Post on 03-Jun-2022
Solution of initial value problems
In the method of lines, time integration is performed using numericalmethods for initial value problems of the form
dudt (t) = f(t, u(t)), t ∈ (tn, tn+1), u(tn) = un
Exact integration∫ tn+1
tndudt (t)dt = un+1 − un =
∫ tn+1
tnf(t, u(t))dt
Evaluation of f(t, u(t)) is impossible for t ∈ (tn, tn+1)
The integral is approximated using numerical quadrature
Multipoint methods
Order barrier: two-level methods are at most second-order accurate
Additional points are needed to construct higher order schemes
Adams methods perform time integration using the points
tn+1, . . . , tn−m, m = 0, 1, . . .
Runge-Kutta methods insert points between tn and tn+1
tn+α ∈ [tn, tn+1], α ∈ [0, 1]
Adams methods
Adams methods are multipoint time integrators which can beinterpreted as Newton-Cotes quadrature rules
A multipoint approximation of p-th order is derived by fitting aLagrange polynomial to the data at p+ 1 time levels
t
n+1
t
present futurepast
f
t
n
t
n�1
t
n�2
Adams methods
Adams-Bashforth methods (explicit)
un+1 = un + f(tn, un)∆t
un+1 = un + 12 [3f(tn, un)− f(tn−1, un−1)]∆t
un+1 = un + 112 [23f(tn, un)− 16f(tn−1, un−1) + 5f(tn−2, un−2)]∆t
Adams-Moulton methods (implicit)
un+1 = un + f(tn+1, un+1)∆t
un+1 = un + 12 [f(tn+1, un+1) + f(tn, un)]∆t
un+1 = un + 112 [5f(tn+1, un+1) + 8f(tn, un)− f(tn−1, un−1)]∆t
Adams methods
Adams-Bashforth method of order p− 1 can be used as predictorfor an Adams-Moulton corrector of order p
Adams methods of any order are easy to derive and implement(as long as the time step ∆t is fixed)
Just one derivative evaluation per time step is required (if solutionvalues calculated at the previous time levels are stored)
Low-order methods are needed to start/restart time integration
Time step is difficult to change (the coefficients are different)
Numerical instabilities and/or spurious oscillations may occur
Runge-Kutta methods
Runge-Kutta methods are predictor-(multi)corrector algorithms
Auxiliary solutions correspond to time levels between tn and tn+1
The second-order Runge-Kutta method employs a forward Eulerpredictor for un+1/2 and a midpoint rule corrector for un+1
un+1/2 = un + 12f(tn, un)∆t
un+1 = un + f(tn+1/2, un+1/2)∆t
Higher order can be achieved using additional corrector steps
Runge-Kutta methods
The classical fourth-order Runge-Kutta method is given by
un+1/2 = un + 12f(tn, un)∆t
un+1/2 = un + 12f(tn+1/2, un+1/2)∆t
un+1 = un + f(tn+1/2, un+1/2)∆t
un+1 = un + 16 [f(tn, un) + 2f(tn+1/2, un+1/2)
+2f(tn+1/2un+1/2) + f(tn+1, un+1)]∆t
Runge-Kutta methods
Self-starting and easy to operate with variable time steps
More stable and accurate than Adams methods of the same order
Higher-order approximations are difficult to derive; a p-stage methodrequires p evaluations of the derivative function per time step
More expensive than Adams methods of comparable order
Time step control
Criteria for the choice of the time step ∆t depend on the type of thetime stepping procedure (explicit or implicit)
In explicit schemes, time steps that are smaller than but sufficientlyclose to the stability bound are usually optimal
In unconditionally stable implicit schemes, the optimal value of ∆tdepends on the magnitude of the time discretization error
Adaptive time step control can be performed using error estimatorsdeveloped in the context of numerical methods for ODEs
Time step control
Let u∆t be a solution calculated using the time step ∆t
u∆t = u+ (∆t)2e(u) +O(∆t)4
um∆t = u+m2(∆t)2e(u) +O(∆t)4
The local truncation error can be estimated as follows:
e(u) ≈ um∆t − u∆t
(∆t)2(m2 − 1)
The time step can be adjusted to satisfy
||u− u∆t|| ≈ TOL
Time step control
The optimal time step ∆t∗ can be determined as follows:
||u− u∆t∗ || ≈(
∆t∗∆t
)2 ||u∆t − um∆t||m2 − 1 = TOL
⇒ (∆t∗)2 = TOL(∆t)2(m2 − 1)||u∆t − um∆t||
Accuracy can be improved using Richardson extrapolation
u = m2u∆t − um∆t
m2 − 1 + O(∆t)4
Time step control
Practical implementation
1 Given the old solution un, calculate un+1m∆t and un+1
∆t
2 Calculate∥∥un+1
∆t − un+1m∆t
∥∥ and the next time step ∆t∗
3 If ∆t∗ is much smaller than ∆t, reset and go back to step 1
4 Accept un+1 = un+1∆t or calculate un+1 = m2un+1
∆t−un+1
m∆t
m2−1
Other error estimators: embedded Runge-Kutta and Adams methods
PID control
1 Monitor the relative changes at the current time step
en = ||un+1 − un||||un+1||
2 If en is too large (en > δ), repeat the time step using
∆t∗ = δ
en∆tn
3 Adjust the time step smoothly to approach the prescribedtolerance TOL for the indicator en
∆tn+1 =(en−1
en
)kP(TOL
en
)kI(
e2n−1
enen−2
)kD
∆tn
Properties of numerical methods
Consistency: local truncation errors are proportional to positivepowers of the time step ∆t and/or mesh size h
Stability: Round-off and iteration iteration errors that are generatedin the numerical solution process are not amplified
Convergence: The global error (measured w.r.t. certain norms)tends to zero in the limit h→ 0 and ∆t→ 0
Stability analysis
Two-level explicit schemes produce linear systems of the form
un+1 = Cun
In practice, the old solution already contains numerical errors
un+1 = Cun
If no new errors are generated at the current step, we obtain
en+1 = Cen, en = un − un
Stability analysis
The discretization is stable if the error does not grow
‖en+1‖ = ‖Cen‖ ≤ ‖en‖
A matrix form consistent with ‖ · ‖ is given by
‖C‖ := supu 6=0
‖Cu‖‖u‖
The error growth can be estimated as follows:
‖en+1‖‖en‖
= ‖Cen‖
‖en‖≤ ‖C‖
Stability analysis
Boundedness of the matrix norm ‖C‖ is sufficient for stability
‖C‖ ≤ 1
If φ(m) is an eigenvector of the matrix C and λ(m) is its associatedeigenvalue then Cφ(m) = λ(m)φ(m). It follows that
|λ(m)| = ‖λ(m)φ(m)‖‖φ(m)‖
= ‖Cφ(m)‖
‖φ(m)‖≤ ‖C‖
The corresponding stability condition for the eigenvalues of C is
|λ(m)| ≤ 1, m = 1, . . . , N
Stability analysis
Convection-diffusion equation∂u
∂t+ v
∂u
∂x= d
∂2u
∂x2
Space discretization (CD)duidt + v
ui+1 − ui−1
2∆x = dui−1 − 2ui + ui+1
(∆x)2
Time discretization (FE)
un+1i − uni
∆t + vuni+1 − uni−1
2∆x = duni−1 − 2uni + uni+1
(∆x)2
Stability analysis
Linear algebraic system un+1 = Cun
un+1i = uni −
ν
2 (uni+1 − uni−1) + δ(uni−1 − 2uni + uni+1)
where ν = v ∆t∆x is the Courant number and δ = d ∆t
(∆x)2
C =
· · ·a b c
a b ca b c· · ·
a = δ + ν
2
b = 1− 2δ
c = δ − ν2
Stability analysis
Eigenvectors
φ(m)j = cos θmj + i sin θmj = eiθmj , j,m = 1, . . . , N
where i is the complex unity and
θm = m πN is the phase angle
low frequen ieshigh frequen ies
m = N m = 0
0�
Cφ(m) = λ(m)φ(m) ⇒ aφ(m)j+1 + bφ
(m)j + cφ
(m)j−1 = λ(m)φ
(m)j
b = 1− (a+ c) ⇒ λ(m) = 1 + a(eiθm − 1) + c(e−iθm − 1)
Stability analysis
Eigenvalues
λ(m) = 1 + 2δ(cos θm − 1) + i2ν sin θm
Stability condition
|λ(m)|2 = [1 + 2δ(cos θm − 1)]2 + 4ν2 sin2 θm ≤ 1
pure convection: δ = 0 ⇒ |λ(m)| ≥ 1 (unstable)
pure diffusion: ν = 0 ⇒ |λ(m)| ≤ 1 for δ ≤ 12
δ = d ∆t(∆x)2 ⇒ ∆t ≤ d (∆x)2
2 (conditionally stable)
Fourier analysis
The eigenvectors φ(m)j = eiθmj form the Fourier basis such that
unj − unj =∑m
anmeiθmj
The von Neumann stability analysis examines the evolution of
anmeiθmj = anm(cos θmj + i sin θmj)
The discretization is stable if the amplitude of the sine and cosinewaves remains bounded for all Fourier modes / phase angles
Fourier analysis
The amplification factor of the m-th mode is defined by
Gm = an+1m
anm= λ(m)
The wave amplitude anm is non-increasing if
|Gm| ≤ 1 ∀m = 1, . . . , N
In general Gm is a complex number. We have
Gm = α+ iβ ⇒ |Gm| =√α2 + β2
1st example
Pure convection equation∂u
∂t+ v
∂u
∂x= 0, v > 0
Space discretization (CD)duidt + v
ui+1 − ui−1
2∆x = 0
Time discretization (FE)
un+1i − uni
∆t + vuni+1 − uni−1
2∆x = 0
1st example
Discrete problem
un+1j = unj + ν
2 (unj+1 − unj−1) = 0, ν = v∆t∆x
Substitution of unj = aneiθj yields an equation for an+1
(an+1 − an)eiθj + ν
2an(eiθ(j+1) − eiθ(j−1)) = 0
⇒ an+1 = an − ν
2an(eiθ − e−iθ)
where eiθ − e−iθ = cos θ + i sin θ − cos θ + i sin θ = 2i sin θ
1st example
Amplification factor
G = an+1
an= 1− iν sin θ
Conclusion: unstable since
|G|2 = 1 + ν2 sin2 θ ≥ 1
0 0.1 0.2 0.3 0.4 0.5 0.6 0.7 0.8 0.9 1−1
−0.8
−0.6
−0.4
−0.2
0
0.2
0.4
0.6
0.8
1x 10
22
2nd example
Pure convection equation∂u
∂t+ v
∂u
∂x= 0, v > 0
Space discretization (BD)duidt + v
ui − ui−1
∆x = 0
Time discretization (FE)
un+1i − uni
∆t + vuni − uni−1
∆x = 0
2nd example
Discrete problem
un+1j = unj + ν(unj − unj−1) = 0, ν = v
∆t∆x
Substitution of unj = aneiθj yields an equation for an+1
(an+1 − an)eiθj + νan(eiθj − eiθ(j−1)) = 0
⇒ an+1 = an − νan(1− e−iθ)
G = an+1
an = 1− ν + νe−iθ = 1− ν + ν(cos θ − i sin θ)
2nd example
Amplification factorRe(G) = 1− ν + ν cos θ, Im(G) = −ν sin θ
Conclusion: stable under theCFL condition
ν = v∆t∆x ≤ 1
0 0.1 0.2 0.3 0.4 0.5 0.6 0.7 0.8 0.9 1−0.2
0
0.2
0.4
0.6
0.8
1
1.2
Initial
Exact
∆ x=10−2
∆ x=10−3
3rd example
Pure convection equation∂u
∂t+ v
∂u
∂x= 0, v > 0
Space discretization (CD)duidt + v
ui+1 − ui−1
2∆x = 0
Time discretization (BE)
un+1i − uni
∆t + vun+1i+1 − u
n+1i−1
2∆x = 0
3rd example
Discrete problem
un+1j = unj + ν
2 (un+1j − un+1
j−1 ) = 0, ν = v∆t∆x
(an+1 − an)eiθj + ν
2an+1(eiθ(j+1) − eiθ(j−1)) = 0
Amplification factorG = 1
1+iν sin θ
Conclusion: stable for all ν since |G|2 = G · G = 11+ν2 sin2 θ ≤ 1
Numerical diffusion
Pure convection equation: BD/FE approximation
∂u
∂t+ v
∂u
∂x= 0 7→ un+1
i − uni∆t + v
uni − uni−1∆x = 0
un+1i−un
i
∆t + vun
i+1−uni−1
2∆x =(v∆x
2) un
i+1−2uni +un
i−1(∆x)2
This corresponds to the 2nd order CD approximation of
∂u
∂t+ v
∂u
∂x=(v∆x
2
)∂2u
∂x2
where v∆x2 is the numerical diffusion coefficient
Example
BD/FE approximation, ε = v∆x2 , ∆t = ∆x/2
∂u
∂t+ v
∂u
∂x= 0, u(x, t) = u0(x− vt), v ≡ 1, T = 0.5
0 0.1 0.2 0.3 0.4 0.5 0.6 0.7 0.8 0.9 1−0.2
0
0.2
0.4
0.6
0.8
1
1.2
Initial
Exact
∆ x=10−2
∆ x=10−3
smooth solution
0 0.1 0.2 0.3 0.4 0.5 0.6 0.7 0.8 0.9 1−0.2
0
0.2
0.4
0.6
0.8
1
1.2
Initial
Exact
∆ x=10−2
∆ x=10−3
discontinuous solution
Modified equations
Let uh be a numerical solution of the time-dependent problem
∂u
∂t+ Lu = 0
This approximation represents the exact solution of another PDE
∂u
∂t+ Lu = ε
Such modified equations provide a useful tool for qualitativeanalysis of accuracy and stability of numerical approximations
Derivation of ME
Double Taylor series expansion about (xi, tn) yields a PDE
The difference between this PDE and the original one is the localtruncation error which contains both space and time derivatives
Time derivatives can be expressed in terms of space derivativesusing the new PDE and its derivatives
The resulting representation of the local truncation error is bettersuited for qualitative analysis of accuracy and stability
Interpretation of MG
Modified equation (Warming & Hyett, 1974)
∂u
∂t+ v
∂u
∂x=∞∑p=1
µ2p∂2pu
∂x2p +∞∑p=1
µ2p+1∂2p+1u
∂x2p+1
µ2p 6= 0 ⇒ amplitude error (numerical diffusion)
µ2p+1 6= 0 ⇒ phase error (numerical dispersion)
Stability condition (necessary, not sufficient)
k := min{p ∈ N : µ2p 6= 0}, sign(µ2k) = (−1)k+1
Examples of ME
Pure convection equation∂u∂t + v ∂u∂x = 0, v > 0
Upwind discretizationun+1
i−un
i
∆t + vun
i −uni−1
∆x = 0
Taylor expansion about (xi, tn)
un+1i = uni + ∆t
(∂u∂t
)ni
+ (∆t)2
2
(∂2u∂t2
)ni
+ (∆t)3
6
(∂3u∂t3
)ni
+ . . .
uni−1 = uni −∆x(∂u∂x
)ni
+ (∆x)2
2
(∂2u∂x2
)ni− (∆x)3
6
(∂3u∂x3
)ni
+ . . .
Examples of ME
Local truncation error
ε = −∆t2
(∂2u∂t2
)ni− (∆t)2
6
(∂3u∂t3
)ni
+ v∆x2
(∂2u∂x2
)ni− v(∆x)2
6
(∂3u∂x3
)ni
+ . . .
Modified equation∂u
∂t+ v
∂u
∂x= v∆x
2 (1− ν)∂2u
∂x2 + v(∆x)2
6 (3ν − 2ν2 − 1)∂3u
∂x3 + . . .
Stability condition (CFL)
0 ≤ ν ≤ 1, ν = v∆t∆x
Examples of ME
Central differences∂u
∂t+ v
∂u
∂x= −v
2∆x2
∂2u
∂x2 −v(∆x)2
6 (1 + 2ν2)∂3u
∂x3 + . . .
Finite elements (P1)∂u
∂t+ v
∂u
∂x= −v
2∆x2
∂2u
∂x2 −v(∆x)2
3 ν2 ∂3u
∂x3 + . . .
µ2 = − v2∆x2 < 0 ⇒ unstable since the ME is ill-posed
(no continuous dependence of the solution on the data)
Lax-Wendroff method
Taylor expansion un+1 = un + ∆tunt + (∆t)2
2 untt +O(∆t)3
∂u∂t + v ∂u∂x = 0 ⇒ ut = −vux, utt = (ut)t = −v(ut)x = v2uxx
Time discretization un+1 = un − v∆tunx + (v∆t)2
2 unxx + O(∆t)3
Lax-Wendroff / central differences
un+1i − uni
∆t + vuni+1 − uni−1
2∆x = (v∆t)2
2uni+1 − 2uni + uni−1
(∆x)2
Stabilization via inherent numerical diffusion, 2nd order accuracy
Taylor-Galerkin methods
Taylor expansion un+1 = un + ∆tunt + (∆t)2
2 untt + (∆t)3
6 unttt +O(∆t)4
uttt = (utt)t = un+1tt − untt
∆t +O(∆t) = v2un+1xx − unxx
∆t +O(∆t)
un+1 = un − v∆tunx + (v∆t)2
2 unxx + (v∆t)2
6 (un+1xx − unxx) + O(∆t)4
Mun+1i − uni
∆t + vuni+1 − uni−1
2∆x = (v∆t)2
2uni+1 − 2uni + uni−1
(∆x)2
Mui = ui+1 + 4ui + ui−1
6 − v2∆t6
ui+1 − 2ui + ui−1
(∆x)2
Modified equations
ME for Lax-Wendroff / CD ⇒ ν2 ≤ 1∂u∂t + v ∂u∂x = −v(∆x)2
6 (1− ν2)∂3u∂x3 − v(∆x)3
8 ν(1− ν2)∂4u∂x4 + . . .
ME for Lax-Wendroff / FE ⇒ ν2 ≤ 13
∂u∂t + v ∂u∂x = v(∆x)2
6 ν2 ∂3u∂x3 − v(∆x)3
24 ν(1− 3ν2)∂4u∂x4 + . . .
ME for Taylor-Galerkin FE ⇒ ν2 ≤ 1∂u∂t + v ∂u∂x = − v(∆x)3
24 ν(1− ν2)∂4u∂x4 + v(∆x)4
180 (1− 5ν2 + 4ν4)∂5u∂x5 + . . .
Example
Lax-Wendroff / Taylor-Galerkin ∆x = 10−2, ν = 0.5∂u
∂t+ v
∂u
∂x= 0, u(x, t) = u0(x− vt), v ≡ 1, T = 0.5
0 0.1 0.2 0.3 0.4 0.5 0.6 0.7 0.8 0.9 1
−0.2
0
0.2
0.4
0.6
0.8
1
1.2Initial
Exact
LW
TG
smooth solution
0 0.1 0.2 0.3 0.4 0.5 0.6 0.7 0.8 0.9 1
−0.2
0
0.2
0.4
0.6
0.8
1
1.2Initial
Exact
LW
TG
discontinuous solution