06 1.14 2.51 1.61e-04 1.05e+00
2. b) sbvpcol, equidistant, m=4, Error of error
H err mesh
err coll
p` mesh
p` coll
const mesh
const coll
1.00e-01 5.77e-03 5.77e-03 5.00e-02 8.73e-04 8.73e-04 2.73 2.73 3.07e+00 3.07e+00 2.50e-02 5.53e-06 5.53e-06 7.30 7.30 2.76e+06 2.76e+06 1.25e-02 1.66e-08 2.04e-08 8.38 8.08 1.48e+08 4.92e+07 6.25e-03 2.36e-10 1.21e-09 6.13 4.07 7.79e+03 1.15e+00
3. b) sbvpcol, equidistant, m=6, Error of error
h err mesh
err coll
p` mesh
p` coll
const mesh
const coll
5.000e-01 1.26e-03 1.26e-03 2.500e-01 3.71e-05 3.71e-05 5.09 5.09 1.54e+02 1.54e+02 1.250e+01 3.11e-07 3.11e-07 6.90 6.90 3.48e+04 3.48e+04 6.250e-02 4.16e-10 4.16e-10 9.55 9.55 6.18e+08 6.18e+08 3.125e-02 1.51e-12 6.01e-13 8.10 9.43 1.10e+06 3.71e+08
· Exact solution is a reference solution for step size h = 3.1250e-03 (m=2,4)
· Exact solution is a reference solution for step size h = 3.9063e-04 (m=6)
107
TABLE 2.11 : Error of error estimate based on h-h/2 for Example 2 - β=1-
1. b) sbvpcol, equidistant, m=2, Error of error
h err mesh
err coll
p` mesh
p` coll
const mesh
const coll
1.00e-01 1.11e-03 1.08e-03 5.00e-02 8.76e-05 8.76e-05 +3.66 +3.66 5.11e+00 5.11e+00 2.50e-02 5.75e-06 5.75e-06 +3.93 +3.93 1.13e+01 1.13e+01 1.25e-02 6.09e-08 6.09e-08 +6.56 +6.56 1.87e+05 1.87e+05 6.25e-03 4.39e-07 4.39e-07 –2.85 –2.85 2.31e–13 2.31e–13
2. b) sbvpcol, equidistant, m=4, Error of error
h err mesh
err coll
p` mesh
p` coll
const mesh
const coll
1.00e-01 2.20e-04 8.20e-05 5.00e-02 8.01e-06 5.46e-07 4.78 7.23 1.32e+01 1.39e+03 2.50e-02 2.03e-08 2.01e-08 8.62 4.76 1.32e+06 8.64e–01 1.25e-02 5.77e-11 1.40e-09 8.46 3.85 7.20e+05 2.91e–02 6.25e-03 3.34e-12 5.60e-11 4.11 4.65 3.89e–03 9.55e–01
3. b) sbvpcol, equidistant, m=6, Error of error
h err mesh
err coll
p` mesh
p` coll
const mesh
const coll
5.000e-01 1.78e-05 1.78e-05 2.500e-01 1.29e-07 1.29e-07 7.10 7.10 2.26e+02 2.26e+02 1.250e+01 4.23e-10 4.23e-10 8.26 8.26 7.13e+03 7.13e+03 6.250e-02 1.10e-12 8.11e-13 8.58 9.03 2.38e+04 1.23e+05 3.125e-02 6.74e-16 1.36e-14 10.7 5.90 2.25e+08 1.35e–01
· Exact solution is a reference solution for step size h = 3.1250e-03 (m=2,4)
· Exact solution is a reference solution for step size h = 3.9063e-04 (m=6)
108
TABLE 2.12 : Error of error estimate based on h-h/2 for Example 2 - β=2-
1. b) sbvpcol, equidistant, m=2, Error of error
h err mesh
Err coll
p` mesh
p` coll
const mesh
const coll
1.00e-01 4.11e-04 4.11e-04 5.00e-02 6.68e-06 6.68e-06 5.94 5.94 3.61e+02 3.61e+02 2.50e-02 1.12e-06 1.12e-06 2.58 2.58 1.51e–02 1.51e–02 1.25e-02 7.73e-07 7.73e-07 0.53 0.53 8.03e–06 8.03e–06 6.25e-03 7.52e-07 7.52e-07 0.04 0.04 9.24e–07 9.24e–07
2. b) sbvpcol, equidistant, m=4, Error of error
h err mesh
err coll
p` mesh
p` coll
const mesh
const coll
1.00e-01 1.42e-04 1.42e-04 5.00e-02 7.77e-07 7.77e-07 7.51 7.51 4.57e+03 4.57e+03 2.50e-02 1.06e-08 6.16e-09 6.19 6.98 8.78e+01 9.36e+02 1.25e-02 2.69e-11 4.19e-10 8.62 3.88 6.95e+05 1.01e–01 6.25e-03 4.79e-13 1.58e-11 5.81 4.73 3.06e+00 4.22e–01
3. b) sbvpcol, equidistant, m=6, Error of error
h err mesh
err coll
p` mesh
p` coll
const mesh
const coll
5.000e-01 1.79e-06 1.79e-06 2.500e-01 1.03e-08 1.03e-08 7.44 7.44 4.93e+01 4.93e+01 1.250e+01 3.30e-11 3.30e-11 8.29 8.29 1.95e+05 1.95e+05 6.250e-02 1.29e-14 2.29e-13 11.3 7.17 4.59e+07 1.02e+01 3.125e-02 2.32e-16 2.33e-15 5.79 6.62 1.35e–03 8.98e–01
· Exact solution is the reference solution for step size h = 3.1250e-03 (m=2,4)
· Exact solution is the reference solution for step size h = 3.9063e-04 (m=6)
109
Figure 3.1 : plot of solution and error using sbvp for Example 3
110
TABLE 3.1 : global error for Example 3
1. sbvpcol, equidistant, m=4
h err mesh
err coll
p mesh
p coll
const mesh
const coll
1.00e-01 1.39e-17 1.39e-17 5.00e-02 1.06e-18 1.06e-18 3.71 3.71 7.13e-14 7.13e-14 2.50e-02 2.83e-20 3.06e-20 5.23 5.11 6.81e-12 1.38e-13 1.25e-02 1.63e-21 1.74e-21 4.12 4.14 1.11e-13 7.46e-15 6.25e-03 9.37e-23 9.95e-23 4.12 4.13 1.13e-13 7.07e-15
2. sbvpcol, Gauss, m=2
h err mesh
err coll
p mesh
p coll
const mesh
const coll
1.00e-01 1.01e-16 1.01e-16 5.00e-02 1.35e-17 1.35e-17 2.90 2.90 8.07e-14 8.07e-14 2.50e-02 1.02e-18 1.44e-18 3.73 3.22 7.18e-14 2.24e-14 1.25e-02 6.16e-20 1.92e-19 4.04 2.91 1.85e-13 8.89e-15 6.25e-03 3.64e-21 2.23e-20 4.08 3.10 2.13e-13 1.78e-14
3. sbvpcol, Gauss, m=3
h err mesh
err coll
p mesh
p coll
const mesh
const coll
1.00e-01 2.77e-17 2.77e-17 5.00e-02 7.17e-19 9.98e-19 5.27 4.80 5.17e-12 1.73e-12 2.50e-02 3.36e-20 8.70e-20 4.42 3.52 3.99e-13 3.79e-14 1.25e-02 6.09e-22 6.38e-21 5.79 3.77 6.25e-11 9.49e-14 6.25e-03 9.89e-24 4.15e-22 5.94 3.94 1.25e-10 2.03e-13
4. sbvpcol, Gauss , m =4
h err mesh
err coll
p mesh
p coll
const mesh
const coll
1.00e-01 1.07e-18 2.56e-18 5.00e-02 1.31e-19 2.12e-19 3.03 3.59 1.15e-15 9.96e-15 2.50e-02 9.95e-22 5.57e-21 7.04 5.25 1.91e-10 1.44e-12 1.25e-02 7.11e-24 2.16e-22 7.13 4.69 2.61e-10 1.82e-13 6.25e-03 3.49e-26 6.96e-24 7.67 4.95 2.81e-09 5.76e-13
· Exact solution is a reference solution for step size h = 3.125e-003
111
TABLE 3.2 : matrix condition estimates for Example 3
1. sbvpcol, equidistant, m=4
h condestDF cond ord. condest
ord. cond
const condest
const cond
1.00e-01 6.20e+04 2.01e+04 5.00e-02 4.28e+05 1.53e+05 -2.79 -2.93 1.01e+02 2.39e+01 2.50e-02 3.16e+06 1.20e+06 -2.88 -2.98 7.60e+01 2.04e+01 1.25e-02 2.42e+07 9.56e+06 -2.94 -2.99 6.19e+01 1.94e+01 6.25e-03 1.89e+08 7.62e+07 -2.97 -3.00 5.44e+01 1.90e+01
2. sbvpcol, Gauss, m=2
h condestDF cond ord. condest
ord. cond
const condest
const cond
1.00e-01 2.22e+05 7.12e+04 5.00e-02 1.52e+06 5.42e+05 -2.78 -2.93 3.68e+02 8.39e+01 2.50e-02 1.12e+07 4.27e+06 -2.88 -2.98 2.69e+02 7.24e+01 1.25e-02 8.61e+07 3.39e+07 -2.94 -2.99 2.20e+02 6.88e+01 6.25e-03 6.74e+08 2.71e+08 -2.97 -3.00 1.93e+02 6.74e+01
3. sbvpcol, Gauss, m=3
h condestDF cond ord. condest
ord. cond
const condest
const cond
1.00e-01 1.65e+06 4.48e+05 5.00e-02 1.08e+07 3.34e+06 -2.71 -2.90 3.23e+03 4.22e+03 2.50e-02 7.70e+07 2.62e+07 -2.83 -2.97 2.24e+03 4.55e+02 1.25e-02 5.78e+08 2.08e+08 -2.91 -2.99 1.69e+03 4.25e+02 6.25e-03 4.47e+09 1.66e+09 -2.95 -3.00 1.40e+03 4.14e+02
4. sbvpcol, Gauss, m=4
h condestDF cond ord. condest
ord. cond
const condest
const cond
1.00e-01 7.93e+06 1.90e+06 5.00e-02 1.64e+07 1.39e+07 -1.05 -2.87 7.11e+05 2.57e+03 2.50e-02 3.43e+08 1.08e+08 -4.39 -2.96 3.20e+01 1.92e+03 1.25e-02 2.52e+09 8.59e+08 -2.88 -2.99 8.37e+03 1.77e+03 6.25e-03 1.93e+10 6.85e+09 -2.94 -3.00 6.54e+03 1.71e+03
· condestDF:=condest(DF,1), cond:=cond(DF,2)
112
TABLE 3.3 : global error for Example 3
1. a) sbvpcol, equidistant, m=2, Error
h err mesh
err coll
p mesh
p coll
const mesh
const coll
1.00e-01 1.78e-16 1.78e-16 5.00e-02 2.42e-17 2.42e-17 2.88 2.88 1.34e-13 1.34e-13 2.50e-02 5.73e-18 5.73e-18 2.08 2.08 1.22e-14 1.22e-14 1.25e-02 1.29e-18 1.34e-18 2.15 2.10 1.58e-14 1.31e-14 6.25e-03 2.55e-19 2.68e-19 2.34 2.32 3.70e-14 3.53e-14
2. a) sbvpcol, equidistant, m=4, Error
h err mesh
err coll
p mesh
p coll
const mesh
const coll
1.00e-01 1.39e-17 1.39e-17 5.00e-02 1.06e-18 1.06e-18 3.71 3.71 7.13e-14 7.13e-14 2.50e-02 2.83e-20 3.06e-20 5.23 5.11 6.81e-12 1.38e-13 1.25e-02 1.63e-21 1.74e-21 4.12 4.14 1.11e-13 7.46e-15 6.25e-03 9.37e-23 9.95e-23 4.12 4.13 1.13e-13 7.07e-15
3. a) sbvpcol, equidistant, m=6, Error
h err mesh
err coll
p mesh
p coll
const mesh
const coll
5.000e-01 2.77e-18 2.77e-18 2.500e-01 3.79e-20 3.79e-20 6.19 6.19 4.31e-12 4.31e-12 1.250e+01 3.61e-22 3.61e-22 6.72 6.72 2.08e-11 2.08e-11 6.250e-02 3.45e-24 4.04e-24 6.71 6.48 2.02e-11 8.71e-12 3.125e-02 4.77e-26 5.70e-26 6.17 6.15 1.94e-12 2.03e-12
· Exact solution is a reference solution for step size h = 3.125e-03
113
TABLE 3.4 : Error of error based on h-h/2 for Example 3
1. b) sbvpcol, equidistant, m=2, Error of error
h err mesh
err coll
p mesh
p coll
const mesh
const coll
1.00e-01 3.14e-17 3.14e-17 5.00e-02 3.93e-18 3.93e-18 3.00 3.00 3.13e-14 3.13e-14 2.50e-02 1.02e-19 1.02e-19 5.27 5.27 2.84e-11 2.84e-11 1.25e-02 7.64e-21 4.80e-20 3.74 1.08 9.80e-14 5.55e-18 6.25e-03 1.72e-21 7.90e-21 2.15 2.60 9.61e-17 4.30e-15
2. b) sbvpcol, equidistant, m=4, Error of error
h err mesh
err coll
p mesh
p coll
const mesh
const coll
1.00e-01 2.86e-19 2.86e-19 5.00e-02 4.07e-20 4.07e-20 2.81 2.81 1.86e-16 1.86e-16 2.50e-02 8.32e-22 4.19e-22 5.61 6.60 8.13e-13 1.59e-11 1.25e-02 2.49e-24 9.64e-24 8.39 5.44 2.26e-08 2.18e-13 6.25e-03 4.06e-26 3.87e-25 5.93 4.64 4.94e-13 6.48e-15
3. b) sbvpcol, equidistant, m=6, Error of error
h err mesh
err coll
p mesh
p coll
const mesh
const coll
1.00e-01 3.26e-20 3.26e-20 5.00e-02 3.03e-22 3.03e-22 6.75 6.75 1.82e-13 1.82e-13 2.50e-02 3.18e-24 3.18e-24 6.58 6.58 1.09e-13 1.09e-13 1.25e-02 5.51e-27 1.13e-26 9.17 8.14 1.58e-09 3.49e-11 6.25e-03 2.33e-29 1.22e-28 8.92 6.53 1.05e-09 3.08e-14
· Exact solution is the reference solution for step size h =1/2560
114
Figure 4.1 : plot of solution and error using sbvp for Example 4
115
TABLE 4.1 : global error using reference solution for Example 4
1. sbvpcol, equidistant, m=4
h err mesh
err coll
p mesh
p coll
const mesh
const coll
1.00e-01 7.73e-05 1.48e-04 5.00e-02 1.00e-04 1.00e-04 –0.37 +0.57 3.28e–05 6.30e–05 2.50e-02 2.19e-05 2.19e-05 +2.19 +2.19 7.11e–02 7.11e–02 1.25e-02 1.31e-06 1.31e-06 +4.06 +4.06 7.09e+01 7.09e+01 6.25e-03 8.24e-08 8.24e-08 +3.99 +3.99 5.13e+01 5.13e+01
2. sbvpcol, Gauss, m=2
h err mesh
err coll
p mesh
p coll
const mesh
const coll
1.00e-01 1.33e-04 3.05e-04 5.00e-02 4.92e-05 7.72e-05 +1.44 +1.98 3.62e-03 2.94e-02 2.50e-02 5.82e-05 5.82e-05 –0.24 +0.41 2.39e-05 2.62e-04 1.25e-02 1.00e-05 1.02e-05 +2.54 +2.52 6.70e-01 6.30e-01 6.25e-03 1.42e-05 1.42e-05 –0.50 –0.48 1.15e-06 1.25e-06
3. sbvpcol, Gauss, m=3
h err mesh
err coll
p mesh
p coll
const mesh
const coll
1.00e-01 9.34e-06 1.43e-04 5.00e-02 4.37e-05 4.37e-05 –2.23 +1.71 5.55e–08 7.40e–07 2.50e-02 3.18e-06 3.18e-06 +3.78 +3.78 3.63e+00 3.63e+00 1.25e-02 1.94e-07 1.94e-07 +4.03 +4.03 9.26e+00 9.26e+00 6.25e-03 1.15e-08 1.15e-08 +4.07 +4.07 1.08e+01 1.08e+01
4. sbvpcol, Gauss, m=4
h err mesh
err coll
p mesh
p coll
const mesh
const coll
1.00e-01 3.56e-04 3.57e-04 5.00e-02 3.99e-05 3.99e-05 3.16 3.16 5.10e–01 5.16e–01 2.50e-02 3.19e-06 3.19e-06 3.65 3.65 2.22e+00 2.22e+00 1.25e-02 1.94e-07 1.94e-07 4.04 4.04 9.42e+00 9.42e+00 6.25e-03 1.15e-08 1.15e-08 4.07 4.07 1.10e+01 1.10e+01
· Exact solution is the reference solution for step size h =1.5626e-03
116
TABLE 4.2 : matrix condition estimates for Example 4 1. sbvpcol, equidistant, m=4
h condestDF cond ord. condest
ord. cond
const condest
const cond
1.00e-01 8.91e+09 2.47e+09 5.00e-02 2.61e+09 7.26e+08 +1.77 +1.76 5.25e+11 1.43e+11 2.50e-02 1.45e+11 6.98e+10 –6.49 –6.59 9.31e+00 1.96e+00 1.25e-02 3.21e+11 1.00e+11 –0.45 –0.52 4.52e+10 1.02e+10 6.25e-03 1.17e+12 3.78e+11 –1.87 –1.92 8.77e+07 2.26e+07
2. sbvpcol, Gauss, m=2
h condestDF cond ord. condest
ord. cond
const condest
const cond
1.00e-01 1.12e+09 4.17e+08 5.00e-02 1.52e+10 5.34e+09 -3.75 -3.68 1.98e+05 8.72e+04 2.50e-02 3.21e+10 1.14e+10 -1.08 -1.09 5.91e+08 2.04e+08 1.25e-02 3.13e+11 1.12e+11 -3.28 -3.30 1.75e+05 5.86e+04 6.25e-03 1.68e+12 6.04e+11 -2.42 -2.43 7.72e+06 2.64e+06
3. sbvpcol, Gauss, m=3
h condestDF cond ord. condest
ord. cond
const condest
const cond
1.00e-01 7.63e+09 2.35e+09 5.00e-02 3.69e+11 1.12e+11 –5.60 –5.58 1.93e+04 6.19e+03 2.50e-02 2.08e+11 6.56e+10 +0.83 +0.78 4.44e+12 1.15e+12 1.25e-02 6.97e+11 2.27e+11 –1.74 –1.79 3.31e+08 8.86e+07 6.25e-03 2.72e+12 9.05e+11 –1.97 –1.99 1.26e+08 3.64e+07
4. sbvpcol, Gauss, m=4
h condestDF cond ord. condest
ord. cond
const condest
const cond
1.00e-01 1.88e+12 5.07e+11 5.00e-02 9.69e+11 2.66e+11 +0.96 +0.92 1.70e+13 4.29e+12 2.50e-02 5.68e+11 1.67e+11 +0.77 +0.68 9.76e+12 2.01e+12 1.25e-02 1.87e+12 5.77e+11 –1.72 –1.79 1.01e+09 2.26e+08 6.25e-03 7.22e+12 2.30e+12 –1.95 –1.99 3.61e+08 9.24e+07
· condestDF:=condest(DF,1), cond:=cond(DF,2)
117
TABLE 4.3 : global error using reference solution for Example 4 1. a) sbvpcol, equidistant, m=2, Error
h err mesh
err coll
p mesh
p coll
const mesh
const coll
1.00e-01 1.07e-02 1.07e-02 5.00e-02 2.45e-03 2.45e-03 2.13 2.13 1.43e+00 1.43e+00 2.50e-02 6.81e-04 6.81e-04 1.85 1.85 6.18e–01 6.18e–01 1.25e-02 1.94e-04 1.94e-04 1.81 1.81 5.38e–01 5.38e–01 6.25e-03 4.70e-05 4.70e-05 2.05 2.05 1.55e+00 1.55e+00
2. a) sbvpcol, equidistant, m=4, Error
h err mesh
err coll
p mesh
p coll
const mesh
const coll
1.00e-01 7.73e-05 1.48e-04 5.00e-02 1.00e-04 1.00e-04 –0.37 0.57 3.28e–05 6.30e–05 2.50e-02 2.19e-05 2.19e-05 +2.19 2.19 7.11e–02 7.11e–02 1.25e-02 1.31e-06 1.31e-06 +4.06 4.06 7.09e+01 7.09e+01 6.25e-03 8.24e-08 8.24e-08 +3.99 3.99 5.13e+01 5.13e+01
3. a) sbvpcol, equidistant, m=6, Error
h err mesh
err coll
p mesh
p coll
const mesh
const coll
1.00e-01 1.24e-04 1.48e-04 5.00e-02 4.03e-05 4.03e-05 1.62 1.87 5.18e–03 1.10e–02 2.50e-02 3.19e-06 3.19e-06 3.66 3.66 2.33e+00 2.33e+00 1.25e-02 1.94e-07 1.94e-07 4.04 4.04 9.44e+00 9.44e+00 6.25e-03 1.15e-08 1.15e-08 4.08 4.08 1.11e+01 1.11e+01
· Exact solution is the reference solution for step size h =1.5626e-03
118
TABLE 4.4 : Error of error estimate based on h-h/2 for Example 4
1. b) sbvpcol, equidistant, m=2, Error of error
h err mesh
err coll
p mesh
p coll
const mesh
const coll
1.00e-01 2.92e-04 2.92e-04 5.00e-02 9.20e-05 9.20e-05 +1.67 +1.67 1.36e−02 1.36e−02 2.50e-02 3.22e-05 3.22e-05 +1.51 +1.51 8.59e−03 8.59e−03 1.25e-02 1.15e-04 1.15e-04 −1.84 −1.84 3.70e−08 3.70e−08 6.25e-03 1.83e-05 1.83e-05 +2.65 +2.65 1.27e+01 1.27e+01
2. b) sbvpcol, equidistant, m=4, Error of error
h err mesh
err coll
p mesh
p coll
const mesh
const coll
1.00e-01 1.06e-05 5.03e-05 5.00e-02 3.00e-05 3.00e-05 −1.51 +0.82 3.27e−07 3.52e−04 2.50e-02 6.30e-08 6.30e-08 +8.90 +8.90 1.13e+07 1.13e+07 1.25e-02 6.23e-10 6.23e-10 +6.66 +6.66 2.92e+03 2.92e+03 6.25e-03 3.33e-10 3.33e-10 +0.91 +0.91 3.29e+08 3.29e+08
3. b) sbvpcol, equidistant, m=6, Error of error
h err mesh
err coll
p mesh
p coll
const mesh
const coll
1.00e-01 4.29e-05 1.93e-05 5.00e-02 3.88e-06 3.88e-06 3.47 2.32 1.26e−01 3.99e−03 2.50e-02 1.47e-07 1.47e-07 4.72 4.72 5.47e+00 5.47e+00 1.25e-02 8.61e-09 8.61e-09 4.09 4.09 5.20e−01 5.20e−01 6.25e-03 4.82e-10 4.82e-10 4.16 4.16 7.06e−01 7.06e−01
· Exact solution is the reference solution for step size h =1.5626e-03
119
Figure 5.1 : plot of solution and error using sbvp for Example 5
120
TABLE 5.1 : global error for Example 5 1. sbvpcol, equidistant, m=4
h err mesh
err coll
p mesh
p coll
const mesh
const coll
1.00e-01 1.10e-04 1.10e-04 5.00e-02 7.83e-06 7.83e-06 3.81 3.81 7.13e–01 7.13e–01 2.50e-02 8.44e-08 9.65e-08 6.54 6.34 2.50e+03 1.39e+03 1.25e-02 4.60e-09 5.21e-09 4.20 4.21 4.45e–01 5.37e–01 6.25e-03 2.69e-10 3.00e-10 4.10 4.11 2.88e–01 3.58e–01
2. sbvpcol, Gauss, m=2
h err mesh
err coll
p mesh
p coll
const mesh
const coll
1.00e-01 2.69e-03 2.69e-03 5.00e-02 1.63e-04 1.63e-04 4.05 4.05 3.03e+01 3.03e+01 2.50e-02 5.66e-06 1.01e-05 4.85 4.02 3.32e+02 2.74e+01 1.25e-02 3.50e-07 1.30e-06 4.01 2.95 1.52e+01 5.38e–01 6.25e-03 2.17e-08 1.62e-07 4.01 3.00 1.49e+01 6.66e–01
3. sbvpcol, Gauss, m=3
h err mesh
err coll
p mesh
p coll
const mesh
const coll
1.00e-01 2.33e-04 2.33e-04 5.00e-02 1.59e-05 1.59e-05 3.87 3.87 1.73e+00 1.73e+00 2.50e-02 1.59e-07 3.66e-07 6.65 5.44 7.09e+03 1.93e+02 1.25e-02 3.04e-09 2.27e-08 5.71 4.00 2.24e+02 9.37e–01 6.25e-03 5.11e-11 1.43e-09 5.89 3.99 5.00e+02 8.90e–01
4. sbvpcol, Gauss, m=4
h err mesh
err coll
p mesh
p coll
const mesh
const coll
1.00e-01 8.28e-05 8.28e-05 5.00e-02 1.10e-06 1.10e-06 6.24 6.24 1.43e+02 1.43e+02 2.50e-02 8.44e-09 1.48e-08 7.02 6.22 1.50e+03 1.35e+02 1.25e-02 5.15e-11 5.02e-10 7.35 4.88 5.16e+03 9.73e–01 6.25e-03 2.45e-13 1.57e-11 7.72 5.00 2.50e+03 1.65e+00
121
TABLE 5.2 : convergence of first derivative for Example 5 1. sbvpcol, equidistant, m=4
h err mesh
err coll
p mesh
p coll
const mesh
const coll
1.00e-01 1.14e-02 1.14e-02 5.00e-02 1.62e-03 1.62e-03 2.81 2.81 7.45e+00 7.45e+00 2.50e-02 4.07e-05 4.07e-05 5.31 5.31 1.33e+04 1.33e+04 1.25e-02 2.66e-06 2.66e-06 3.94 3.94 8.29e+01 8.29e+01 6.25e-03 1.56e-07 1.56e-07 4.09 4.09 1.61e+02 1.61e+02
2. sbvpcol, Gauss, m=2
h err mesh
err coll
p mesh
p coll
const mesh
const coll
1.00e-01 1.59e-01 1.59e-01 5.00e-02 2.58e-02 2.58e-02 2.62 2.62 6.68e+01 6.68e+01 2.50e-02 5.21e-03 5.21e-03 2.31 2.31 2.60e+01 2.60e+01 1.25e-02 1.21e-03 1.21e-03 2.11 2.11 1.23e+01 1.23e+01 6.25e-03 2.86e-04 2.86e-04 2.08 2.08 2.61e+00 2.61e+00
3. sbvpcol, Gauss, m=3
h err mesh
err coll
p mesh
p coll
const mesh
const coll
1.00e-01 2.77e-02 2.77e-02 5.00e-02 3.80e-03 3.80e-03 2.86 2.86 2.02e+01 2.02e+01 2.50e-02 3.38e-04 3.38e-04 3.49 3.49 1.33e+02 1.33e+02 1.25e-02 3.59e-05 3.59e-05 3.24 3.24 5.20e+01 5.20e+01 6.25e-03 4.11e-06 4.11e-06 3.13 3.13 3.19e+01 3.19e+01
4. sbvpcol, Gauss, m=4
h err mesh
err coll
p mesh
p coll
const mesh
const coll
1.00e-01 1.65e-02 1.65e-02 5.00e-02 5.57e-04 5.57e-04 4.89 4.89 1.28e+03 1.28e+03 2.50e-02 2.18e-05 2.18e-05 4.68 4.68 6.74e+02 6.74e+02 1.25e-02 1.23e-06 1.23e-06 4.14 4.14 9.43e+01 9.43e+01 6.25e-03 6.70e-08 6.70e-08 4.20 4.20 1.23e+02 1.23e+02
122
TABLE 5.3 : convergence of second derivative for Example 5 1. sbvpcol, equidistant, m=4
h err mesh
err coll
p mesh
p coll
const mesh
const coll
1.00e-01 9.52e-01 9.52e-01 5.00e-02 2.25e-01 2.25e-01 2.08 2.08 1.15e+02 1.15e+02 2.50e-02 1.58e-02 1.58e-02 3.83 3.83 2.16e+04 2.16e+04 1.25e-02 2.20e-03 2.20e-03 2.84 2.84 5.63e+02 5.63e+02 6.25e-03 2.73e-04 2.73e-04 3.01 3.01 1.19e+03 1.19e+03
2. sbvpcol, Gauss, m=2
h err mesh
err coll
p mesh
p coll
const mesh
const coll
1.00e-01 4.89e+00 4.89e+00 5.00e-02 2.15e+00 2.15e+00 1.18 1.18 73.8 73.8 2.50e-02 1.09e+00 1.09e+00 0.98 0.98 41.1 41.1 1.25e-02 5.36e–01 5.36e–01 1.03 1.03 48.4 48.4 6.25e-03 2.63e–01 2.63e–01 1.03 1.03 48.1 48.1
3. sbvpcol, Gauss, m=3
h err mesh
err coll
p mesh
p coll
const mesh
const coll
1.00e-01 1.37e+00 1.37e+00 5.00e-02 4.35e–01 4.35e–01 1.65 1.65 6.10e+01 6.10e+01 2.50e-02 1.27e–01 1.27e–01 1.77 1.77 8.79e+01 8.79e+01 1.25e-02 3.11e–02 3.11e–02 2.03 2.03 2.30e+02 2.30e+02 6.25e-03 7.49e–03 7.49e–03 2.05 2.05 2.52e+02 2.52e+02
4. sbvpcol, Gauss, m=4
h err mesh
err coll
p mesh
p coll
const mesh
const coll
1.00e-01 1.47e+00 1.47e+00 5.00e-02 1.36e–01 1.36e–01 3.43 3.43 3.96e+03 3.96e+03 2.50e-02 1.24e–02 1.24e–02 3.46 3.46 4.32e+03 4.32e+03 1.25e-02 1.72e–03 1.72e–03 2.84 2.84 4.42e+02 4.42e+02 6.25e-03 2.03e–04 2.03e–04 3.09 3.09 1.32e+03 1.32e+03
123
TABLE 5.4 : matrix condition estimates for Example 5 1. sbvpcol, equidistant, m=4
h condestDF cond ord. condest
ord. cond
const condest
const cond
1.00e-01 4.41e+03 2.04e+03 5.00e-02 1.27e+04 6.22e+03 -1.53 -1.61 1.31e+02 4.99e+01 2.50e-02 5.61e+04 2.45e+04 -2.14 -1.98 2.09e+01 1.66e+01 1.25e-02 1.90e+05 9.77e+04 -1.76 -2.00 8.43e+01 1.55e+01 6.25e-03 4.56e+05 3.91e+05 -1.26 -2.00 7.51e+02 1.53e+01
2. sbvpcol, Gauss, m=2
h condestDF cond ord. condest
ord. cond
const condest
const cond
1.00e-01 5.97e+03 2.98e+03 5.00e-02 1.79e+04 7.80e+03 -1.59 -1.39 1.55e+02 1.22e+02 2.50e-02 4.97e+04 2.01e+04 -1.47 -1.37 2.19e+02 1.29e+02 1.25e-02 1.32e+05 7.19e+04 -1.41 -1.84 1.10e+02 2.30e+02 6.25e-03 3.44e+05 2.88e+05 -1.38 -2.00 3.12e+02 1.12e+01
3. sbvpcol, Gauss, m=3
h condestDF cond ord. condest
ord. cond
const condest
const cond
1.00e-01 2.69e+04 1.22e+04 5.00e-02 7.75e+04 3.51e+04 -1.53 -1.52 8.02e+02 3.66e+02 2.50e-02 2.41e+05 9.39e+04 -1.64 -1.42 2.61e+02 5.01e+02 1.25e-02 6.81e+05 2.45e+05 -1.50 -1.39 9.59e+02 5.66e+02 6.25e-03 1.83e+06 9.40e+05 -1.43 -1.94 1.31e+03 5.03e+01
4. sbvpcol, Gauss, m=4
h condestDF cond ord. condest
ord. cond
const condest
const cond
1.00e-01 1.02e+05 3.77e+04 5.00e-02 1.08e+05 1.08e+05 -0.08 -1.52 8.48e+04 1.15e+03 2.50e-02 7.63e+05 3.08e+05 -2.82 -1.51 2.29e+01 1.15e+03 1.25e-02 2.33e+06 8.23e+05 -1.61 -1.42 2.01e+03 1.65e+03 6.25e-03 6.52e+06 2.39e+06 -1.49 -1.54 3.48e+03 9.82e+02
· condestDF:=condest(DF,1), cond:=cond(DF,2)
124
TABLE 5.5 : global error for Example 5 1. a) sbvpcol, equidistant, m=2, Error
h err mesh
err coll
p mesh
p coll
const mesh
const coll
1.00e-01 4.71e-03 4.71e-03 5.00e-02 3.51e-04 3.51e-04 3.75 3.75 2.62e+01 2.62e+01 2.50e-02 5.00e-05 5.29e-05 2.81 2.73 1.60e+00 8.30e+00 1.25e-02 1.25e-05 1.29e-05 2.00 2.04 8.02e–02 9.86e–02 6.25e-03 3.12e-06 3.17e-06 2.00 2.02 8.08e–02 8.98e–02
2. a) sbvpcol, equidistant, m=4, Error
h err mesh
err coll
p mesh
p coll
const mesh
const coll
1.00e-01 1.10e-04 1.10e-04 5.00e-02 7.83e-06 7.83e-06 3.81 3.81 7.13e–01 7.13e–01 2.50e-02 8.44e-08 9.65e-08 6.54 6.34 2.50e+03 1.39e+03 1.25e-02 4.60e-09 5.21e-09 4.20 4.21 4.45e–01 5.37e–01 6.25e-03 2.69e-10 3.00e-10 4.10 4.11 2.88e–01 3.58e–01
3. a) sbvpcol, equidistant, m=6, Error
h err mesh
err coll
p mesh
p coll
const mesh
const coll
5.000e-01 2.94e-05 2.94e-05 2.500e-01 2.59e-07 2.59e-07 6.83 6.83 1.98e+02 1.98e+02 1.250e+01 6.57e-10 6.57e-10 8.62 8.62 4.29e+04 4.29e+04 6.250e-02 8.13e-12 8.13e-12 6.34 6.34 9.28e+00 9.28e+00 3.125e-02 8.98e-14 1.14e-13 6.50 6.16 1.91e+01 4.22e+00
125
TABLE 5.6 : Error of error estimate based on h-h/2 for Example 5
2. b) sbvpcol, equidistant, m=2, Error of error
h err mesh
err coll
p` mesh
p` coll
const mesh
const coll
1.00e-01 1.10e-03 1.10e-03 5.00e-02 1.07e-04 1.07e-04 3.37 3.37 1.07e+00 1.07e+00 2.50e-02 9.05e-08 1.57e-06 10.2 6.08 2.00e+09 8.72e+03 1.25e-02 6.72e-09 2.03e-07 3.75 2.95 9.24e−02 8.47e−02 6.25e-03 4.21e-10 2.56e-08 4.00 2.99 2.73e−01 1.01e−01
2. b) sbvpcol, equidistant, m=4, Error of error
h err mesh
err coll
p` mesh
p` coll
const mesh
const coll
1.00e-01 9.88e-07 9.88e-07 5.00e-02 4.61e-07 4.61e-07 1.10 1.10 1.24e−05 1.24e−05 2.50e-02 1.87e-09 1.99e-09 7.95 7.86 1.01e+04 7.66e+03 1.25e-02 1.98e-11 6.41e-11 6.56 4.96 6.08e+01 1.75e−01 6.25e-03 3.18e-13 2.42e-12 5.96 4.73 4.38e+00 6.31e−02
3. b) sbvpcol, equidistant, m=6, Error of error
h err mesh
err coll
p` mesh
p` coll
const mesh
const coll
1.00e-01 7.15e-07 7.15e-07 5.00e-02 3.66e-09 3.66e-09 7.61 7.61 2.91e+01 2.91e+01 2.50e-02 1.01e-11 1.01e-11 8.50 8.50 4.15e+02 4.15e+02 1.25e-02 4.78e-14 4.78e-14 7.73 7.73 2.43e+01 2.43e+01 6.25e-03 1.45e-16 4.24e-16 8.36 6.81 3.89e+01 4.53e−01
126
Figure 6.1 : plot of solution and error using sbvp for Example 6 1. a) Solution, α =1 b) Error, α =1
2. a) Solution, α =2 b) Error, α = 2
3. a) Solution, α =3 b) Error, α = 3
127
TABLE 6.1 : global error for Example 6 -α=1-
1. sbvpcol, equidistant, m=4
h err mesh
err coll
p mesh
p coll
const mesh
const coll
1.00e-01 3.69e-08 3.71e-08 5.00e-02 2.31e-09 2.32e-09 4.00 4.00 3.69e-04 3.70e-04 2.50e-02 1.45e-10 1.45e-10 4.00 4.00 3.69e-04 3.73e-04 1.25e-02 9.05e-12 9.06e-12 4.00 4.00 3.69e-04 3.72e-04 6.25e-03 5.62e-13 5.62e-13 4.01 4.01 3.88e-04 3.90e-04
2. sbvpcol, Gauss, m=2
h err mesh
err coll
p mesh
p coll
const mesh
const coll
1.00e-01 5.63e-05 5.63e-05 5.00e-02 6.99e-06 6.99e-06 3.00 3.00 5.76e-02 5.76e-02 2.50e-02 8.71e-07 8.71e-07 3.00 3.00 5.67e-02 5.67e-02 1.25e-02 1.09e-07 1.09e-07 3.00 3.00 5.62e-02 5.62e-02 6.25e-03 1.36e-08 1.36e-08 3.00 3.00 5.59e-02 5.59e-02
3. sbvpcol, Gauss, m=3
h err mesh
err coll
p mesh
p coll
const mesh
const coll
1.00e-01 1.17e-07 2.69e-07 5.00e-02 6.90e-09 1.73e-08 4.08 3.96 1.41e-03 2.45e-03 2.50e-02 4.19e-10 1.09e-09 4.04 3.98 1.24e-03 2.61e-03 1.25e-02 2.58e-11 6.87e-11 4.02 3.99 1.15e-03 2.69e-03 6.25e-03 1.60e-12 4.31e-12 4.01 4.00 1.10e-03 2.81e-03
4. sbvpcol, Gauss, m=4
h err mesh
err coll
p mesh
p coll
const mesh
const coll
1.00e-01 1.45e-09 1.45e-09 5.00e-02 4.50e-11 4.50e-11 5.01 5.01 1.49e-04 1.49e-04 2.50e-02 1.40e-12 1.40e-12 5.00 5.00 1.46e-04 1.46e-04 1.25e-02 4.44e-14 4.44e-14 5.00 5.00 1.32e-04 1.32e-04 6.25e-03 1.15e-14 1.24e-14 1.94 1.84 2.22e-10 1.39e-10
128
TABLE 6.2 : global error for Example 6 -α=2-
1. sbvpcol, equidistant, m=4
h err mesh
err coll
p mesh
p coll
const mesh
const coll
1.00e-01 3.25e-08 3.27e-08 5.00e-02 2.03e-09 2.04e-09 4.00 4.00 3.26e-04 3.28e-04 2.50e-02 1.27e-10 1.27e-10 4.00 4.00 3.26e-04 3.28e-04 1.25e-02 7.94e-12 7.95e-12 3.99 4.00 3.25e-04 3.26e-04 6.25e-03 4.89e-13 4.90e-13 4.02 4.02 3.54e-04 3.56e-04
2. sbvpcol, Gauss, m=2
h err mesh
err coll
p mesh
p coll
const mesh
const coll
1.00e-01 2.08e-04 2.08e-04 5.00e-02 3.23e-05 3.23e-05 2.68 2.68 9.96e-02 9.96e-02 2.50e-02 5.06e-06 5.06e-06 2.68 2.68 9.91e-02 9.91e-02 1.25e-02 7.94e-07 7.94e-07 2.67 2.67 9.59e-02 9.59e-02 6.25e-03 1.25e-07 1.25e-07 2.67 2.67 9.57e-02 9.57e-02
3. sbvpcol, Gauss, m=3
h err mesh
err coll
p mesh
p coll
const mesh
const coll
1.00e-01 3.62e-07 3.62e-07 5.00e-02 1.73e-08 1.73e-08 4.39 4.39 8.89e-03 8.89e-03 2.50e-02 8.06e-10 1.09e-09 4.42 3.98 9.74e-03 2.61e-03 1.25e-02 5.25e-11 6.87e-11 3.94 3.99 1.65e-03 2.69e-03 6.25e-03 3.18e-12 4.31e-12 4.04 4.00 2.56e-03 2.81e-03
4. sbvpcol, Gauss, m=4
h err mesh
err coll
p mesh
p coll
const mesh
const coll
1.00e-01 6.40e-09 6.40e-09 5.00e-02 2.37e-10 2.37e-10 4.76 4.76 3.64e-04 3.64e-04 2.50e-02 9.17e-12 9.17e-12 4.69 4.69 3.01e-04 3.01e-04 1.25e-02 3.60e-13 3.60e-13 4.67 4.67 2.80e-04 2.80e-04 6.25e-03 1.29e-14 1.29e-14 4.80 4.80 4.99e-04 4.99e-04
129
TABLE 6.3 : global error for Example 6 -α=3-
1. sbvpcol, equidistant, m=4
h err mesh
err coll
p mesh
p coll
const mesh
const coll
1.00e-01 3.74e-08 3.74e-08 5.00e-02 1.82e-09 1.83e-09 4.36 4.36 8.61e-04 8.50e-04 2.50e-02 1.14e-10 1.14e-10 4.00 4.00 2.91e-04 2.93e-04 1.25e-02 7.11e-12 7.11e-12 4.00 4.00 2.92e-04 2.94e-04 6.25e-03 4.43e-13 4.43e-13 4.00 4.00 2.96e-04 2.96e-04
2. sbvpcol, Gauss, m=2
h err mesh
err coll
p mesh
p coll
const mesh
const coll
1.00e-01 3.81e-04 3.81e-04 5.00e-02 6.63e-05 6.63e-05 2.52 2.52 1.27e-01 1.27e-01 2.50e-02 1.16e-05 1.16e-05 2.51 2.51 1.24e-01 1.24e-01 1.25e-02 2.04e-06 2.04e-06 2.51 2.51 1.22e-01 1.22e-01 6.25e-03 3.58e-07 3.58e-07 2.51 2.51 1.20e-01 1.20e-01
3. sbvpcol, Gauss, m=3
h err mesh
err coll
p mesh
p coll
const mesh
const coll
1.00e-01 3.11e-07 3.11e-07 5.00e-02 1.48e-08 1.73e-08 4.39 4.17 7.63e-03 4.60e-03 2.50e-02 8.59e-10 1.09e-09 4.11 3.98 3.29e-03 2.61e-03 1.25e-02 5.27e-11 6.87e-11 4.02 3.99 2.37e-03 2.69e-03 6.25e-03 3.27e-12 4.31e-12 4.01 4.00 2.26e-03 2.81e-03
4. sbvpcol, Gauss, m=4
h err mesh
err coll
p mesh
p coll
const mesh
const coll
1.00e-01 1.20e-08 1.20e-08 5.00e-02 5.23e-10 5.23e-10 4.52 4.52 4.03e-04 4.03e-04 2.50e-02 2.29e-11 2.29e-11 4.51 4.51 3.90e-04 3.90e-04 1.25e-02 1.01e-12 1.01e-12 4.50 4.50 3.76e-04 3.76e-04 6.25e-03 4.35e-14 4.35e-14 4.53 4.53 4.31e-04 4.31e-04
130
TABLE 6.4 : convergence of first derivative for Example 6 -α=0-
1. sbvpcol, equidistant, m=4
h error mesh
error coll *
p mesh
p coll *
const mesh
const coll*
1.00e-01 8.46e-07 9.17e-08 5.00e-02 5.39e-08 5.83e-09 3.97 3.97 7.91e-03 8.66e-04 2.50e-02 3.40e-09 3.68e-10 3.99 3.99 8.31e-03 8.99e-04 1.25e-02 2.22e-10 2.31e-11 3.94 3.99 6.87e-03 9.13e-04 6.25e-03 2.23e-11 9.57e-12 3.32 1.27 4.58e-04 6.11e-09
2. sbvpcol, Gauss, m=2
h error mesh
error coll *
p mesh
p coll *
const mesh
const coll*
1.00e-01 4.38e-03 4.20e-05 5.00e-02 1.11e-03 5.35e-06 1.97 2.97 4.11e-01 3.95e-02 2.50e-02 2.81e-04 6.75e-07 1.99 2.99 4.29e-01 4.11e-02 1.25e-02 7.05e-05 8.47e-08 1.99 2.99 4.39e-01 4.21e-02 6.25e-03 1.77e-05 1.06e-08 2.00 3.00 4.45e-01 4.28e-02
3. sbvpcol, Gauss, m=3
h error mesh
error coll *
p mesh
p coll *
const mesh
const coll*
1.00e-01 4.37e-05 2.69e-07 5.00e-02 5.56e-06 1.73e-08 2.97 3.96 4.11e-02 2.47e-03 2.50e-02 7.01e-07 1.09e-09 2.99 3.98 4.28e-02 2.61e-03 1.25e-02 8.81e-08 6.87e-11 2.99 3.99 4.38e-02 1.70e-04 6.25e-03 1.10e-08 5.13e-12 3.00 3.742 4.44e-02 9.12e-04
4. sbvpcol, Gauss, m=4
h error mesh
error coll *
p mesh
p coll *
const mesh
const coll*
1.00e-01 3.11e-07 1.27e-09 5.00e-02 1.98e-08 4.07e-11 +3.97 +4.96 2.90e-03 1.17e-04 2.50e-02 1.25e-09 1.84e-12 +3.98 +4.47 3.02e-03 2.63e-05 1.25e-02 7.97e-11 3.29e-12 +3.97 –0.83 2.92e-03 8.39e-14 6.25e-03 1.48e-11 6.88e-12 +2.43 –1.06 3.40e-06 3.09e-14
* error computed only in collocations points (‘ mesh ‘ – mesh and coll. and mesh )
131
TABLE 6.5 : convergence of first derivative for Example 6 -α=1-
1. sbvpcol, equidistant, m=4
h error mesh
error coll *
p mesh
p coll *
const mesh
const coll*
1.00e-01 7.99e-07 2.11e-07 5.00e-02 5.10e-08 1.46e-08 3.97 3.86 7.45e-03 1.52e-03 2.50e-02 3.22e-09 9.98e-10 3.98 3.87 7.78e-03 1.58e-03 1.25e-02 2.18e-10 6.79e-11 3.88 3.88 5.38e-03 1.63e-03 6.25e-03 1.85e-11 8.37e-12 3.56 3.02 1.31e-03 3.78e-05
2. sbvpcol, Gauss, m=2
h error mesh
error coll *
p mesh
p coll *
const mesh
const coll*
1.00e-01 4.38e-03 1.17e-03 5.00e-02 1.11e-03 2.92e-04 1.98 2.00 4.13e-01 1.17e-01 2.50e-02 2.81e-04 7.30e-05 1.99 2.00 4.29e-01 1.18e-01 1.25e-02 7.05e-05 1.82e-05 1.99 2.00 4.39e-01 1.17e-01 6.25e-03 1.77e-05 4.55e-06 2.00 2.00 4.45e-01 1.17e-01
3. sbvpcol, Gauss, m=3
h error mesh
error coll *
p mesh
p coll *
const mesh
const coll*
1.00e-01 4.37e-05 2.80e-06 5.00e-02 2.80e-06 3.17e-07 2.97 3.14 4.11e-02 3.87e-03 2.50e-02 7.01e-07 3.76e-08 2.99 3.08 4.28e-02 3.18e-03 1.25e-02 8.81e-08 4.58e-09 2.99 3.04 4.38e-02 2.78e-03 6.25e-03 1.10e-08 5.64e-10 3.00 3.02 4.44e-02 2.59e-03
4. sbvpcol, Gauss, m=4
h error mesh
error coll *
p mesh
p coll *
const mesh
const coll*
1.00e-01 3.11e-07 9.51e-08 5.00e-02 1.98e-08 5.91e-09 3.97 4.01 2.90e-03 9.67e-04 2.50e-02 1.25e-09 3.68e-10 3.98 4.00 3.02e-03 9.58e-04 1.25e-02 7.97e-11 2.29e-11 3.97 4.01 2.92e-03 9.71e-04 6.25e-03 1.30e-11 6.67e-12 2.61 1.78 7.46e-06 5.57e-08
* error computed only in collocations points (‘ mesh ‘ – mesh and coll. and mesh )
132
TABLE 6.6 : convergence of first derivative for Example 6 -α=2-
1. sbvpcol, equidistant, m=4
h error mesh
error coll *
p mesh
p coll *
const mesh
const coll*
1.00e-01 2.02e-06 2.34e-07 5.00e-02 1.60e-07 1.84e-08 3.66 3.67 9.15e-03 1.09e-03 2.50e-02 1.26e-08 1.45e-09 3.67 3.67 9.60e-03 1.11e-03 1.25e-02 9.99e-10 1.15e-10 3.65 3.65 9.01e-03 1.00e-03 6.25e-03 5.57e-11 8.65e-12 4.16 3.74 8.42e-02 1.51e-03
2. sbvpcol, Gauss, m=2
h error mesh
error coll *
p mesh
p coll *
const mesh
const coll*
1.00e-01 1.28e-02 6.82e-03 5.00e-02 3.99e-03 2.15e-03 1.69 1.67 6.22e-01 3.18e-01 2.50e-02 1.24e-03 6.77e-04 1.68 1.66 6.15e-01 3.14e-01 1.25e-02 3.89e-04 2.14e-04 1.68 1.66 6.06e-01 3.12e-01 6.25e-03 1.22e-04 6.76e-05 1.68 1.66 6.00e-01 3.11e-01
3. sbvpcol, Gauss, m=3
h error mesh
error coll *
p mesh
p coll *
const mesh
const coll*
1.00e-01 4.75e-05 1.88e-05 5.00e-02 5.56e-06 1.71e-06 3.09 3.46 5.90e-02 5.38e-02 2.50e-02 7.01e-07 1.52e-07 2.99 3.50 4.28e-02 6.07e-02 1.25e-02 8.81e-08 2.01e-08 2.99 2.92 4.38e-02 7.12e-03 6.25e-03 1.10e-08 2.37e-09 3.00 3.08 4.45e-02 1.47e-02
4. sbvpcol, Gauss, m=4
h error mesh
error coll *
p mesh
p coll *
const mesh
const coll*
1.00e-01 1.29e-06 5.93e-07 5.00e-02 9.57e-08 4.39e-08 3.75 3.76 7.35e-03 3.38e-03 2.50e-02 7.41e-09 3.41e-09 3.69 3.69 6.08e-03 2.76e-03 1.25e-02 5.78e-10 2.67e-10 3.68 3.67 5.83e-03 2.62e-03 6.25e-03 1.95e-11 1.95e-11 3.78 3.77 9.04e-03 4.03e-03
* error computed only in collocations points (‘ mesh ‘ – mesh and coll. and mesh )
133
TABLE 6.7 : convergence of first derivative for Example 6 -α=3-
1. sbvpcol, equidistant, m=4
h error mesh
error coll *
p mesh
p coll *
const mesh
const coll*
1.00e-01 3.96e-06 4.53e-07 5.00e-02 3.46e-07 4.00e-08 3.52 3.50 1.30e-02 1.43e-03 2.50e-02 3.03e-08 3.55e-09 3.51 3.50 1.28e-02 1.42e-03 1.25e-02 2.66e-09 3.13e-10 3.51 3.50 1.28e-02 1.44e-03 6.25e-03 2.23e-10 2.74e-11 3.58 3.52 1.71e-02 1.54e-03
2. sbvpcol, Gauss, m =2
h error mesh
error coll *
p mesh
p coll *
const mesh
const coll*
1.00e-01 2.34e-02 1.28e-02 5.00e-02 8.09e-03 4.49e-03 1.53 1.51 7.97e-01 4.16e-01 2.50e-02 2.81e-03 1.58e-03 1.52 1.51 7.69e-01 4.09e-01 1.25e-02 9.86e-04 5.58e-04 1.52 1.50 7.58e-01 4.05e-01 6.25e-03 3.46e-04 1.97e-04 1.51 1.50 7.38e-01 4.02e-01
3. sbvpcol, Gauss, m =3
h error mesh
error coll *
p mesh
p coll *
const mesh
const coll*
1.00e-01 4.37e-05 1.59e-05 5.00e-02 5.56e-06 1.44e-06 2.97 3.47 4.12e-02 4.64e-02 2.50e-02 7.01e-07 1.64e-07 2.99 3.13 4.28e-02 1.69e-02 1.25e-02 8.81e-08 2.02e-08 2.99 3.02 4.38e-02 1.15e-02 6.25e-03 1.10e-08 2.51e-09 3.00 3.01 4.44e-02 1.08e-02
4. sbvpcol, Gauss, m =4
h error mesh
error coll *
p mesh
p coll *
const mesh
const coll*
1.00e-01 2.43e-06 1.12e-06 5.00e-02 2.11e-07 9.77e-08 3.53 3.52 8.20e-03 3.70e-03 2.50e-02 1.84e-08 8.58e-09 3.52 3.51 7.92e-03 3.60e-03 1.25e-02 1.62e-09 7.57e-10 3.51 3.50 7.67e-03 3.51e-03 6.25e-03 1.40e-10 6.56e-11 3.54 3.53 8.64e-03 3.91e-03
* error computed only in collocations points (‘ mesh ‘ – mesh and coll. and mesh )
134
TABLE 6.8 : convergence of second derivative for Example 6 -α=0-
1. sbvpcol, equidistant, m=4
h error mesh
error coll *
p mesh
p coll *
const mesh
const coll*
1.00e-01 8.72e-05 1.04e-05 5.00e-02 1.11e-05 1.33e-06 2.97 2.97 8.18e-02 9.64e-03 2.50e-02 1.40e-06 1.68e-07 2.99 2.99 8.55e-02 1.02e-02 1.25e-02 1.80e-07 2.31e-08 2.96 2.86 7.69e-02 6.34e-03 6.25e-03 4.02e-08 1.77e-08 2.16 0.38 2.38e-03 1.24e-07
2. sbvpcol, Gauss, m=2
h error mesh
error coll *
p mesh
p coll *
const mesh
const coll*
1.00e-01 2.64e-01 1.51e-01 5.00e-02 1.34e-01 7.69e-02 0.98 0.97 1.40e+00 2.52e+00 2.50e-02 6.75e-02 3.88e-02 0.99 0.99 1.48e+00 2.60e+00 1.25e-02 3.39e-02 1.95e-02 0.99 0.99 1.51e+00 2.64e+00 6.25e-03 1.70e-02 9.78e-03 1.00 1.00 1.53e+00 2.66e+00
3. sbvpcol, Gauss, m=3
h error mesh
error coll *
p mesh
p coll *
const mesh
const coll*
1.00e-01 5.25e-03 2.61e-03 5.00e-02 1.34e-03 6.66e-04 1.97 1.97 4.90e-01 2.44e-01 2.50e-02 3.37e-04 1.68e-04 1.99 1.99 5.20e-01 2.59e-01 1.25e-02 8.46e-05 4.23e-05 1.99 1.99 5.20e-01 2.59e-01 6.25e-03 2.12e-05 1.06e-05 2.00 2.00 5.41e-01 2.71e-01
4. sbvpcol, Gauss, m=4
h error mesh
error coll *
p mesh
p coll *
const mesh
const coll*
1.00e-01 6.22e-05 2.93e-05 5.00e-02 7.93e-06 3.74e-06 2.97 2.97 5.83e-02 2.73e-02 2.50e-02 1.00e-06 4.72e-07 2.98 2.98 6.07e-02 2.85e-02 1.25e-02 1.27e-07 6.02e-08 2.98 2.97 5.94e-02 2.73e-02 6.25e-03 3.29e-08 1.90e-08 1.95 1.67 6.48e-04 8.94e-05
* error computed only in collocations points (‘ mesh ‘ – mesh and coll. and mesh )
135
TABLE 6.9 : convergence of second derivative for Example 6 -α=1-
1. sbvpcol, equidistant, m=4
h error mesh
error coll *
p mesh
p coll *
const mesh
const coll*
1.00e-01 8.72e-05 1.04e-05 5.00e-02 1.11e-05 1.33e-06 2.97 2.97 8.19e-02 9.74e-03 2.50e-02 1.40e-06 1.68e-07 2.99 2.98 8.53e-02 1.02e-02 1.25e-02 1.85e-07 2.42e-08 2.93 2.80 6.84e-02 5.09e-03 6.25e-03 3.38e-08 2.10e-08 2.45 0.20 8.44e-03 5.89e-08
2. sbvpcol, Gauss, m=2
h error mesh
error coll *
p mesh
p coll *
const mesh
const coll*
1.00e-01 2.64e-01 1.51e-01 5.00e-02 1.34e-01 7.69e-02 0.98 0.97 2.52e+00 1.40e+00 2.50e-02 6.75e-02 3.88e-02 0.99 0.99 2.60e+00 1.48e+00 1.25e-02 3.39e-02 1.95e-02 0.99 0.99 2.64e+00 1.51e+00 6.25e-03 1.70e-02 9.78e-03 1.00 1.00 2.66e+00 1.53e+00
3. sbvpcol, Gauss, m=3
h error mesh
error coll *
p mesh
p coll *
const mesh
const coll*
1.00e-01 5.25e-03 2.61e-03 5.00e-02 1.34e-03 6.66e-04 1.97 1.97 4.90e-01 2.44e-01 2.50e-02 3.37e-04 1.68e-04 1.99 1.99 5.20e-01 2.59e-01 1.25e-02 8.46e-05 4.23e-05 1.99 1.99 5.20e-01 2.59e-01 6.25e-03 2.12e-05 1.06e-05 2.00 2.00 5.41e-01 2.71e-01
4. sbvpcol, Gauss, m=4
h error mesh
error coll *
p mesh
p coll *
const mesh
const coll*
1.00e-01 6.22e-05 2.93e-05 5.00e-02 7.93e-06 3.74e-06 2.97 2.97 5.83e-02 2.73e-02 2.50e-02 1.00e-06 4.72e-07 2.98 2.98 6.07e-02 2.85e-02 1.25e-02 1.27e-07 6.02e-08 2.98 2.97 5.94e-02 2.73e-02 6.25e-03 2.78e-08 1.59e-08 2.19 1.92 1.88e-03 2.72e-04
* error computed only in collocations points (‘ mesh ‘ – mesh and coll. and mesh )
136
TABLE 6.10 : convergence of second derivative for Example 6 -α=2-
1. sbvpcol, equidistant, m=4
h error mesh
error coll *
p mesh
p coll *
const mesh
const coll*
1.00e-01 1.48e-04 4.22e-05 5.00e-02 2.33e-05 6.76e-06 2.67 2.64 6.85e-02 1.84e-02 2.50e-02 3.64e-06 1.07e-06 2.68 2.66 7.15e-02 1.96e-02 1.25e-02 5.75e-07 1.71e-07 2.66 2.65 6.72e-02 1.85e-02 6.25e-03 6.58e-08 2.15e-08 3.12 2.99 5.17e-01 8.43e-02
2. sbvpcol, Gauss, m=2
h error mesh
error coll *
p mesh
p coll *
const mesh
const coll*
1.00e-01 3.06e-01 2.63e-02 5.00e-02 1.85e-01 1.64e-02 0.72 0.68 1.63e+00 1.27e+00 2.50e-02 1.13e-01 1.02e-02 0.72 0.68 1.58e+00 1.27e+00 1.25e-02 6.89e-02 6.36e-02 0.71 0.68 1.54e+00 1.26e+00 6.25e-03 4.24e-02 3.97e-02 0.70 0.68 1.48e+00 1.25e+00
3. sbvpcol, Gauss, m=3
h error mesh
error coll *
p mesh
p coll *
const mesh
const coll*
1.00e-01 5.25e-03 2.61e-03 5.00e-02 1.34e-03 6.66e-04 1.97 1.97 4.90e-01 2.44e-01 2.50e-02 3.37e-04 1.68e-04 1.99 1.99 5.20e-01 2.59e-01 1.25e-02 8.46e-05 4.23e-05 1.99 1.99 5.20e-01 2.59e-01 6.25e-03 2.12e-05 1.06e-05 2.00 2.00 5.41e-01 2.71e-01
4. sbvpcol, Gauss, m=4
h error mesh
error coll *
p mesh
p coll *
const mesh
const coll*
1.00e-01 1.21e-04 8.13e-05 5.00e-02 1.80e-05 1.21e-05 2.76 2.75 6.94e-02 4.57e-02 2.50e-02 2.77e-06 1.87e-06 2.70 2.69 5.84e-02 3.83e-02 1.25e-02 4.29e-07 2.92e-07 2.69 2.68 5.62e-02 3.69e-02 6.25e-03 6.20e-08 4.24e-08 2.79 2.78 8.77e-02 5.76e-02
* error computed only in collocations points (‘ mesh ‘ – mesh and coll. and mesh )
137
TABLE 6.11 : convergence of second derivative for Example 6 -α=3-
1. sbvpcol, equidistant, m=4
h error mesh
error coll *
p mesh
p coll *
const mesh
const coll*
1.00e-01 2.86e-04 8.44e-05 5.00e-02 4.95e-05 1.49e-05 2.53 2.50 9.67e-02 2.70e-02 2.50e-02 8.63e-06 2.63e-06 2.52 2.50 9.43e-02 2.68e-02 1.25e-02 1.51e-06 4.63e-07 2.52 2.50 9.35e-02 2.71e-02 6.25e-03 2.51e-07 7.79e-08 2.59 2.57 1.26e-01 3.59e-02
2. sbvpcol, Gauss, m=2
h error mesh
error coll *
p mesh
p coll *
const mesh
const coll*
1.00e-01 5.24e-01 4.81e-01 5.00e-02 3.52e-01 3.30e-01 0.58 0.54 1.97e+00 1.68e+00 2.50e-02 2.39e-01 2.29e-01 0.55 0.53 1.85e+00 1.62e+00 1.25e-02 1.65e-01 1.59e-01 0.54 0.52 1.75e+00 1.56e+00 6.25e-03 1.14e-01 1.12e-01 0.53 0.51 1.67e+00 1.52e+00
3. sbvpcol, Gauss, m=3
h error mesh
error coll *
p mesh
p coll *
const mesh
const coll*
1.00e-01 5.25e-03 2.61e-03 5.00e-02 1.34e-03 6.66e-04 1.97 1.97 4.90e-01 2.44e-01 2.50e-02 3.37e-04 1.68e-04 1.99 1.99 5.20e-01 2.59e-01 1.25e-02 8.46e-05 4.23e-05 1.99 1.99 5.20e-01 2.59e-01 6.25e-03 2.12e-05 1.06e-05 2.00 2.00 5.41e-01 2.71e-01
4. sbvpcol, Gauss, m=4
h error mesh
error coll *
p mesh
p coll *
const mesh
const coll*
1.00e-01 2.26e-04 1.54e-04 5.00e-02 3.88e-05 2.66e-05 2.54 2.53 7.85e-02 5.21e-02 2.50e-02 6.74e-06 4.64e-06 2.53 2.52 7.52e-02 5.03e-02 1.25e-02 1.18e-06 8.16e-07 2.51 2.51 7.21e-02 4.86e-02 6.25e-03 2.02e-07 1.40e-07 2.54 2.54 8.18e-02 5.52e-02
* error computed only in collocations points (‘ mesh ‘ – mesh and coll. and mesh )
138
TABLE 6.12 : matrix condition estimates for Example 6 -α=0-
1. sbvpcol, equidistant, m=4
h condestDF cond ord. condest
ord. cond
const condest
const cond
1.00e-01 7.95e+01 3.17e+01 5.00e-02 1.34e+02 6.15e+01 -0.75 -0.95 1.42e+01 3.53e+00 2.50e-02 2.42e+02 1.22e+02 -0.85 -0.98 1.03e+01 3.23e+00 1.25e-02 4.57e+02 2.42e+02 -0.92 -0.99 8.09e+00 3.11e+00 6.25e-03 8.89e+02 4.83e+02 -0.96 -1.00 6.85e+00 3.06e+00
2. sbvpcol, Gauss, m=2
h condestDF cond ord. condest
ord. cond
const condest
const cond
1.00e-01 6.45e+01 3.23e+01 5.00e-02 1.19e+02 6.30e+01 -0.88 -0.97 8.47e+00 3.48e+00 2.50e-02 2.27e+02 1.25e+02 -0.94 -0.99 7.27e+00 3.28e+00 1.25e-02 4.43e+02 2.49e+02 -0.96 -0.99 6.47e+00 3.19e+00 6.25e-03 8.75e+02 4.97e+02 -0.98 -1.00 6.00e+00 3.15e+00
3. sbvpcol, Gauss, m=3
h condestDF cond ord. condest
ord. cond
const condest
const cond
1.00e-01 7.19e+01 3.06e+01 5.00e-02 1.26e+02 5.96e+01 -0.81 -0.96 1.94e+01 3.34e+00 2.50e-02 2.34e+02 1.18e+02 -0.89 -0.99 8.71e+00 3.11e+00 1.25e-02 4.50e+02 2.35e+02 -0.94 -1.00 7.25e+00 3.00e+00 6.25e-03 8.82e+02 4.70e+02 -0.97 -1.00 6.41e+00 2.97e+00
4. sbvpcol, Gauss, m=4
h condestDF cond ord. condest
ord. cond
const condest
const cond
1.00e-01 7.95e+01 2.98e+01 5.00e-02 1.34e+02 5.78e+01 -0.74 -0.96 1.42e+01 3.25e+00 2.50e-02 2.42e+02 1.14e+02 -0.85 -0.98 10.3e+00 3.02e+00 1.25e-02 4.57e+02 2.28e+02 -0.92 -0.99 8.08e+00 2.92e+00 6.25e-03 8.89e+02 4.55e+02 -0.96 -1.00 6.85e+00 2.88e+00
· condestDF:=condest(DF,1), cond:=cond(DF,2)
139
TABLE 6.13 : matrix condition estimates for Example 6 -α=1-
1. sbvpcol, equidistant, m=4
h condestDF cond ord. condest
ord. cond
const condest
const cond
1.00e-01 2.28e+02 4.90e+01 5.00e-02 3.62e+02 9.28e+01 -0.67 -0.91 4.91e+01 6.07e+00 2.50e-02 6.31e+02 1.82e+02 -0.80 -0.99 3.30e+01 4.80e+00 1.25e-02 1.17e+03 3.63e+02 -0.89 -1.00 2.38e+01 4.59e+00 6.25e-03 2.24e+03 7.26e+02 -0.94 -1.00 1.89e+01 4.54e+00
2. sbvpcol, Gauss, m=2
h condestDF cond ord. condest
ord. cond
const condest
const cond
1.00e-01 1.17e+02 3.43e+01 5.00e-02 2.07e+02 6.76e+01 -0.82 -0.98 1.76e+01 3.59e+00 2.50e-02 3.87e+02 1.35e+02 -0.90 -1.00 1.39e+01 3.40e+00 1.25e-02 7.47e+02 2.70e+02 -0.95 -1.00 1.17e+01 3.36e+00 6.25e-03 1.47e+03 5.41e+02 -0.97 -1.00 1.05e+01 3.36e+00
3. sbvpcol, Gauss, m=3
h condestDF cond ord. condest
ord. cond
const condest
const cond
1.00e-01 2.25e+02 5.77e+01 5.00e-02 3.75e+02 1.10e+02 -0.74 -0.93 4.12e+01 6.80e+00 2.50e-02 6.75e+02 2.18e+02 -0.85 -0.99 2.96e+01 5.61e+00 1.25e-02 1.28e+03 4.36e+02 -0.92 -1.00 2.29e+01 5.47e+00 6.25e-03 2.48e+03 8.73e+02 -0.96 -1.00 1.92e+01 5.45e+00
4. sbvpcol, Gauss, m=4
h condestDF cond ord. condest
ord. cond
const condest
const cond
1.00e-01 3.91e+02 1.40e+02 5.00e-02 6.21e+02 1.69e+02 -0.67 -0.27 8.41e+01 7.44e+01 2.50e-02 1.08e+03 3.34e+02 -0.80 -0.99 5.66e+01 8.78e+00 1.25e-02 2.00e+03 6.67e+02 -0.89 -1.00 4.08e+01 8.43e+00 6.25e-03 3.84e+03 1.33e+03 -0.94 -1.00 3.24e+01 8.34e+00
· condestDF:=condest(DF,1), cond:=cond(DF,2)
140
TABLE 6.14 : matrix condition estimates for Example 6 -α=2-
1. sbvpcol, equidistant, m=4
h condestDF cond ord. condest
ord. cond
const condest
const cond
1.00e-01 4.41e+03 2.04e+03 5.00e-02 1.66e+04 6.22e+03 -1.91 -1.61 5.45e+01 4.99e+01 2.50e-02 5.61e+04 2.45e+04 -1.76 -1.98 8.49e+01 1.66e+01 1.25e-02 2.04e+05 9.77e+04 -1.86 -2.00 5.79e+01 1.55e+01 6.25e-03 7.76e+05 3.91e+05 -1.93 -2.00 4.39e+01 1.53e+01
2. sbvpcol, Gauss, m=2
h condestDF cond ord. condest
ord. cond
const condest
const cond
1.00e-01 5.97e+03 2.98e+03 5.00e-02 1.79e+04 7.80e+03 -1.59 -1.39 1.55e+02 1.22e+02 2.50e-02 4.97e+04 2.01e+04 -1.47 -1.37 2.19e+02 1.29e+02 1.25e-02 1.32e+05 7.19e+04 -1.41 -1.84 2.72e+02 2.30e+01 6.25e-03 3.44e+05 2.88e+05 -1.38 -2.00 3.08e+02 1.12e+01
3. sbvpcol, Gauss, m=3
h condestDF cond ord. condest
ord. cond
const condest
const cond
1.00e-01 2.69e+04 1.22e+04 5.00e-02 7.75e+04 3.51e+04 -1.53 -1.52 8.02e+02 3.66e+02 2.50e-02 2.41e+05 9.39e+04 -1.64 -1.42 5.75e+02 5.01e+02 1.25e-02 6.81e+05 2.45e+05 -1.50 -1.39 9.59e+02 5.66e+02 6.25e-03 1.83e+06 9.40e+05 -1.43 -1.94 1.31e+03 5.03e+01
4. sbvpcol, Gauss, m=4
h condestDF cond ord. condest
ord. cond
const condest
const cond
1.00e-01 1.02e+05 3.77e+04 5.00e-02 2.50e+05 1.08e+05 -1.29 -1.52 5.17e+03 1.15e+03 2.50e-02 7.63e+05 3.08e+05 -1.61 -1.51 2.02e+03 1.15e+03 1.25e-02 2.33e+06 8.23e+05 -1.61 -1.42 2.01e+03 1.65e+03 6.25e-03 6.52e+06 2.39e+06 -1.49 -1.54 3.48e+03 9.82e+02
· condestDF:=condest(DF,1), cond:=cond(DF,2)
141
TABLE 6.15 : matrix condition estimates for Example 6 -α=3-
1. sbvpcol, equidistant, m=4
h condestDF cond ord. condest
ord. cond
const condest
const cond
1.00e-01 4.87e+05 1.82e+05 5.00e-02 2.72e+06 9.77e+05 -2.48 -2.42 1.61e+03 6.90e+02 2.50e-02 1.53e+07 5.36e+06 -2.49 -2.46 1.56e+03 6.25e+02 1.25e-02 8.63e+07 3.27e+07 -2.50 -2.61 1.52e+03 3.55e+02 6.25e-03 4.90e+08 2.61e+08 -2.50 -3.00 1.49e+03 6.40e+01
2. sbvpcol, Gauss, m=2
h condestDF cond ord. condest
ord. cond
const condest
const cond
1.00e-01 6.70e+05 2.89e+05 5.00e-02 3.95e+06 1.67e+06 -2.56 -2.53 1.85e+03 8.49e+02 2.50e-02 2.28e+07 9.61e+06 -2.53 -2.52 2.01e+03 8.73e+02 1.25e-02 1.31e+08 5.49e+07 -2.52 -2.52 2.09e+03 8.95e+02 6.25e-03 7.47e+08 3.13e+08 -2.51 -2.51 2.16e+03 9.13e+02
3. sbvpcol, Gauss, m=3
h condestDF cond ord. condest
ord. cond
const condest
const cond
1.00e-01 6.92e+06 2.71e+06 5.00e-02 4.24e+07 1.60e+07 -2.62 -2.56 1.68e+04 7.39e+03 2.50e-02 2.49e+08 9.30e+07 -2.55 -2.54 2.03e+04 8.00e+03 1.25e-02 1.44e+09 5.35e+08 -2.53 -2.52 2.20e+04 8.40e+03 6.25e-03 8.24e+09 3.06e+09 -2.52 -2.52 2.31e+04 8.71e+03
4. sbvpcol, Gauss, m=4
h condestDF cond ord. condest
ord. cond
const condest
const cond
1.00e-01 4.10e+07 1.55e+07 5.00e-02 2.63e+08 9.29e+07 -2.68 -2.58 8.47e+04 4.05e+04 2.50e-02 1.57e+09 5.45e+08 -2.57 -2.55 1.18e+05 4.45e+04 1.25e-02 9.12e+09 3.15e+09 -2.54 -2.53 1.34e+05 4.78e+04 6.25e-03 5.25e+10 1.81e+10 -2.52 -2.52 1.43e+05 5.03e+04
· condestDF:=condest(DF,1), cond:=cond(DF,2)
142
TABLE 6.16 : global error for Example 6 -α=0-
1. a) sbvpcol, equidistant, m=2, Error
h err mesh
err coll
p mesh
p coll
const mesh
const coll
1.00e-01 1.25e-03 1.25e-03 5.00e-02 3.12e-04 3.12e-04 2.00 2.00 1.25e-01 1.25e-01 2.50e-02 7.81e-05 7.81e-05 2.00 2.00 1.25e-01 1.25e-01 1.25e-02 1.95e-05 1.95e-05 2.00 2.00 1.25e-01 1.25e-01 6.25e-03 4.88e-06 4.88e-06 2.00 2.00 1.25e-01 1.25e-01
2. a) sbvpcol, equidistant, m=4, Error
h err mesh
err coll
p mesh
p coll
const mesh
const coll
5.000e-01 6.06e-05 6.06e-05 2.500e-01 3.73e-06 3.73e-06 4.02 4.02 9.84e-04 9.84e-04 1.250e+01 2.32e-07 2.32e-07 4.01 4.01 9.62e-04 9.62e-04 6.250e-02 1.45e-08 1.45e-08 4.00 4.00 9.53e-04 9.53e-04 3.125e-02 9.06e-10 9.06e-10 4.00 4.00 9.51e-04 9.51e-04
3. a) sbvpcol, equidistant, m=6, Error
h err mesh
err coll
p mesh
p coll
const mesh
const coll
5.000e-01 5.29e-08 5.29e-08 2.500e-01 8.15e-10 8.15e-10 6.02 6.02 3.43e-06 3.43e-06 1.250e+01 1.27e-11 1.27e-11 6.01 6.01 3.36e-06 3.36e-06 6.250e-02 1.97e-13 1.97e-13 6.01 6.01 3.38e-06 3.38e-06 3.125e-02 4.44e-15 4.44e-15 5.47 5.47 7.66e-07 7.66e-07
143
TABLE 6.17 : global error for Example 6 -α=1-
1. a) sbvpcol, equidistant, m=2, Error
h err mesh
err coll
p mesh
p coll
const mesh
const coll
1.00e-01 4.86e-04 4.86e-04 5.00e-02 1.22e-04 1.22e-04 1.99 1.99 4.80e-02 4.79e-02 2.50e-02 3.05e-05 3.05e-05 2.00 2.00 4.86e-02 4.87e-02 1.25e-02 7.62e-06 7.63e-06 2.00 2.00 4.87e-02 4.89e-02 6.25e-03 1.91e-06 1.91e-06 2.00 2.00 4.88e-02 4.89e-02
2. a) sbvpcol, equidistant, m=4, Error
h err mesh
err coll
p mesh
p coll
const mesh
const coll
5.000e-01 2.52e-05 2.52e-05 2.500e-01 1.40e-06 1.40e-06 4.17 4.17 4.55e-04 4.55e-04 1.250e-01 9.00e-08 9.00e-08 3.95 3.95 3.35e-04 3.35e-04 6.250e-02 5.65e-09 5.65e-09 3.99 3.99 3.64e-04 3.64e-04 3.125e-02 3.53e-10 3.53e-10 4.00 4.00 3.69e-04 3.69e-04
3. a) sbvpcol, equidistant, m=6, Error
h err mesh
err coll
p mesh
p coll
const mesh
const coll
5.000e-01 2.19e-08 2.19e-08 2.500e-01 3.07e-10 3.07e-10 6.16 6.16 1.56e-06 1.56e-06 1.250e-01 4.93e-12 4.93e-12 5.96 5.96 1.19e-06 1.19e-06 6.250e-02 7.47e-14 7.47e-14 6.04 6.04 1.42e-06 1.42e-06 3.125e-02 3.11e-15 3.11e-15 4.59 4.59 2.49e-08 2.49e-08
144
TABLE 6.18 : global error for Example 6 -α=2-
1. a) sbvpcol, equidistant, m=2, Error
h err mesh
err coll
p mesh
p coll
const mesh
const coll
1.00e-01 4.29e-04 4.28e-04 5.00e-02 1.07e-04 1.07e-04 2.00 2.00 4.30e-02 4.26e-02 2.50e-02 2.67e-05 2.68e-05 2.00 2.00 4.31e-02 4.31e-02 1.25e-02 6.68e-06 6.69e-06 2.00 2.00 4.29e-02 4.30e-02 6.25e-03 1.67e-06 1.67e-06 2.00 2.00 4.28e-02 4.29e-02
2. a) sbvpcol, equidistant, m=4, Error
h err mesh
err coll
p mesh
p coll
const mesh
const coll
1.00e-01 3.25e-08 3.27e-08 5.00e-02 2.03e-09 2.04e-09 4.00 4.00 3.26e-04 3.28e-04 2.50e-02 1.27e-10 1.27e-10 4.00 4.00 3.26e-04 3.28e-04 1.25e-02 7.94e-12 7.95e-12 3.99 4.00 3.25e-04 3.26e-04 6.25e-03 4.89e-13 4.90e-13 4.02 4.02 3.54e-04 3.56e-04
3. a) sbvpcol, equidistant, m=6, Error
h err mesh
err coll
p mesh
p coll
const mesh
const coll
5.000e-01 2.71e-08 2.71e-08 2.500e-01 2.90e-10 2.90e-10 6.55 6.55 2.54e-06 2.54e-06 1.250e-01 4.34e-12 4.36e-12 6.06 6.06 1.29e-06 1.28e-06 6.250e-02 6.82e-14 6.82e-14 5.99 6.00 1.12e-06 1.14e-06 3.125e-02 7.55e-15 7.55e-15 3.17 3.17 4.53e-10 4.53e-10
145
TABLE 6.19 : global error for Example 6 -α=3-
1. a) sbvpcol, equidistant, m=2, Error
h err mesh
err coll
p mesh
p coll
const mesh
const coll
1.00e-01 5.55e-04 5.55e-04 5.00e-02 9.60e-05 9.62e-05 2.53 2.53 1.88e-01 1.87e-02 2.50e-02 2.40e-05 2.40e-05 2.00 2.00 3.88e-02 3.88e-02 1.25e-02 5.99e-06 5.99e-06 2.00 2.00 3.84e-02 3.86e-02 6.25e-03 1.50e-06 1.50e-06 2.00 2.00 3.84e-02 3.84e-02
2. a) sbvpcol, equidistant, m=4, Error
h err mesh
err coll
p mesh
p coll
const mesh
const coll
1.00e-01 3.74e-08 3.74e-08 5.00e-02 1.82e-09 1.83e-09 4.36 4.36 8.61e-04 8.50e-04 2.50e-02 1.14e-10 1.14e-10 4.00 4.00 2.91e-04 2.93e-04 1.25e-02 7.11e-12 7.11e-12 4.00 4.00 2.92e-04 2.94e-04 6.25e-03 4.43e-13 4.43e-13 4.00 4.00 2.96e-04 2.96e-04
3. a) sbvpcol, equidistant, m=6, Error
h err mesh
err coll
p mesh
p coll
const mesh
const coll
5.000e-01 3.96e-08 3.96e-08 2.500e-01 4.97e-10 4.97e-10 6.32 6.32 3.16e-06 3.16e-06 1.250e+01 5.19e-12 5.19e-12 6.58 6.58 4.57e-06 4.57e-06 6.250e-02 6.20e-14 6.28e-14 6.39 6.37 3.04e-06 2.93e-06 3.125e-02 3.11e-14 3.11e-14 1.00 1.01 9.82e-13 1.05e-12
146
TABLE 6.20 : Error of error estimate based on h-h/2 for Example 6 -α=0-
1. b) sbvpcol, equidistant, m=2, Error of error
h err mesh
err coll
p` mesh
p` coll
const mesh
const coll
1.00e-01 3.13e-07 3.13e-07 5.00e-02 1.96e-08 1.96e-08 4.00 4.00 3.12e-03 3.12e-03 2.50e-02 1.23e-09 1.23e-09 4.00 4.00 3.13e-03 3.13e-03 1.25e-02 7.66e-11 7.66e-11 4.00 4.00 3.14e-03 3.14e-03 6.25e-03 4.77e-12 4.77e-12 4.00 4.00 3.20e-03 3.20e-03
2. b) sbvpcol, equidistant, m=4, Error of error
h err mesh
err coll
p` mesh
p` coll
const mesh
const coll
5.000e-01 6.11e-08 6.11e-08 2.500e-01 9.59e-10 9.59e-10 5.99 5.99 3.90e-06 3.90e-06 1.250e+01 1.50e-11 1.50e-11 6.00 6.00 3.92e-06 3.92e-06 6.250e-02 2.37e-13 2.37e-13 6.00 6.00 3.82e-06 3.82e-06 3.125e-02 4.86e-15 4.86e-15 5.61 5.61 1.34e-06 1.34e-06
3. b) sbvpcol, equidistant, m=6, Error of error
h err mesh
err coll
p` mesh
p` coll
const mesh
const coll
5.000e-01 1.17e-11 1.17e-11 2.500e-01 4.58e-14 4.58e-14 +8.00 +8.00 2.99e-09 2.99e-09 1.250e+01 1.13e-15 1.13e-15 +5.34 +5.34 7.56e-11 7.56e-11 6.250e-02 1.38e-15 1.38e-15 −0.29 −0.29 6.13e-16 6.13e-16 3.125e-02 3.91e-15 3.69e-15 −1.50 −1.42 2.16e-17 2.76e-18
147
TABLE 6.21 : Error of error estimate based on h-h/2 for Example 6 -α=1-
1. b) sbvpcol, equidistant, m=2, Error of error
h err mesh
err coll
p` mesh
p` coll
const mesh
const coll
1.00e-01 5.70e-07 1.90e-07 5.00e-02 3.17e-08 9.41e-08 4.17 1.01 8.44e-03 1.94e-06 2.50e-02 1.75e-09 1.59e-08 4.18 2.56 8.62e-03 2.04e-04 1.25e-02 9.49e-11 2.27e-09 4.20 2.81 9.52e-03 5.10e-04 6.25e-03 5.09e-12 2.96e-10 4.22 2.94 1.02e-02 8.84e-04
2. b) sbvpcol, equidistant, m=4, Error of error
h err mesh
err coll
p` mesh
p` coll
const mesh
const coll
5.000e-01 8.80e-07 8.80e-07 2.500e-01 1.85e-09 1.34e-09 8.89 9.36 4.19e-04 2.00e+03 1.250e+01 2.48e-11 1.05e-11 6.22 7.00 1.02e-05 2.19e-05 6.250e-02 3.77e-13 9.64e-13 6.04 3.45 7.13e-06 1.37e-08 3.125e-02 9.30e-15 3.47e-14 5.34 4.80 1.02e+06 5.77e-07
3. b) sbvpcol, equidistant, m=6, Error of error
h err mesh
err coll
p` mesh
p` coll
const mesh
const coll
5.000e-01 1.83e-10 1.83e-10 2.500e-01 8.11e-14 4.03e-14 +11.1 12.1 4.11e-07 2.54e+02 1.250e+01 2.74e-15 3.12e-15 +4.89 3.69 7.10e-11 6.76e−12 6.250e-02 2.88e-15 2.88e-15 −0.07 0.12 2.37e-15 4.75e−15 3.125e-02 2.64e-15 2.76e-15 +0.12 0.06 4.04e-15 3.72e−15
148
TABLE 6.22 : Error of error estimate based on h-h/2 for Example 6 -α=2-
1. b) sbvpcol, equidistant, m=2, Error of error
h err mesh
err coll
p` mesh
p` coll
const mesh
const coll
1.00e-01 1.59e-07 1.59e-07 5.00e-02 4.79e-08 1.69e-07 +1.73 −0.09 8.64e-06 1.29e-07 2.50e-02 3.38e-09 2.03e-08 +3.83 +3.06 4.55e-03 1.62e-03 1.25e-02 2.11e-10 2.49e-09 +4.00 +3.02 8.63e-03 1.42e-03 6.25e-03 1.31e-11 3.09e-10 +4.01 +3.01 9.01e-03 1.35e-03
2. b) sbvpcol, equidistant, m=4, Error of error
h err mesh
err coll
p` mesh
p` coll
const mesh
const coll
5.000e-01 1.40e-06 1.40e-06 2.500e-01 1.72e-09 1.28e-09 9.67 10.1 1.14e-03 1.76e+04 1.250e+01 2.17e-12 3.42e-11 9.63 5.22 1.08e-03 1.78e−06 6.250e-02 1.45e-13 1.31e-12 3.90 4.71 7.22e-09 6.14e−07 3.125e-02 4.51e-16 4.32e-14 8.33 4.92 1.54e-03 1.10e−06
3. b) sbvpcol, equidistant, m=6, Error of error
h err mesh
err coll
p` mesh
p` coll
const mesh
const coll
5.000e-01 1.36e-10 1.36e-10 2.500e-01 2.27e-12 2.27e-12 +5.90 +5.90 8.13e-09 8.13e−09 1.250e+01 1.04e-15 9.91e-16 +11.1 +11.2 1.09e-05 7.57e+02 6.250e-02 1.06e-15 6.31e-16 −0.03 +0.65 9.72e-16 1.09e−14 3.125e-02 4.18e-15 4.18e-15 −1.98 −2.73 4.40e-18 4.05e−21
149
TABLE 6.23 : Error of error estimate based on h-h/2 for Example 6 -α=3-
1. b) sbvpcol, equidistant, m=2, Error of error
h err mesh
err coll
p` mesh
p` coll
const mesh
const coll
1.00e-01 5.76e-05 5.76e-05 5.00e-02 9.08e-06 8.96e-06 2.66 2.68 2.66e−02 2.78e−02 2.50e-02 4.77e-09 2.15e-08 10.9 8.70 1.36e+09 1.87e+06 1.25e-02 3.00e-10 2.54e-09 3.99 3.08 1.18e−02 1.88e−03 6.25e-03 1.86e-11 3.10e-10 4.01 3.03 1.29e−02 1.51e−03
2. b) sbvpcol, equidistant, m=4, Error of error
h err mesh
err coll
p` mesh
p` coll
const mesh
const coll
5.000e-01 2.79e-08 2.79e-08 2.500e-01 5.86e-08 5.86e-08 −1.07 −1.07 1.33e−08 1.33e−08 1.250e+01 2.00e-09 2.00e-09 +4.88 +4.88 5.05e−05 5.05e−05 6.250e-02 8.86e-11 8.86e-11 +4.49 +4.49 2.28e−05 2.28e−05 3.125e-02 1.58e-15 4.07e-14 +15.8 +11.1 8.85e+08 1.08e+01
3. b) sbvpcol, equidistant, m=6, Error of error
h err mesh
err coll
p` mesh
p` coll
const mesh
const coll
5.000e-01 1.24e-10 1.24e-10 2.500e-01 2.62e-12 2.62e-12 +5.56 +5.56 5.87e-09 5.87e-09 1.250e+01 7.24e-14 7.24e-14 +5.18 +5.18 3.43e-09 3.43e-09 6.250e-02 2.68e-15 1.90e-15 +4.76 +5.25 1.44e-09 1.89e-05 3.125e-02 6.72e-15 6.72e-15 −1.33 −1.82 6.71e-17 6.40e-19
150
Figure 7.1 : plot of solution and error using sbvp for Example 7
151
TABLE 7.1 : global error for Example 7
1. sbvpcol, equidistant, m=4
h
err mesh
err coll
p mesh
p coll
const mesh
const coll
1.00e-01 2.06e-17 2.06e-17 5.00e-02 4.26e-18 4.26e-18 2.27 2.27 3.85e-15 3.85e-15 2.50e-02 1.65e-19 1.65e-19 4.69 4.69 5.34e-12 5.34e-12 1.25e-02 6.22e-21 6.22e-21 4.73 4.73 6.31e-12 6.31e-12 6.25e-03 3.74e-22 3.84e-22 4.06 4.02 3.27e-13 2.75e-13
2. sbvpcol, Gauss, m=2
h
err mesh
err coll
p mesh
p coll
const mesh
const coll
1.00e-01 7.84e-17 7.84e-17 5.00e-02 1.64e-17 1.64e-17 2.26 2.26 1.43e-14 1.43e-14 2.50e-02 1.02e-18 1.13e-18 4.00 3.85 2.64e-12 1.69e-12 1.25e-02 8.19e-20 1.88e-19 3.64 2.59 6.88e-13 1.56e-14 6.25e-03 5.10e-21 2.24e-20 4.00 3.07 3.44e-12 1.30e-13
3. sbvpcol, Gauss, m =3
h
err mesh
err coll
p mesh
p coll
const mesh
const coll
1.00e-01 1.49e-17 1.49e-17 5.00e-02 2.74e-18 2.74e-18 2.44 2.44 4.09e-15 4.09e-15 2.50e-02 1.08e-19 1.32e-19 4.66 4.37 3.18e-12 1.33e-12 1.25e-02 2.70e-21 1.18e-20 5.32 3.49 3.65e-11 5.10e-14 6.25e-03 6.72e-23 8.06e-22 5.33 3.87 3.76e-11 2.80e-13
4. sbvpcol, Gauss, m=4
h
err mesh
err coll
p mesh
p coll
const mesh
const coll
1.00e-01 1.73e-17 1.73e-17 5.00e-02 8.09e-19 8.09e-19 4.42 4.42 4.53e-13 4.53e-13 2.50e-02 1.64e-20 3.32e-20 5.62 4.61 1.68e-11 7.94e-13 1.25e-02 2.78e-22 9.47e-22 5.88 5.13 4.35e-11 5.55e-12 6.25e-03 1.70e-24 4.15e-23 7.34 4.51 2.70e-08 3.67e-13
· Exact solution is a reference solution for step size h = 3.125e-03
152
TABLE 7.2 : matrix condition estimates for Example 7 1. sbvpcol, equidistant, m=4
h condestDF cond ord. condest
ord. cond
const condest
const cond
1.00e-01 1.80e+04 5.59e+03 5.00e-02 5.39e+04 2.05e+04 -1.58 -1.87 4.69e+02 7.52e+01 2.50e-02 1.47e+05 8.02e+04 -1.44 -1.97 7.15e+02 5.58e+01 1.25e-02 5.25e+05 3.20e+05 -1.84 -1.99 1.65e+02 5.12e+01 6.25e-03 1.98e+06 1.28e+06 -1.91 -2.00 1.20e+02 5.01e+01
2. sbvpcol, Gauss, m=2
h condestDF cond ord. condest
ord. cond
const condest
const cond
1.00e-01 7.41e+03 3.84e+03 5.00e-02 2.43e+04 1.48e+04 -1.72 -1.95 1.43e+02 4.34e+01 2.50e-02 8.77e+04 5.88e+04 -1.85 -1.99 9.54e+01 3.80e+01 1.25e-02 3.32e+05 2.35e+05 -1.92 -2.00 7.36e+01 3.68e+01 6.25e-03 1.29e+06 9.42e+05 -1.96 -2.00 6.22e+01 3.66e+01
3. sbvpcol, Gauss, m=3
h condestDF cond ord. condest
ord. cond
const condest
const cond
1.00e-01 4.53e+04 1.52e+04 5.00e-02 1.13e+05 4.88e+04 -1.31 -1.68 2.21e+03 3.17e+02 2.50e-02 4.01e+05 1.93e+05 -1.83 -1.98 4.66e+02 1.29e+02 1.25e-02 1.48e+06 7.69e+05 -1.89 -2.00 3.81e+02 1.22e+02 6.25e-03 4.59e+06 3.08e+06 -1.63 -2.00 1.16e+03 1.20e+02
4. sbvpcol, Gauss, m=4
h condestDF cond ord. condest
ord. cond
const condest
const cond
1.00e-01 1.70e+05 5.43e+04 5.00e-02 4.17e+05 1.30e+05 -1.29 -1.26 8.66e+03 3.02e+03 2.50e-02 9.05e+05 4.91e+05 -1.12 -1.92 1.46e+04 1.55e+03 1.25e-02 3.25e+06 1.96e+06 -1.84 -1.99 1.02e+03 3.13e+02 6.25e-03 1.51e+07 7.82e+06 -2.22 -2.00 1.93e+02 3.07e+02
· condestDF:=condest(DF,1), cond:=cond(DF,2), Max. number of function evaluations exceeded
153
TABLE 7.3 : global error for Example 7 1. a) sbvpcol, equidistant, m=2, Error
h
err mesh
err coll
p mesh
p coll
const mesh
const coll
1.00e-01 1.59e-16 1.59e-16 5.00e-02 1.64e-17 1.64e-17 3.28 3.28 3.02e-13 3.02e-13 2.50e-02 3.71e-18 4.09e-18 2.14 2.00 1.00e-15 6.60e-15 1.25e-02 9.27e-19 9.70e-19 2.00 2.08 5.96e-15 8.72e-15 6.25e-03 1.85e-19 1.96e-19 2.33 2.31 2.52e-14 2.42e-14
2. a) sbvpcol, equidistant, m=4, Error
h
err mesh
err coll
p mesh
p coll
const mesh
const coll
1.00e-01 2.06e-17 2.06e-17 5.00e-02 4.26e-18 4.26e-18 2.27 2.27 3.85e-15 3.85e-15 2.50e-02 1.65e-19 1.65e-19 4.69 4.69 5.34e-12 5.34e-12 1.25e-02 6.22e-21 6.22e-21 4.73 4.73 6.31e-12 6.31e-12 6.25e-03 3.74e-22 3.84e-22 4.06 4.02 3.27e-13 2.75e-13
3. a) sbvpcol, equidistant, m=6, Error
h
err mesh
err coll
p mesh
p coll
const mesh
const coll
1.00e-01 4.44e-17 4.44e-17 5.00e-02 1.34e-18 1.34e-18 5.06 5.06 5.05e-12 5.05e-12 2.50e-02 1.67e-20 1.67e-20 6.31 6.31 2.22e-10 2.22e-10 1.25e-02 1.66e-22 1.66e-22 6.66 6.66 7.75e-10 7.75e-10 6.25e-03 2.20e-24 2.23e-24 6.24 6.22 1.22e-10 1.14e-10
· Exact solution is a reference solution for step size h = 3.125e-03
154
TABLE 7.4 : Error of error estimate based on h-h/2 for Example 7 1. a) sbvpcol, equidistant, m=2, Error of error
h
err mesh
err coll
p mesh
p coll
const mesh
const coll
1.00e-01 4.85e-17 4.85e-17 5.00e-02 2.95e-18 2.95e-18 4.04 4.04 5.26e-13 5.26e-13 2.50e-02 9.93e-20 2.57e19 4.89 3.52 6.83e-12 1.12e-13 1.25e-02 2.73e-21 3.79e-20 5.19 2.76 2.03e-11 6.86e-15 6.25e-03 1.08e-21 6.31e-21 1.34 2.58 9.77e-19 3.13e-15
2. a) sbvpcol, equidistant, m=4, Error of error
h
err mesh
err coll
p mesh
p coll
const mesh
const coll
1.00e-01 5.92e-18 5.92e-18 5.00e-02 3.54e-19 3.54e-19 4.06 4.06 6.86e-14 6.86e-14 2.50e-02 1.24e-20 1.24e-20 3.35 3.35 8.07e-15 8.07e-15 1.25e-02 8.35e-24 8.35e-24 7.31 7.31 6.27e-09 6.27e-09 6.25e-03 5.21e-25 2.85e-24 2.77 1.07 1.59e-18 9.23e-22
3. a) sbvpcol, equidistant, m=6, Error of error
h
err mesh
err coll
p mesh
p coll
const mesh
const coll
1.00e-01 6.74e-20 6.74e-20 5.00e-02 3.80e-20 3.80e-20 0.83 0.83 4.52e-19 4.52e-19 2.50e-02 2.81e-22 2.81e-22 7.08 7.08 6.18e-11 6.18e-11 1.25e-02 3.83e-25 3.83e-25 9.52 9.52 4.99e-07 4.99e-07 6.25e-03 2.11e-26 1.31e-28 4.18 7.98 3.38e-17 5.89e-10
· Exact solution is the reference solution for step size h = 3.90625e-004
155
TABLE 8.1 : global error for g(t)=sin(t10)cos(15t)
-α=1-
1. sbvpcol, equidistant, m=4
h err mesh
err coll
p mesh
p coll
const mesh
const coll
1.00e-01 3.50e-03 3.69e-03 5.00e-02 1.88e-04 1.89e-04 4.22 4.30 5.83e+01 7.33e+01 2.50e-02 1.08e-05 1.09e-05 4.11 4.11 4.23e+01 4.13e+01 1.25e-02 6.63e-07 6.67e-07 4.03 4.03 3.09e+01 3.12e+01 6.25e-03 4.13e-08 4.14e-08 4.01 4.01 2.78e+01 2.86e+01
2. sbvpcol, Gauss, m=2
h err mesh
err coll
p mesh
p coll
const mesh
const coll
1.00e-01 3.03e-02 2.28e-02 5.00e-02 1.47e-04 5.10e-03 4.35 2.16 6.77e+01 3.30e+00 2.50e-02 8.41e-06 7.88e-04 4.13 2.69 3.42e+01 1.63e+01 1.25e-02 5.12e-07 1.08e-04 4.04 2.86 2.47e+01 3.06e+01 6.25e-03 3.18e-08 2.28e-02 4.01 2.94 2.17e+01 4.24e+01
3. sbvpcol, Gauss, m=3
h err mesh
err coll
p mesh
p coll
const mesh
const coll
1.00e-01 8.00e-05 5.20e-03 5.00e-02 1.41e-06 4.41e-04 5.83 3.56 5.37e+01 1.89e+01 2.50e-02 2.21e-08 2.82e-05 5.99 3.96 8.87e+01 6.35e+01 1.25e-02 3.45e-10 1.77e-06 6.00 3.99 9.07e+01 7.02e+01 6.25e-03 5.40e-12 1.11e-07 6.00 4.00 9.01e+01 7.15e+01
4. sbvpcol, Gauss, m=4
h err mesh
err coll
p mesh
p coll
const mesh
const coll
1.00e-01 4.75e-07 4.33e-04 5.00e-02 6.61e-10 1.31e-05 9.49 5.04 1.47e+03 4.81e+01 2.50e-02 6.56e-12 4.43e-07 6.65 4.88 2.99e-01 2.97e+01 1.25e-02 2.75e-14 1.40e-08 7.90 4.98 2.98e+01 4.23e+01 6.25e-03 6.11e-16 4.71e-10 5.49 4.90 7.77e-04 2.96e+01
156
TABLE 8.2 : global error for g(t)=sin(t10)cos(15t) -α=2-
1. sbvpcol, equidistant, m=4
h err mesh
err coll
p mesh
p coll
const mesh
const coll
1.00e-01 3.40e-03 3.67e-03 5.00e-02 1.86e-04 1.86e-04 4.19 4.30 5.31e+01 7.39e+01 2.50e-02 1.07e-05 1.08e-05 4.11 4.11 4.19e+01 4.10e+01 1.25e-02 6.56e-07 6.60e-07 4.03 4.03 3.07e+01 3.07e+01 6.25e-03 4.09e-08 4.10e-08 4.01 4.01 2.76e+01 2.83e+01
2. sbvpcol, Gauss, m=2
h err mesh
err coll
p mesh
p coll
const mesh
const coll
1.00e-01 2.90e-03 2.28e-02 5.00e-02 1.43e-04 5.10e-03 4.34 2.16 6.39e+01 3.30e+00 2.50e-02 8.13e-06 7.88e-04 4.13 2.69 3.43e+01 1.63e+01 1.25e-02 4.96e-07 1.08e-04 4.03 2.86 2.36e+01 3.06e+01 6.25e-03 3.08e-08 1.41e-05 4.01 2.94 2.11e+01 4.24e+01
3. sbvpcol, Gauss, m=3
h err mesh
err coll
p mesh
p coll
const mesh
const coll
1.00e-01 7.87e-05 5.20e-03 5.00e-02 1.39e-06 4.41e-04 5.82 3.56 5.17e+01 1.89e+01 2.50e-02 2.19e-08 2.82e-05 5.99 3.96 8.72e+01 6.35e+01 1.25e-02 3.43e-10 1.77e-06 6.00 3.99 8.84e+01 7.02e+01 6.25e-03 5.36e-12 1.11e-07 6.00 4.00 9.03e+01 7.15e+01
4. sbvpcol, Gauss, m=4
h err mesh
err coll
p mesh
p coll
const mesh
const coll
1.00e-01 4.89e-07 4.33e-04 5.00e-02 5.75e-10 1.31e-05 9.73 5.04 2.64e+03 4.81e+01 2.50e-02 6.34e-12 4.43e-07 6.50 4.88 1.66e–01 2.97e+01 1.25e-02 2.68e-14 1.40e-08 7.89 4.98 27.6e+01 4.23e+01 6.25e-03 6.11e-16 4.71e-10 5.45 4.90 6.39e–04 2.96e+01
157
TABLE 8.3 : global error for g(t)=sin(t10)cos(15t) -α=3-
1. sbvpcol, equidistant, m=4
h err mesh
err coll
p mesh
p coll
const mesh
const coll
1.00e-01 3.30e-03 3.65e-03 5.00e-02 1.84e-04 1.84e-04 4.17 4.31 4.84e+01 7.48e+01 2.50e-02 1.06e-05 1.06e-05 4.11 4.11 4.14e+01 4.08e+01 1.25e-02 6.49e-07 6.53e-07 4.03 4.03 3.04e+01 3.03e+01 6.25e-03 4.04e-08 4.05e-08 4.01 4.01 2.75e+01 2.79e+01
2. sbvpcol, Gauss, m=2
h err mesh
err coll
p mesh
p coll
const mesh
const coll
1.00e-01 2.70e-02 2.28e-02 5.00e-02 1.38e-04 5.10e-03 4.29 2.16 5.24e+01 3.30e+00 2.50e-02 7.78e-06 7.88e-04 4.15 2.69 3.47e+01 1.63e+01 1.25e-02 4.77e-07 1.08e-04 4.03 2.86 2.20e+01 3.06e+01 6.25e-03 2.96e-08 1.41e-05 4.01 2.94 2.04e+01 4.24e+01
3. sbvpcol, Gauss, m=3
h err mesh
err coll
p mesh
p coll
const mesh
const coll
1.00e-01 7.77e-05 5.20e-03 5.00e-02 1.38e-06 4.41e-04 5.81 3.56 5.03e+01 1.89e+01 2.50e-02 2.18e-08 2.82e-05 5.99 3.96 8.62e+01 6.35e+01 1.25e-02 3.42e-10 1.77e-06 5.99 3.99 8.64e+01 7.02e+01 6.25e-03 5.35e-12 1.11e-07 6.00 4.00 9.00e+01 7.15e+01
4. sbvpcol, Gauss, m=4
h err mesh
err coll
p mesh
p coll
const mesh
const coll
1.00e-01 4.98e-07 4.33e-04 5.00e-02 5.68e-10 1.31e-05 9.77 5.04 2.96e+03 4.81e+01 2.50e-02 6.16e-12 4.43e-07 6.52 4.88 1.76e–01 2.97e+01 1.25e-02 2.61e-14 1.40e-08 7.88 4.98 2.61e+01 4.23e+01 6.25e-03 5.55e-16 4.71e-10 5.56 4.90 9.87e–04 2.96e+01
158
Figure 9.1 : plot of solution and error using sbvp for Example 1
1. a) Solution, α =1 b) Error, α =1
2. a) Solution, α =2 b) Error, α = 2
3. a) Solution, α =3 b) Error, α = 3
159
TABLE 9.1 : global error for g(t)=cos(15t) -α=1-
1. sbvpcol, equidistant, m=4
h err mesh
err coll
p mesh
p coll
const mesh
const coll
1.00e-01 1.57e-03 1.70e-03 5.00e-02 1.08e-04 1.08e-04 3.89 3.97 1.23e+01 1.60e+01 2.50e-02 6.67e-06 6.67e-06 4.02 4.02 1.84e+01 1.84e+01 1.25e-02 4.15e-07 4.17e-07 4.01 4.00 1.74e+01 1.70e+01 6.25e-03 2.60e-08 2.60e-08 4.00 4.00 1.70e+01 1.72e+01
2. sbvpcol, Gauss, m=2
h err mesh
err coll
p mesh
p coll
const mesh
const coll
1.00e-01 1.62e-02 2.84e-02 5.00e-02 1.08e-03 3.46e-03 3.88 3.02 1.23e+02 2.98e+01 2.50e-02 6.83e-05 4.29e-04 4.01 3.03 1.81e+02 3.06e+01 1.25e-02 4.29e-06 5.33e-05 3.99 3.01 1.71e+02 2.81e+01 6.25e-03 2.68e-07 6.63e-06 4.00 3.01 1.74e+02 2.82e+01
3. sbvpcol, Gauss, m=3
h err mesh
err coll
p mesh
p coll
const mesh
const coll
1.00e-01 2.12e-03 2.56e-03 5.00e-02 1.56e-04 1.64e-04 3.75 3.99 1.17e+01 2.52e+01 2.50e-02 1.02e-05 1.03e-05 3.94 4.00 2.11e+01 2.59e+01 1.25e-02 6.42e-07 6.44e-07 3.99 4.00 2.47e+01 2.62e+01 6.25e-03 4.02e-08 4.02e-08 4.00 4.00 2.59e+01 2.63e+01
4. sbvpcol, Gauss, m=4
h err mesh
err coll
p mesh
p coll
const mesh
const coll
1.00e-01 1.03e-04 1.84e-04 5.00e-02 1.73e-06 5.82e-06 5.90 4.98 8.15e+01 1.77e+01 2.50e-02 2.74e-08 1.82e-07 5.98 5.00 1.03e+02 1.84e+01 1.25e-02 4.30e-10 5.71e-09 5.99 5.00 1.10e+02 1.85e+01 6.25e-03 6.73e-12 1.78e-10 6.00 5.00 1.12e+02 1.90e+01
160
TABLE 9.2 : global error for g(t)=cos(15t) -α=2-
1. sbvpcol, equidistant, m=4
h err mesh
err coll
p mesh
p coll
const mesh
const coll
1.00e-01 1.59e-03 1.69e-03 5.00e-02 1.07e-04 1.07e-04 3.90 3.97 1.27e+01 1.61e+01 2.50e-02 6.60e-06 6.60e-06 4.02 4.02 1.83e+01 1.83e+01 1.25e-02 4.11e-07 4.12e-07 4.01 4.00 1.72e+01 1.69e+01 6.25e-03 2.57e-08 2.57e-08 4.00 4.00 1.69e+01 1.70e+01
2. sbvpcol, Gauss, m=2
h err mesh
err coll
p mesh
p coll
const mesh
const coll
1.00e-01 7.75e-02 7.75e-02 5.00e-02 1.00e-03 1.00e-03 2.95 2.95 6.97e+01 6.97e+01 2.50e-02 1.09e-03 1.09e-03 3.18 3.18 1.39e+02 1.39e+02 1.25e-02 1.12e-04 1.12e-04 3.29 3.29 2.06e+02 2.06e+02 6.25e-03 1.13e-05 1.13e-05 3.31 3.31 2.29e+02 2.29e+02
3. sbvpcol, Gauss, m=3
h err mesh
err coll
p mesh
p coll
const mesh
const coll
1.00e-01 7.31e-03 7.31e-03 5.00e-02 4.83e-04 4.83e-04 3.92 3.92 6.04e+01 6.04e+01 2.50e-02 2.24e-05 2.24e-05 4.43 4.43 2.81e+02 2.81e+02 1.25e-02 1.44e-06 1.44e-06 3.96 3.96 5.01e+02 5.01e+02 6.25e-03 8.34e-08 8.34e-08 4.11 4.11 9.39e+02 9.39e+02
4. sbvpcol, Gauss, m=4
h err mesh
err coll
p mesh
p coll
const mesh
const coll
1.00e-01 5.18e-04 5.18e-04 5.00e-02 1.25e-05 1.25e-05 5.37 5.37 1.21e+02 1.21e+02 2.50e-02 3.19e-07 3.19e-07 5.30 5.30 9.74e+01 9.74e+01 1.25e-02 7.90e-09 7.90e-09 5.34 5.34 1.13e+02 1.13e+02 6.25e-03 1.95e-10 1.95e-10 5.34 5.34 1.16e+02 1.16e+02
161
TABLE 9.3 : global error for g(t)=cos(15t) -α=3-
1. sbvpcol, equidistant, m=4
h err mesh
err coll
p mesh
p coll
const mesh
const coll
1.00e-01 2.78e-03 2.78e-03 5.00e-02 1.38e-04 1.38e-04 4.34 4.34 6.10e+01 6.10e+01 2.50e-02 6.52e-06 6.52e-06 4.41 4.41 7.49e+01 7.49e+01 1.25e-02 4.06e-07 4.07e-07 4.01 4.00 1.70e+01 1.67e+01 6.25e-03 2.54e-08 2.54e-08 4.00 4.00 1.67e+01 1.68e+01
2. sbvpcol, Gauss, m=2
h err mesh
err coll
p mesh
p coll
const mesh
const coll
1.00e-01 1.05e-01 1.05e-01 5.00e-02 2.43e-02 2.43e-02 2.11 2.11 1.37e+01 1.37e+01 2.50e-02 4.14e-03 4.14e-03 2.57 2.57 5.32e+01 5.32e+01 1.25e-02 6.04e-04 6.04e-04 2.76 2.76 1.10e+02 1.10e+02 6.25e-03 8.14e-05 8.14e-05 2.89 2.89 1.91e+02 1.91e+02
3. sbvpcol, Gauss, m=3
h err mesh
err coll
p mesh
p coll
const mesh
const coll
1.00e-01 1.09e-02 1.09e-02 5.00e-02 3.08e-04 3.08e-04 5.14 5.14 1.52e+03 1.52e+03 2.50e-02 2.09e-05 2.09e-05 3.88 3.88 3.49e+01 3.49e+01 1.25e-02 1.29e-06 1.29e-06 4.01 4.01 5.61e+01 5.61e+01 6.25e-03 8.05e-08 8.05e-08 4.00 4.00 5.38e+01 5.38e+01
4. sbvpcol, Gauss, m=4
h err mesh
err coll
p mesh
p coll
const mesh
const coll
1.00e-01 1.01e-03 1.01e-03 5.00e-02 4.32e-05 4.32e-05 4.53 4.53 3.40e+01 3.40e+01 2.50e-02 1.58e-06 1.58e-06 4.77 4.77 6.94e+01 6.94e+01 1.25e-02 5.36e-08 5.36e-08 4.88 4.88 1.06e+02 1.06e+02 6.25e-03 1.74e-09 1.74e-09 4.94 4.94 1.37e+02 1.37e+02
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