Transport Schemes on a Sphere Using Radial

Basis Functions

Natasha Flyer a,∗, Grady B. Wright b,1,

aNational Center for Atmospheric Research, Scientific Computing Division, 1850

Table Mesa Dr., Boulder, CO 80305 USA

bDepartment of Mathematics, University of Utah, Salt Lake City, UT 84112-0090,

USA

Abstract

The aim of this work is to introduce the physics community to the high per-formance of radial basis functions (RBFs) compared to other spectral methods formodeling transport (pure advection) and to provide the first known application ofthe RBF methodology to hyperbolic partial differential equations on a sphere. First,it is shown that even when the advective operator is posed in spherical coordinates(thus having singularities at the poles), the RBF formulation of it is completelysingularity-free. Then, two classical test cases are conducted: 1) linear advection,where the initial condition is simply transported around the sphere and 2) defor-mational flow (idealized cyclogenesis), where an angular velocity is applied to theinitial condition, spinning it up about an axis of rotation. The results show thatRBFs require a much lower spatial resolution (i.e. number of nodes) while beingable to take unusually large time-steps to achieve the same accuracy as comparedto other commonly used spectral methods on a sphere such as spherical harmon-ics, double Fourier series, and spectral element methods. Futhermore, RBFs arealgorithmically much simpler to program.

Key words: Radial Basis Functions, Hyperbolic PDEs, Spherical Geometry,Mesh-Free1991 MSC: 58J45, 41A21, 41A15, 41A63, 65D25, 65D30, 65M20, 65M70, 76M22,86A10

∗ Corresponding author.Email addresses: flyer@ucar.edu, wright@math.utah.edu (Grady B. Wright ).

1 The work was supported by NSF VIGRE grant DMS-0091675.

Preprint submitted to Elsevier Science 28 July 2006

1 Introduction

The purpose of this paper is to introduce the prospect of using radial basisfunctions (RBFs), a novel numerical methodology that does not require anymesh or grid, for geophysical modeling in spherical domains. It has the ad-vantage of achieving spectral accuracy in multi-dimensions for arbitrary nodelayouts with extreme algorithmic simplicity. For the purposes of interpolatingmulti-dimensional surfaces, the methodology has been around for approxi-mately 30 years. However, it is only in the last 15 years that it has beenapplied to solving partial differential equations (PDEs) containing parabolicand/or elliptic operators. (cite articles). Furthermore, it is only in the last 3years or so that it has been considered in spherical domains. Thus, the aim ofthis article is two-fold: 1) to present the elegance and power of this method-ology to the physics community and 2) to provide the first known applicationof it to purely hyberbolic PDEs in spherical domains.

Since geophysical fluid motions on all scales are dominated by the advectionprocess, the numerical solution to the advection problem is therefore funda-mentally important for the overall accuracy of the flow solver. Thus, in thecurrent paper, we will show striking results for advection tests with respect toaccuracy and time stability, requiring only a low number of degrees of freedomfor spatial discretization while being able to take much larger time-steps thanmethods currently employed in geophysical modeling (e.g. spherical harmon-ics, spectral elements). We consider two well-known test cases that probe thesuitability of a new numerical methodology for modeling advection in sphericalgeometries. The first is the classical convection of a cosine bell with compactsupport over a sphere at different angles of rotation [1]. The second is a cyclo-genesis test problem with a deformational flow that describes the wrap-up of avortex with increasingly stronger gradients over time, which is a simple modelfor the observed evolution of cold and warm frontal zones [2]. An overviewof the paper is as follows: Section 2 gives an introduction to RBFs; Section3 discusses node distributions on a sphere and the convergence rates of RBFinterpolants; Section 4 derives the RBF formulation of the advection operator;Section 5 and 6 are the numerical tests and results for convergence and timestability.

2 Introduction to Radial Basis Functions

The motivation of the RBF methodology originated with R.L. Hardy [3] ask-ing the question, ‘Given a set of sparse scattered data, {fj}N

j=1, at the nodelocations {xj}N

j=1 in multi-dimensions, can an interpolant be constructed thatadequately represents the unknown surface?’. It was shown by Haar [4] that,

2

in more than one dimension, interpolation by an expansion in basis functions,{ψj(x)}N

j=1, x ∈ Rd, that are independent of the node locations is not well-

posed. That is, there exists an infinite number of node configurations thatwill yield the interpolation problem singular. Hardy bypassed this singularityproblem with a novel approach in which the interpolant is constructed fromlinear combinations a single basis function that is radially symmetric aboutits center and whose argument is dependent on the node locations. In lieu ofgiving up orthogonality, well-posedness of the interpolant and its derivativesfor any set of distinct scattered nodes in any dimension is gained.

Commonly used RBFs are given in Figure .1, where r = ‖x − xj‖ is theEuclidean or ℓ2 norm. The piecewise smooth RBFs feature a jump in somederivative at x = xj and thus can only lead to algebraic convergence. For in-stance, the |r|3 has a jump in the third derivative and in 1-D will lead to fourthorder convergence, in 2-D sixth order convergence and so forth (c.f.[5]). Onthe other hand, infinitely smooth RBFs will lead to spectral convergence [6,7].Notice that the infinitely smooth RBFs depend on a shape parameter ε. It wasfirst shown by Driscoll and Fornberg [8] that, in 1-D, in the limit of ε → 0 (i.e.flat RBFs) the RBF methodology reproduces pseudospectral methods (PS) ifthe nodes are accordinging placed (i.e. equi-spaced nodes for Fourier methods,Gauss-Chebyshev nodes for Chebshev methods, etc.).

A comparison between the concept of PS methods and RBFs with differingvalues of the shape parameter is given in Figure .2. First, a PS expansion isalways along a given coordinate direction, making them inherently 1-D ob-jects (with expansions in higher dimensions represented by a tensor productwith the respective 1-D basis expansions). By contrast, the scalar argumentr of the RBF does not depend on a coordinate system, but is simply the dis-tance between two nodes that are defined in d-dimensional space. Secondly, PSmethods approximate a function by linear combinations of orthogonal func-tions that become more oscillatory as the degree of the polynomial increases,resulting in very clearly linearly independent functions as can be seen in Fig-ure .2. By contrast, RBFs approximate a function with an expansion of of oneradially symmetric function whose only variation is the location at which itis centered. While increasing the order of the polynomial expansion improvesaccuracy for PS, accuracy of an RBF approximation can be improved by in-creasing number of terms in the exansion or decreasing the shape parameterε [9]. In either case, the RBFs in the expansion become indistinguishable fromone another as can clearly be seen in the case for ε = 0.01 in Figure .2, whichleads to ill-conditioning. It will be shown, however, that for even a moderatenumber of terms in the expansion and values of ε, RBFs outperform othermethods that require much higher spatial and temporal resolution to achievethe same accuracy and which are much more more algorithmically complex.

It should be noted that the Contour-Pade algorithm [10] can be used to bypass

3

the RBF ill-conditioning mentioned above for the case of a fixed (relativelysmall) number of terms and increasingly small values of ε (even ε = 0). Fur-thermore, Fornberg and Piret [11] have recently discovered an algorithm forbypassing the ill-conditioning for RBF interpolation on the surface of thesphere for either the case of fixed ε and increasingly large number of terms, orfixed number of terms and increasingly small ε. We will, however, not pursuethese algorithms in this study.

Although we will be solving hyperbolic PDEs, a good way to introduce theRBF methodology is through interpolation since at each time-step (in theexplicit scheme) the exact spatial derivative operator is applied to the RBFinterpolant to arrive at the derivative of the function values at the node points.As mentioned above, RBFs approximate a function f(x) sampled at some setof N distinct node locations by translates of a single radially symmetric func-tion φ that is dependent on these nodes. For example, given the nodes {xj}N

j=1

and corresponding (scalar) function values {fj}Nj=1, the RBF interpolant s(x)

to the data is defined by

s(x) =N∑

j=1

cj φ(‖x− xj‖), (1)

where the expansion coefficients, {cj}Nj=1, are found by enforcing the colloca-

tion conditions such that the residual is zero at the data locations. This isequivalent to solving the symmetric linear system of equations

φ(‖x1 − x1‖) φ(‖x1 − x2‖) · · · φ(‖x1 − xN‖)φ(‖x2 − x1‖) φ(‖x2 − x2‖) · · · φ(‖x2 − xN‖)

......

. . ....

φ(‖xN − x1‖) φ(‖xN − x2‖) · · · φ(‖xN − xN‖)

︸ ︷︷ ︸

A

c1

c2...

cN

=

f1

f2

...

fN

, (2)

where A is the interpolation matrix. The concept of RBF collocation is illus-trated in Figure .3 and .4 for 1-D and 2-D, respectively. The well-posednessof (2) is discussed in [12, Ch. 12–16] for all the RBFs listed in Figure .1.

To obtain a discrete RBF derivative operator, the exact differential operator isapplied to (1) and then evaluated at the data locations. The discrete operatorwill be spectrally accurate if the RBF is infinitely differentiable as discussedin the next section.

4

3 Node Distribution and Convergence of RBF Interpolants

Since RBFs only depend on the scalar distance between nodes and not on agrid, the basis functions are not linked to any geometry or dimension. In otherwords, there is nothing inherently built into the RBFs to shout out “sphericalgeometry”. In fact, they are unaware of the north or south poles. Studies haveshown that if the shape parameter, ε, is kept fixed throught the domain (aswill be done in the current study—variable shape parameter is needed whenimplementing local mesh refinement [13–15]) best results are achieved witha roughly evenly distributed nodes (cite Iske optimal pt dist.). Since onlya maximum of 20 nodes can be evenly distributed on a sphere, there are amultitude of algorithms to define “even” distribution for larger numbers ofnodes, such as equal partitioned area, convex hull approaches, electrostaticrepulsion, etc. [16]. Although any of these will suffice, we have decided to usean electrostatic repulsion approach since the nodes do not line up along anyvertices or lines, emphasizing the arbitrary node layout and coordinate-freenature of a RBF methodology. The nodes are also readily available for manydifferent numbers [17].

Assuming {xj}Nj=1 are N nodes on the unit sphere, this approach, also known

as the minimum energy (ME) point distribution on the sphere S2, provides a

quasi-uniform distribution on the sphere by maximizing the minimum distancebetween nodes according to the measure

h = maxx∈S2

min1≤i≤N

dist(x, xi), (3)

where dist is the geodesic distance from x to xi. This quantity is referred toas the mesh-norm [18,17] and, geometrically, it represents the radius of thelargest cap that covers the space between any subset of nodes on the sphere.The ME node sets have the property that h decays approximately uniformlylike the inverse of the square root of the number of nodes N , i.e.

h ∼ 1√N.

Thus, they are similar to a uniform discretization of the unit square. In Figure.5, the distribution for 1849 nodes on the unit sphere is displayed.

The mesh norm is also of practical importance since it appears in many proofsof error bounds for RBF interpolation on the sphere (e.g. [18,19]). Indeed,in the context of infinitely smooth RBFs, it is shown in [18] that, providedthe underlying function being interpolated is sufficiently smooth, RBF inter-polants converge (in the L∞ norm) like h−1/2e−c/4h, i.e at a geometric rate,for some constant c > 0 that depends on the RBF. For the ME node sets,convergence will thus proceed like N1/4e−c

√N/4. In the experiments that fol-

5

low we will demonstrate that this error bound seems to also hold for the RBFmethod-of-lines approximation of the two test cases.

4 RBF formulation of the advection operator

Physical phenomena are naturally not associated with any coordinate system.However, scientists impose coordinate systems to model their PDEs. In spher-ical geometries, this results in the spatial operator being singular at the poles.For example, the gradient on the surface of a unit sphere in spherical coordi-nates (λ is longitude, θ is latitude and measured from the equator) is givenby

∇ =1

cos θ

∂

∂λλ +

∂

∂θθ, (4)

which is singular at θ = ±π2

, the north and south pole, respectively. Since

RBFs depend only on the Euclidean distance between nodes, the basis func-tions are thus not associated with any coordinate system and therefore do not“feel” the effects of the geometry of the domain. As a result, RBFs do notrecognize the singularities naturally inherent in the coordinate system and allremnants of such singularities vanish when the methodology is implemented.This allows the scientist to directly connect the physics to the numerics. Sincethe test cases considered both use the advection operator, we will now see howit becomes nonsingular in spherical coordinates when formulated with RBFs.

Let x = (x, y, z), xj = (xj , yj, zj) be two points on the surface of the unitsphere and (λ, θ), (λj, θj) the corresponding spherical coordinates, i.e.

x = cosλ cos θ,

y = sinλ cos θ, (5)

z = sin θ.

(Note that this differs from traditional spherical coordinates in that we mea-sure θ from the equator rather than from the north pole). Then, the Euclideandistance from x to xj is

r = ‖x− xj‖ =√

(x− xj)2 + (y − yj)2 + (z − zj)2

=√

2(1 − cos θ cos θj cos(λ− λj) − sin θ sin θj).

It is important to note that the distances are not great circle arcs measuredalong the surface but are the Euclidean distance measured straight throughthe sphere.

Let φj(r) = φ(‖x− xj‖) be an RBF centered at xj , then using the chain rule,the partial derivatives of the RBF φj(r) with respect to λ and θ are given by

6

∂

∂λφj(r)= cos θ cos θj sin(λ− λj)

(

1

r

dφj

dr

)

, (6)

∂

∂θφj(r)= (sin θ cos θj cos(λ− λj) − cos θ sin θj)

(

1

r

dφj

dr

)

(7)

Note that (6) and (7) are well defined for r = 0 (i.e. x = xj since φj(r) is C∞

and radially symmetric about xj. Inserting (6) and (7) into (4), we have theaction of the gradient operator on the RBF scalar function:

∇φj(r) =[

cos θj sin(λ− λj)λ+

(cos θj cos(λ− λj) sin θ − cos θ sin θj)θ](

1

r

dφj

dr

)

. (8)

As noted at the beginning of this section, the gradient has a singularity inthe λ direction at the poles unless the derivative (with respect to λ) of theunderlying function also vanishes at the poles. We see from (8) that this isexactly what happens when using an RBF.

Now, we have all the components that are necessary to build the action of theadvection operator on an RBF representation of a geophysical field. Supposewe want to advect some scalar quantity, say a given height field h(λ, θ), wherethe components of the advecting wind U are given by U = u(λ, θ)λ+v(λ, θ)θ.Let {xj}N

j=1 = {(λj, θj)}Nj=1 be the node locations where h(λ, θ) is known. We

first represent h(λ, θ) as an RBF expansion given by

h(λ, θ) =N∑

j=1

cjφj(r) (9)

where φj(r) is again the RBF centered at the node xj = (λj , θj). We thenapply the exact differential operator

(U · ∇) =u(λ, θ)

cos θ

∂

∂λλ + v(λ, θ)

∂

∂θθ

to (9) and evaluate it at the node locations:

(U · ∇)h(λi, θi)=N∑

j=1

cj [(U · ∇)φj(r)]|(λ,θ)=(λi,θi)︸ ︷︷ ︸

Components of B

(i = 1, . . . , N)

=Bc

= (BA−1)h

=DNh,

7

where h contains the N discrete values of h at the nodes, c contains the Ndiscrete expansion coefficients and is formally given by c = A−1h, where A−1

is the inverse of the RBF interpolation matrix defined in (2). The discreteoperator DN = BA−1 is referred to as the RBF differentiation matrix, andthe components of the matrix B are explicitly given by

Bi,j = {u(λi, θi) cos θj sin(λi − λj)+

v(λi, θi)[sin θi cos θj sin(λi − λj) + cos θi cos θj ]}(

1

r

dφ

dr

)∣∣∣∣∣r=‖xi−xj‖

,

(10)

for i, j = 1, . . . , N . Notice that this operator (10) is no where singular on thesphere, remembering of course that the velocity field is completely smooth.Although the computation of DN is O(N3) operations, it is a pre-processingstep that needs to be done only once.

5 Numerical Test Case 1: Solid Body Rotation

The first test case (solid body rotation or passive advection), using the setupgiven in [1], simulates the advection of a height field, h(λ, θ), over the suface ofa sphere at an angle α relative to the standard longitude-latitude (λ-θ) grid asmeasured between the equatorial lines (see Figure .6). The PDE to be solvedis the advection equation, which in spherical coordinates is given by

∂h

∂t+

u

a cos θ

∂h

∂λ+v

a

∂h

∂θ= 0, (11)

with the advecting wind being

u= u0(cos θ cosα + sin θ cosλ sinα), (12)

v=−u0 sinλ sinα, (13)

where a is the radius of the earth, 6.37122 ·106m and u0 = 2πa/(12 days = 288hours)

We will consider two initial conditions (which are also the solution for alltime) that are to be advected without distortion by the above steady wind(12-13). As illustrated in Figure .7, the first (a) is the classical test case in theliterature [1], a cosine bell profile that is C1 and centered at (λc, θc):

h(λ, θ) =

h0

2

[

1 + cos(

π ρR

)]

ρ < R,

0 ρ ≥ R,(14)

8

where h0 = 1000m, R = a/3 and ρ = a arccos(sin θc sin θ + cos θc cos θ cos(λ−λc)). The second (b) is an exceptionally steep Gaussian profile that is C∞:

h(λ, θ) = h0e−(2.25ρ/R)2 , (15)

where h0, ρ, and R are the same as (a). This profile is used to demonstratethat the RBF method is indeed spectral.

The center of the bell is initially taken to be at the equator, (λc, θc) = (0, 0).In testing previously used methods such as spherical harmonics, double fourierseries, or spectral element methods, the object is rotated at various angles αwith regard to the polar axis of the spherical coordinate system, with rotationover the poles being the most severe test case. This is to see how the method-ology handles the “pole singularity” inherent in a spherical coordinate system.Since RBFs and the node layout are free of any coordinate system, the erroris invariant to the angle of rotation (see the Appendix for a rigorous proof).As a result, the choice of α is irrelevant. We choose α = π/2 (i.e. flow rightover the poles) only for comparison reasons to other methods, since it is theangle for which the error is most quoted in the literature.

The method-of-lines RBF formulation for (11) is given by

∂h

∂t+DNh = 0, (16)

where the differentiation matrix DN represents the discretized advection oper-ator derived in the previous section. The standard fourth-order Runge-Kuttascheme (RK4) is used to advance the solution in time.

5.1 Comparative Results for Solid Body Rotation

We first consider the cosine bell test case using GA RBFs (c.f. Figure .1) withε = 8.2, N = 4096 ME nodes, and a time-step of ∆t = 30 minutes. The choicefor these values is to obtain error norm results that are comparable to othermethods used in climate modeling, and yet point out the strength of the RBFmethod in terms of time stability and spatial resolution requirements.

Figure 8(a) displays a surface plot of the numerical solution after one fullrevolution around the sphere (t = 12 days). Comparing this to the true so-lution (also the initial condition) in Figure 7(a), we see that the numericalRBF solution is visibly identical to the exact solution. Figure 8(b) displaysan orthographic projection of the error in the RBF solution. The figure showsthat the error is dominate in the region where the bell transitions to zero andthat there are is no dispersive wave-train flowing the bell. At the conclusion

9

of this revolution, the ℓ2 error is 6.18 · 10−3 and the ℓ∞ error is 2.27 · 10−3.Figure .9 shows the error as a function of time for the 12 day simulation.

Table .1 compares the performance of commonly used spectral methods on thesphere: spherical harmonics (SH), double Fourier (DF), and spectral elementsimplemented with a discontinous Galerkin approach (DGSE) on the elementboundaries (an advance to classic spectral elements on a sphere). A commonbasis of comparison for all the methods considered that could be found inthe literature was the specifications (spatial and temporal) needed for eachmethod to achieve an ℓ2 error of approximately 0.005 [20–23]. Studying thetable there are five points that clearly stand out:

(1) The number of nodes needed for the RBF method is approximately halfthat needed for the DGSE method and 8 times less than needed for bothSH and DF.

(2) The time-step taken for the RBF method is 5 times larger than the DGSEand 20 times larger than that taken for both SH and DF.

(3) The algorithmic complexity of implementing an RBF method is essen-tially trivial in comparison to other methods. Our code from start tofinish is exactly 37 lines of MATLAB, using NO built-in MATLAB rou-tines except Gaussian elimination (i.e. mldivide). The beauty, however,is that the complexity of the code would not change if the geometry ischanged or the dimension of the problem to be solved is increased, whichcan not be said for any other method.

(4) Only the DGSE and RBF method allow for local mesh refinement [13–15].(5) The RBF method has the highest computational cost, requiring a matrix-

vector multiply per time-step, where the matrix is full. However, fastalgorithms are available that have the potential reducing this cost toO(N logN) or possibly O(N) (c.f. [24–26]).

In order to better understand these results, we will perform a convergencestudy with regard to: (a) h-refinement, the number of nodes used N (remem-bering that 1/

√N is proportional to the spacing of the node distributions)

and (b) ε-refinement, the shape parameter of the RBF. Next, we will do astability and eigenvalue study with regard to these parameters, illustratingwhy such large time-steps can be taken. Our objective is to show trends inconvergence and stability which can be illustrated with a much lower numberof nodes. We choose N = 1849, the same number of degrees of freedom as theNational Center for Atmospheric Research (NCAR) T42 spherical harmoniccommunity climate model (i.e. 42 spherical harmonics), which is much quotedin the literature for comparison purposes [20–22].

10

5.2 Convergence Study

5.2.1 h-refinement

Figure 10(a) shows the convergence rates in both the ℓ∞ and ℓ2 norm for thecosine bell test case on a log-log plot. A time-step of ∆t = 30 minutes was usedfor each node set in the RK4 integration so that spatial error dominate. Similarto all spectral methods, the RBF methodology results in spectral convergenceif the initial condition is C∞. However, in this case the cosine bell is only aC1 function, which results in low-order algebraic convergence as illustrated inthe figure by the dashed lines.

To demonstrate the the RBF method is indeed spectral, we instead advectedthe steep Gaussian bell (15) (c.f. Figure 7(b)) that is similar to the cosine bell,but is C∞. Figure 10(b) (a log-linear plot) shows that indeed the convergencerates are spectral in both the ℓ∞ and ℓ2 norm for this case. Time steps of 30,30, 10, 10, and 5 minutes were used for the respective ME node sets N = 529,1024, 2025, 3136, and 4096 in the RK4 time integration so that spatial errorsdominate.

It should be noted that there is nothing special about the node layout used.Similar error norms would be achieved if we were using equal-area node dis-tributions or those laid out according to the golden ratio, as in a sunflowerpattern. The only requirement is that the nodes are roughly equally distributedin some sense on the surface of the sphere as close clustering can lead to ill-conditioning.

5.2.2 ε-refinement

[GRADY: REWRITE THIS SECTION] As noted in Section 2, accuracy greatlyimproves in the limit as ε → 0, i.e. as in the limit of flat RBFs. Of course,in this limit the RBF interpolation matrix becomes severely ill-conditioned.However, we are able to obtain very good results for even moderate values of ε.As an example, the convergence results illustrated in Figures .10 for N = 1849were obtained with ε = 6 for the cosine bell test case and ε = 3 for the Gaus-sian bell test case. Figures 11(a) and 11(b) show the error as function of εwith N = 1849 and a time step ∆t = 50 minutes (in the next subsection wewill see why such very large time-steps can be taken). The fact that the errorincreases after an optimal value of ε, which was used in each test case, is mostlikely due to ill-conditioning setting in. Currently, the only way to determinethe optimal value of epsilon to use is experimentation. As mentioned earlier,Fornberg and Piret in an upcoming paper have developed an algorithm thatallows for stable computation of the RBF interpolation matrix in the limitas ε → 0 for nodes scattered on a the surface of a sphere. Theoretically, this

11

should allow for even higher numerical accuracy to be achieved while keepingthe number of nodes used constant.

5.3 Eigenvalue Stability

Since the RBF differentiation matrices are not normal, classical eigenvaluestability theory is theoretically insufficient. However, in practice, as with manypseudospectral methods, it is still a very good predictor for the maximumstable time-step as we shall see below.

Let us first examine why such high accuracy can be achieved with a largetime step compared to other methods as reported in the previous section. Ifthe eigenvalues of the differentiation matrix DN for N = 1849 are plottedas in Figure .12, we see that they lie exactly on the imaginary axis. Notone eigenvalue lies in the right half plane, which would eventually lead tonurmerical instability, nor in the left half plane, leading to dissipation. Thisresult follows from the fact thatDN = BA−1, the product of an anti-symmtericB and a positive definite matrix A. Although the product of the two is notantisymmetric, it preserves the property of antisymmetry that all eigenvalueslie on the imaginary axis [27].

The size of the maximum time step to maintain stability depends on theeigenvalues of DN fitting within the stability domain of the RK4 method.The maximum eigenvalue of DN , in turn, depends on the two parameters Nand ε as shown in Figure .15. As the accuracy of the method increases, ei-ther by increasing N with fixed ε (h-refinement) or decreasing ε with fixed N(ε-refinement), the maximum eigenvalue will increase, implying that the max-imum time step that can be taken must decrease. Note, however, that changesin these two variables have different impacts on the maximum eigenvalue (orallowable time step). The maximum eigenvalue linearly increases with spatialresolution up/down the imaginary axis, which is the classical result for linearhyperbolic PDEs. In contrast, decreasing ε changes the maximum eigenvaluein a manner that is reminiscent of what happens to finite difference approxima-tions to derivative operators as the order increases for fixed N . For example,increasing the accuracy of the first derivative operator from second order cen-tered finite differences to Fourier PS, increases the maximum eigenvalue by π(p.41-42 [28]). In a similar manner, as shown in Figure .15b, the maximumeigenvalue increases by 4 as ε varies from 9 to zero. However, the reader shouldbe reminded that the RBF method is spectral for all values of ε. It is just thatas ε decreases to some (typically non-zero) optimal value, the accuracy im-proves (e.g. in the case of the cosine bell it is ε=6, for the Gaussian bell itis ε=3). Furthermore, even though RBFs reproduce classical PS methods inlimit of ε = 0 [8,11,29], it is for nonzero values of the parameter that RBFs

12

outperform PS as was shown in the comparative study of the previous section.

The results shown in Figure .15a & b are also given in Table .2 and Table.3 that translate the maximum eigenvalue into the maximum allowable timestep for the RK4 integrator (i.e such that the eigenvalue falls with the RK4stability domain). For the case N = 1849 or ε = 6, the theory predicts thatthe maximum time step is 208 minutes (3 hours 28 minutes). To test this,we plot the time evolution of the ℓ∞ error for the cosine bell test case usingε = 6 and N = 1849 with a time-step of 208 minutes and 209 minutes in theRK4 integrator. After 36 days of integration (3 full revolutions around thesphere), the test run using a 208 minute time step is completely stable whilethe use of a 209 minute time step has caused numerical instability to set in atapproximately 20 days, verifying the predicted results from classical eigenvaluestability theory. However, for best computational efficiency one should usetime steps so that temporal and spatial errors match. Such is illustrated inFigure .14 which shows that for ε = 6 and N = 1849 the ℓ∞ error of thesolution after a 12 day single revolution around the earth is steady up to a50 minute time step. Larger time steps, although stable as just demonstrated,will cause discretization errors in time to dominate over those in space. Forthe comparative study of the previous section with N = 4096, this breakingpoint was a 30 minute time step.

6 Numerical Test Case 2: Deformational flow

The second test case involves no translational motion, but instead an angu-lar velocity field spins up the initial condition, resulting in two diametricallyopposed vortices. This test for idealized cyclogenesis was first described byNair et al. [2]. Let (λ′, θ′) be the rotated coordinate system with north poleat (λp, θp) with respect to the regular spherical coordinate system (λ, θ). Inthese rotated coordinates, the PDE to simulate is given by

∂h

∂t+

u′

cos θ′∂h

∂λ′= 0 (17)

where u′ is the tangential velocity field in the rotated coordinates and is givenby

u′ = ω(θ′) cos θ′.

Note that since the vortices are steady there is no velocity in the normaldirection v′ of the rotated coordinate system. The angular velocity ω′ of thevortex field is given by

ω(θ′) =

3√

32ρ(θ′)

sech2(ρ(θ′)) tanh(ρ(θ′)) if ρ(θ′) 6= 0

0 if ρ(θ′) = 0,

13

where ρ(θ′) = ρ0 cos θ′ is the radial distance of the vortex. The exact solutionat time t is given as

h(λ′, θ′, t) = 1 − tanh

(

ρ(θ′)

γsin (λ′ − ω(θ′)t)

)

, (18)

where γ is a parameter defining the characteristic width of the frontal zone.Note that (18) is independent of any units.

Many, if not all, of the current methods for simulating this vortex flow (e.g.[2,23,30]) require transforming (17) to regular (λ, θ) spherical coordinate sys-tem since these methods are dependent on this orientation. Upon transforma-tion (17) becomes

∂h

∂t+

u

cos θ

∂h

∂λ+ v

∂h

∂θ= 0,

where

u=ω(θ′)(sin θp cos θ + cos θp cos(λ− λp) sin θ),

v=ω(θ′) cos θp sin(λ− λp).

This form is much more complicated than the original since we have nowintroduced flow in the normal direction to the (λ, θ) grid and a non-physicalsingularity at the poles.

Since the RBF method is completely independent of how the underlying co-ordinate system is oriented, we can simply simulate (17) in its rotated (λ′, θ′)form. The method-of-lines RBF formulation of (17) is given by

∂h

∂t= −DNh, (19)

where the differentiation matrix DN = BA−1, with A given by (2) and

Bi,j = ω(θ′i) cos θ′i cos θ′j sin(λ′i − λ′j)

(

1

r

dφ

dr

)∣∣∣∣∣r=‖xi−xj‖

, i, j = 1, . . . , N.

Here (λ′i, θ′i), i = 1, 2, . . . , N , are the original nodes of the standard (λ, θ)

discretization rotated to the new coordinate system (λ′, θ′):

λ′i = arctan

(

sin(λi − λp)

sinλp cos(λi − λp) − cos θp tan θi

)

, (20)

θ′i = arcsin (sin θi sin θp + cos θ cos θp cos(λi − λp)) . (21)

As in the solid body rotation problem, the RBF formulation is completely freeof any coordinate singularities.

14

6.1 Results for Deformational Flow

The analysis for deformational flow will be similar to that of solid body ro-tation. We will perform a convergence and eigenvalue/stability study to un-derstand the high performance of the RBF method. However, since this is anewer test that is performed in the numerical climate modeling community,the only results in the spectral methods literature that have been publishedis for SE methods and thus the comparison is limited to this method.

6.1.1 Convergence study

Figures .16a and .16b shows the initial condition and Figures .16c and .16dthe respective results after 6 days of integration for N = 3136 and ε = 6.8.The error in the ℓ1 and 2 norms are 0.0002 and 0.0006, respectively GRADYNEW RESULTS. To achieve an error of 0.0001 and 0.0006 in the ℓ1 and ℓ2norms, respectively, the DGSE method [23] required 3456 nodes SEE HOWCOMPARES. Figure XXX (a log-linear plot) shows the convergence rates ofthe RBF method are spectral as demonstrated in the ℓ1 and ℓ2 norm.

6.1.2 Eigenvalue analysis

Since DN is a non-normal matrix, the eigenvalue stability of the semi-discreteRBF approximation (??) will be governed by eigenvalues of the matrix −ω(θ)DN .Figure ??a displays these eigenvalues for the N = 3136 minimum energy nodeson the sphere and the GA radial function. As we can see, the eigenvalue spec-trum is no longer purely imaginary. Plotting the eigenvalues with the RK4stability domain, as done in Figure ??b, shows that:

(1) Even with a 30 minute time-step, the spectrum is far from RK4’s imagi-nary axis limitation, indicating that larger time-steps can easily be taken.

(2) There are eigenvalues with small real parts that lie outside RK4’s stabilitydomain, indicating that stability may be an issue for long-time integrationof the system.

The first remark can be addressed by Figure .17. The second remark can beaddressed by Figure 20(a) that shows how the relative ℓ1 and ℓ2 errors growwith time for ∆t = 30 and N = 3136 nodes. We can see that the error inthe solution does not strongly increase until around 45 days when instabilitybegins to set in. If the solution at 45 days is graphed as in Figure 20(b),we see that the solution is very highly oscillatory, implying that there arereally no adverse affects of the small real parts of the eigenvalue spectrumuntil the solution features have become too fine to be resolvable with thegiven spatial discretization. Close examination of the figure shows that there

15

are less than the necessary theoretical limit of two nodes per wavelength. Asa result, standard high-frequency filtering implemented about once a monthwill keep accuracy for modes that theoretically can be resolved, and thus fullyrestore long-term stability.

7 Summary

The main goal of this paper is to illustrate the effectiveness and performance ofthe RBF methodology for solving purely hyperbolic PDEs on a sphere, usingtest cases that are standard in the numerical climate modeling communitygiven by solid body rotation and deformational flow. The detailed results ofthe previous sections can be simply summed up as follows:

(1) For these test cases, the RBF methodology outperforms all currently usedspectral methods in terms of the number of nodes and time-step neededas well as algorithmic simplicity to achieve a given accuracy.(a) In either test case, the code is very short and simple - less than 40

lines of MATLAB, using no built-in MATLAB routines.(b) Eigenvalues lie on imaginary axis (solid body) or have very small real

parts (deformational flow) which gives the ability to take exception-ally large time-steps.

(c) The method is spectrally accurate for smooth initial conditions, re-quiring a much low number of nodes compared to other commonlyused spectral methods on a sphere.

(2) Differentiation matrices are entirely free of any ’pole’ singularities andinvariant of the orientation of the original coordinate system.

(3) Connects mathematics directly to the physics without the interference ofa surface-based grid

Furthermore, algorithmic complexity does not increase with the dimension ofthe problem, since RBFs only depend on the Euclidean distance between nodeswhich is always a scalar independent of dimension (e.g. if linear advection wereposed in 3D, the code used in 2D for the solid body rotation test case wouldnot change in length or complexity),These results illustrate that RBFs canprovide a promising new approach to modeling in spherical geometries.

Rotational invariance of DN for solid body rotation

Although we used a spherical coordinate system which was measured fromthe equator, we will now show how the RBF derivative matrix DN for thespatial derivatives of (11) is rotationally invariant, i.e. completely independentof how the original spherical coordinate system was oriented in space. Thus,when implementing the advective operator with wind (12)–(13) using RBFs

16

in spherical coordinates, the result is not only singularity free, but is alsocompletely independent of the spherical coordinate orientation.

An equivalent demonstration is to keep the spherical coordinate system fixed,and instead turn (in unison) both the axis of solid body rotation and all thepoint locations by an angle α. Let Ci,j be the expression inside the curlybrackets for Bi,j in (10) with u and v given by (12) and (13), respectively. IfCi,j remains invariant with α, then we are finished.

Each entry Ci,j involves only two node points, located at (λi, θi) and (λj, θj).After the axis of solid body rotation is turned by an angle α, we obtain thenew coordinates (λ′, θ′) for a point (λ, θ) by the transformation

sin λ′ cos θ′ =sinλ cos θ cosα + sin θ sinα , (.1)

sin θ′ =− sin λ cos θ sinα + sin θ cosα . (.2)

The goal is to show that

Ci,j = cosα cos θ′i cos θ′j sin(λ′i − λ′j)+

sinα[cos θ′i cosλ′i sin θ′j − cos θ′j cosλ′j sin θ′i] (.3)

depends only on (λi, θi) and (λj , θj) (i.e. that it is independent of α).

The analysis turns out to be fairly simple if we consider the node points inCartesian coordinates. So, we let the points (λ′i, θ

′i) and (λ′j , θ

′j) corresponds

to the points x′i = (x′i, y′i, z

′i) and x′j = (x′j , y

′j, z

′j), respectively. Using the

relationship between Cartesian and spherical coordinates (5), we can rewrite(.3) as follows:

Ci,j = (x′jy′i − x′iy

′j) cosα + (x′iz

′j − x′jz

′i) sinα (.4)

Now, the counterpart to (.1) and (.2) in Cartesian coordinates is given by thetransformation

x′

y′

z′

=

1 0 0

0 cosα sinα

0 − sinα cosα

x

y

z

.

Using this transformation we can relate the rotated points in (.4) to the orig-inal points. After some algebra, we find the simple expression

Ci,j = xjyi − xiyj

= cos θi cos θj sin(λi − λj) .

Thus, Ci,j is completely independent of the angle α, which shows the RBFmethodology for (11) is completely independent of how the original sphericalcoordinate system was oriented in space.

17

References

[1] D. L. Williamson, J. B. Drake, J. J. Hack, R. Jakob, P. N. Swarztrauber, Astandard test set for numerical approximations to the shallow water equationsin spherical geometry, J. Comput. Phys. 102 (1992) 211–224.

[2] R. D. Nair, J. Cote, A. Staniforth, Cascade interpolation for semi-Lagrangianadvection over the sphere, Quart. J. Roy. Meteor. Soc. 125 (1999) 1445–1468.

[3] R. L. Hardy, Multiquadric equations of topograpy and other irregular surfaces,J. Geophy. Res. 76 (1971) 1905–1915.

[4] A. Haar, Die minkowskische geometrie und die ann an stetige funktionen, Math.Ann. 18 (1918) 294–311.

[5] B. Fornberg, N. Flyer, S. Hovde, C. Piret, Localization properties of RBFexpansions for cardinal interpolation. i. equispaced nodes, Adv. Comp. Math.Submitted (2006).

[6] W. R. Madych, S. A. Nelson, Bounds on multivariate polynomials andexponential error estimates for multiquadric interpolation, J. Approx. Theory70 (1992) 94–114.

[7] J. Yoon, Spectral approximation orders of radial basis function interpolationon the Sobolev space, SIAM J. Math. Anal. 23 (4) (2001) 946–958.

[8] T. A. Driscoll, B. Fornberg, Interpolation in the limit of increasingly flat radialbasis functions, Comput. Math. Appl. 43 (2002) 413–422.

[9] E. Larsson, B. Fornberg, A numerical study of some radial basis function basedsolution methods for elliptic PDEs, Comput. Math. Appl. 46 (2003) 891–902.

[10] B. Fornberg, G. Wright, Stable computation of multiquadric interpolants for allvalues of the shape parameter, Comput. Math. Appl. 48 (2004) 853–867.

[11] B. Fornberg, C. Piret, A stable algorithm for flat radial basis functions on asphere To be submitted (2006).

[12] E. W. Cheney, W. A. Light, A Course in Approximation Theory, Brooks/Cole,New York, 2000.

[13] B. Fornberg, J. Zuev, The Runge phenomenon and spatially variable shapeparameters in RBF interpolation, Comput. Math. Appl. Submitted (2006).

[14] T. A. Driscoll, A. Heryundono, Adaptive residual subsampling methods forradial basis function interpolation and collocation problems, Comput. Math.Appl. Submitted (2006).

[15] J. Wertz, E. J. Kansa, L. Ling, The role of multiquadric shape parameters insolving elliptic partial differential equations, Comput. Math. Appl. Submitted(2006).

18

[16] A. Sherwood, How can I arrange N points evenly on a sphere?,http://www.ogre.nu/sphere.htm.

[17] R. S. Womersley, I. H. Sloan, Interpolation and cubature on the sphere,http://web.maths.unsw.edu.au/~rsw/Sphere/.

[18] K. Jetter, J. Stockler, J. D. Ward, Error estimates for scattered datainterpolation on spheres, Math. Comput. 68 (1999) 733–747.

[19] S. Hubbert, T. Morton, Lp-error estimates for radial basis function interpolationon the sphere, J. Approx. Theory 129 (2004) 58–77.

[20] R. Jakob-Chien, J. J. Hack, D. L. Williamson, Spectral transform solutions tothe shallow water test set, J. Comp. Phys. 119 (1995) 164–187.

[21] M. Taylor, J. Tribbia, M. Iskandarani, The spectral element method for theshallow water equations on the sphere, J. Comp. Phys. 130 (1997) 92–108.

[22] W. F. Spotz, M. A. Taylor, P. N. Swarztrauber, Fast shallow water equationsolvers in latitude-longitude coordinates, J. Comp. Phys. 145 (1998) 432–444.

[23] R. D. Nair, S. J. Thomas, R. D. Loft, A discontinuous Galerkin transport schemeon the cubed-sphere, Mon. Wea. Rev. 133 (2005) 814–828.

[24] R. K. Beatson, J. B. Cherrie, C. T. Mouat, Fast fitting of radial basis functions:Methods based on preconditioned GMRES iteration, Adv. Comput. Math. 11(1999) 253–270.

[25] J. B. Cherrie, R. K. Beatson, G. N. Newsam, Fast evaluation of radial basisfunctions: methods for generalized multiquadrics Rn, SIAM J. Sci. Comput.23 (5) (2002) 1549–1571.

[26] G. Roussos, B. J. C. Baxter, Rapid evaluation of radial basis functions, J.Comput. Appl. Math. 180 (2005) 51–70.

[27] R. Platte, T. A. Driscoll, Eigenvalue stability of radial basis functiondiscretizations for time-dependent problems, Comp. Math. Appl. to appear.

[28] B. Fornberg, A Practical Guide to Pseudospectral Methods, CambridgeUniversity Press, Cambridge, 1996.

[29] B. Fornberg, G. Wright, E. Larsson, Some observations regarding interpolantsin the limit of flat radial basis functions, Comput. Math. Appl. 47 (2004) 37–55.

[30] R. D. Nair, B. Machenhauer, The mass-conservative cell-integrated semi-Lagrangian advection scheme on the sphere, Mon. Wea. Rev. 130 (2002) 649–667.

19

Piecewise smooth

0 0.5 1 1.5 20

2

4

6

8

10

RC: |r|3

0 0.5 1 1.5 2−1

0

1

2

3

TPS: r2 log |r|

Infinitely smooth

0 0.5 1 1.5 20

1

2

3

MQ:√

1 + (εr)2

0 0.5 1 1.5 20

0.5

1

1.5

GA: e−(εr)2

0 0.5 1 1.5 20

0.5

1

1.5

IMQ: (1 + (εr)2)−1

2

0 0.5 1 1.5 20

0.5

1

1.5

IQ: (1 + (εr)2)−1

Fig. .1. Examples of commonly used RBFs φ(r): RC = radial cubic, TPS = thinplate spline, MQ = multiquadric, GA = Gaussian, IMQ = inverse multiquadric, IQ= inverse quadratic. The variable ε in the infinitely smooth RBFs is known as theshape parameter.

20

Chebyshev polynomials GA RBFs

−1 −0.5 0 0.5 1−1

−0.5

0

0.5

1

T0(x)

−1 −0.5 0 0.5 1

0

0.2

0.4

0.6

0.8

1

ε = 10

−1 −0.5 0 0.5 1−1

−0.5

0

0.5

1

T4(x)

−1 −0.5 0 0.5 1

0

0.2

0.4

0.6

0.8

1

ε = 2

−1 −0.5 0 0.5 1−1

−0.5

0

0.5

1

T10(x)

−1 −0.5 0 0.5 1

0

0.2

0.4

0.6

0.8

1

ε = 1/2

−1 −0.5 0 0.5 1−1

−0.5

0

0.5

1

T20(x)

−1 −0.5 0 0.5 1

0

0.2

0.4

0.6

0.8

1

ε = 1/100

Fig. .2. Comparison of 1-D Chebyshev PS basis, Tk(x), and Gaussian (GA) RBFbasis, φ(r) = e−(εr)2 (for this case r = x).

21

f1

f2

f3 f

4fk

x1

x2 x

3x

4x

k

Fig. .3. Example of RBF interpolation in 1-D.

22

Fig. .4. Example of RBF interpolation in 2-D.

23

−1 −0.5 0 0.5 1

−0.8

−0.6

−0.4

−0.2

0

0.2

0.4

0.6

0.8

Fig. .5. Minimum energy (ME) point distribution on the sphere, N = 1849.

24

Fig. .6. Solid body rotation over the surface of a sphere at at an angle α (darkersolid lines) relative to the standard longitude-latitude grid (lighter solid lines).

25

(a)

(b)

Fig. .7. (a) Cosine bell and (b) Gaussian bell initial conditions and exact solutionafter 1 revolution (t = 12 days) on an ‘unrolled’ sphere. Solid black circles mark theN = 4096 ME node points.

26

(a)

(b)

Fig. .8. (a) Numerical solution of the cosine bell test case after 1 revolution(t = 12 days) on an ‘unrolled’ sphere (c.f. Figure 7(a)). Solid black circles markthe N = 4096 ME node points. (b) Orthographic projection of the error (exact -numerical) for part (a). Solid contour lines mark the location of the exact solutionand are plotted with a uniform interval of 100 m. Results are for the GA RBF andRK4 for the time integration with a 30 minute time-step (spatial errors dominate).

27

0 2 4 6 8 10 12

2

4

6

8

10x 10

−3

Time (days)

Nor

mal

ized

err

or

Cos ine bell tes t

ℓ2

ℓ∞

Fig. .9. The ℓ∞ and ℓ2 error for the cosine bell test case as a function of time forthe N = 4096 ME node set. Advection of the height field is directly over the poles.Results for the GA RBF, and ∆t = 30 minutes in the RK4 integration.

28

23 32 45 56 6410

−3

10−2

10−1

√N

Norm

alize

der

ror

Cosine bell test, t = 12 days

ℓ2

ℓ∞

(a)

20 30 40 50 6010

−7

10−6

10−5

10−4

10−3

10−2

10−1

√N

Norm

alize

der

ror

Gaussian bell test, t = 12 days

ℓ2

ℓ∞

(b)

Fig. .10. (a) The ℓ∞ and ℓ2 error at t = 12 days for the cosine bell test case as afunction of the spacing of the ME node sets (h ∼ N−1/2) on a log-log plot. Thedashed line near the ℓ2 error is a plot of N−3/2, showing that the convergence iscubic. (b) Same as (a) but for the Gaussian test case on a log-linear plot. Resultsare for the GA RBF and RK4 for the time integration.

29

2 4 6 8

10−2

10−1

ǫ

Nor

mal

ized

err

or

Cos ine bell tes t, t=12 days

ℓ2ℓ∞

(a)

2 4 6 810

−4

10−3

10−2

ε

Nor

mal

ized

err

or

Gaussian bell test, t=12 days

ℓ2

ℓ∞

(b)

Fig. .11. Normalized ℓ∞ and ℓ2 error measured after one full revolution (t = 12 days)as a function of ε for the (a) cosine bell and (b) Gaussian bell initial conditions.Results are for N = 1849 ME node set, the GA RBF, and ∆t = 50 minutes in theRK4 integration.

30

−1 0 1

x 10−8

−25

−20

−15

−10

−5

0

5

10

15

20

25

Re(λ)

Im(λ

)

Fig. .12. Eigenvalues of the RBF differentation matrix for the solid body rotationtest case. Results are for N = 1849 ME node set and the GA RBF with ε = 6.

31

0 5 10 15 20 25 30 35

10−2

10−1

100

101

Time (days)

Nor

mal

ized

err

or

Cosine bell test

time-s tep =208 min.

time-s tep =209 min.

Fig. .13. The ℓ∞ error for the cosine bell test for 0 ≤ t ≤ 36 days (3 full revolutions)using a stable and unstable time-step with RK4 (see Table .2). Results are for theN = 1849 ME node set and the GA RBF with ε = 6.

32

50 100 150 200

10−2

10−1

Time-step ∆t (minutes)

Norm

alize

der

ror

Cosine bell test, t = 12 days

ℓ∞

Fig. .14. The ℓ∞ error at t = 12 days for the cosine bell test as a function of thetime-step used in the RK4 integration. Results are for the N = 1849 ME node setand the GA RBF with ε = 6.

33

20 30 40 50 6010

15

20

25

30

35

√N

Max

. im

agin

ary

eige

nval

ue

(a)

0 2 4 6 8 1017

17.5

18

18.5

19

19.5

20

20.5

21

21.5

ε

Max

. im

agin

ary

eige

nval

ue

(b)

Fig. .15. Maximum eigenvalue of the differentiation matrix for the solid body rota-tion test as a function of (a) the spacing of the ME node sets (h ∼ N−1/2) and (b)ε where N = 1849.

34

(a) (b)

(c) (d)

(e) (f)

Fig. .16. (a) and (b) initial condition; (c) and (d) solution of deformational flow testafter 6 days of integration.

35

0.05 0.1 0.15 0.2 0.25 0.3

10−4

10−3

10−2

10−1

Time-step ∆t

Norm

alize

der

ror

Deform. flow test, t = 3

ℓ2

ℓ1

Fig. .17. The ℓ1 and ℓ2 error at t = 3 for the deformational flow test as a function ofthe time-step used in the RK4 integration. Results are for the N = 3136 ME nodeset and the GA RBF with ε = 6.45.

36

20 30 40 50 6010

−6

10−5

10−4

10−3

10−2

√N

Norm

alize

der

ror

Deform. flow test, t = 3

ℓ2

ℓ1

Fig. .18. The ℓ1 and ℓ2 error at t = 3 time units for the deformational flow testas a function of the spacing of the ME node sets (h ∼ N−1/2) on a log-linear plot.Results are for the GA RBF and RK4 for the time integration with ∆t = 1/10(spatial errors dominate).

37

−3 −2.5 −2 −1.5 −1 −0.5 0 0.5−3

−2

−1

0

1

2

3

∆t = 3/14

Re(λ)

Im(λ

)

(a)

−3 −2.5 −2 −1.5 −1 −0.5 0 0.5−3

−2

−1

0

1

2

3

∆t = 3/15

Re(λ)Im

(λ)

(b)

−3 −2.5 −2 −1.5 −1 −0.5 0 0.5−3

−2

−1

0

1

2

3

∆t = 3/18

Re(λ)

Im(λ

)

(c)

Fig. .19. Stability domain of RK4 together with the eigenvalues of the differentiationmatrix for the deformational flow test scaled by (a) ∆t = 3/14, (b) ∆t = 3/15, and(c) ∆t = 3/18.

38

0 20 40 60 80

10−6

10−4

10−2

100

102

104

Time

Norm

alize

der

ror

∆t = 3/14

∆t = 3/18

∆t = 3/15

ℓ1

ℓ2

(a)

(b)

Fig. .20. (a) ℓ1 and ℓ2 error for the RBF approximation of the deformational flowtest as a function of time for different ∆t. (b) Exact solution of the deformationalflow test at t = 35, the time at which the RBF approximation becomes unstableusing a time step of ∆t = 3/18. Results are for N = 3136 ME node set and the GARBF.

39

MethodCost per

ℓ2 error Time-stepNumber of Code length Local mesh

time-step grid points (# of lines) refinement

RBF O(N2) 0.006 1/2 hour 4096 < 40 yes

SH O(M3/2) 0.005 90 seconds 32768 > 500 no

DF O(N log N) 0.005 90 seconds 32768 > 100 no

DGSE O(kN3/2e ) 0.005 6 minutes 7776 > 1000 yes

Table .1Performance comparsion between commonly used spectral methods in order toachievce a ℓ2 error of approximately 0.005. WHAT DO N AND M MEAN?

40

ε Max. eigenvalue ∆tmax (minutes)

1 21.3 191

2 21.2 192

3 20.9 195

4 20.6 198

5 20.2 202

6 19.6 208

7 19.0 215

8 18.2 223

9 17.4 234

Table .2A comparison between ε and the corresponding maximum eigenvalue of the differ-entiation matrix for the cosine bell test together with the maximum allowable time-step for RK4 according to eigenvalue stability theory. Results are for the N = 1849ME node set and the GA RBF.

41

N√

N Max. eigenvalue ∆tmax (minutes)

529 23 10.6 383

1024 32 15.2 267

1849 43 19.6 208

3136 56 27.1 150

4096 64 30.8 132

Table .3A comparison between

√N , N = number of nodes, and the maximum eigenvalue

of the differentiation matrix for the cosine bell test together with the maximumallowable time-step for the RK4 integrator according to eigenvalue stability theory.√

N is inversely proportional to the quasi-uniform spacing of nodes h. Results arefor the GA RBFs

42