Solving PDEs with Radial Basis Functions Bengt Fornberg * Department of Applied Mathematics University of Colorado Boulder, CO 80309, USA Natasha Flyer † Institute for Mathematics Applied to Geosciences National Center for Atmospheric Research Boulder, CO 80305, USA September 22, 2014 Abstract Finite differences was the first numerical approach that permitted large-scale simulations in many applications areas, such as geophysical fluid dynamics. As accuracy and integration time requirements gradually increased, the focus shifted from finite differences to a variety of different spectral methods. During the last few years, radial basis functions, in particular in their ‘local’ RBF-FD form, have taken the major step from being mostly a curiosity approach for small scale PDE ‘toy problems’ to becoming a major contender also for very large simulations on advanced distributed memory computer systems. Being entirely mesh-free, RBF-FD discretizations are also particularly easy to implement, even when local refinements are needed. This article gives some background to this development, and highlights some recent results. Contents 1 Introduction 2 2 Background to RBFs for PDEs 3 2.1 Different RBF types .................................. 5 2.1.1 Piecewise smooth RBFs ............................ 5 2.1.2 Infinitely smooth RBFs ............................ 5 2.2 Non-singularity theorems ............................... 6 2.2.1 Gaussian RBFs ................................. 6 2.2.2 Some other RBF types ............................. 7 3 Near-flat RBFs 8 3.1 The ill-conditioning of the A-matrix ......................... 8 3.2 Overview of some stable algorithms .......................... 9 3.2.1 Contour-Pad´ e algorithm ............................ 10 * Email : [email protected]† Email : fl[email protected]1
Transcript
Solving PDEs with Radial Basis Functions
Bengt Fornberg ∗
Department of Applied MathematicsUniversity of Colorado
Boulder, CO 80309, USA
Natasha Flyer †
Institute for Mathematics Applied to GeosciencesNational Center for Atmospheric Research
Boulder, CO 80305, USA
September 22, 2014
Abstract
Finite differences was the first numerical approach that permitted large-scale simulations inmany applications areas, such as geophysical fluid dynamics. As accuracy and integration timerequirements gradually increased, the focus shifted from finite differences to a variety of differentspectral methods. During the last few years, radial basis functions, in particular in their ‘local’RBF-FD form, have taken the major step from being mostly a curiosity approach for small scalePDE ‘toy problems’ to becoming a major contender also for very large simulations on advanceddistributed memory computer systems. Being entirely mesh-free, RBF-FD discretizations arealso particularly easy to implement, even when local refinements are needed. This article givessome background to this development, and highlights some recent results.
The present article is motivated by the recent successes of RBFs in the field of computationalgeoscience. This is quite far from how the RBF methodology first originated. It was proposedby R. Hardy around 1970 in connection with a cartography application that required multivariatescattered-node interpolation [42]. A key non-singularity proof by C.A. Micchelli in 1986 [60] accel-erated the further development and acceptance of RBFs. Pioneering work by M.J.D. Powell and hiscollaborators at University of Cambridge played also a major role in the early history of RBFs [63].In 1990, E.J. Kansa suggested that taking analytic derivatives of RBF interpolants could providea numerical solution approach for PDEs [46, 47].
Several monographs on RBFs or with extensive RBF content appeared between 2003 and 2007, inchronological order by Buhmann [9], Iske [44], Wendland [84] and Fasshauer [17]. Acta Numericafeatured RBF articles in 2000 [8] and in 2006 [70]. These works reflected a growing use of RBFsas a practical computational procedure for increasingly larger scale applications. Like [17], thebrief monograph [10] from 2014 discussed certain RBF approaches for solving PDEs. The per-spective presented in this article (as well as in the SIAM monograph [28] scheduled for 2015, bythe present authors) is quite different, and will additionally describe the RBF-FD (RBF-generatedfinite difference) approach.
We will here omit quite large areas of RBF theory that are well described in the previous mono-graphs, and in particular results that are not directly needed for effectively solving PDEs. Attentionwill however be given to ‘flat’ (or near-flat) basis functions, to the use of RBFs for creating weights
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for RBF-FD formulas, and to the application of RBF / RBF-FD discretizations for solving large-scale benchmark problems, mostly from the geosciences.
2 Background to RBFs for PDEs
PDE discretizations in more than 1-D are often based on meshes, which may be either structuredor unstructured, with the latter case best known in the context of finite elements. In the formercase, it is relatively easy to approximate derivatives to high orders of accuracy by making finitedifference (FD) stencils increasingly wide. That limit provides an alternate way to understand anduse pseudospectral (PS) methods [7, 24, 81]. A more common way to implement PS methods isvia expansions in basis functions, such as tensor products of 1-D Fourier or Chebyshev expansions.The computational efficiency of the resulting procedure can in some cases become very high, butthis comes at the price of severe regularity constraints on the shape of the computational domain.Spectral element approaches, involving domain decomposition into rectangles (when in 2-D), to-gether with curvilinear mappings can overcome some of this, and can also permit local refinementin critical areas. However, their implementation is complex and the small node spacing that be-comes necessary near internal (artificial) boundaries often severely hurts time stepping stabilityconditions.
When solving PDEs, it is very desirable to use entirely mesh-free node distributions, i.e. to beable to scatter computational nodes (collocation points) just as needed to fit boundaries and tosatisfy spatially variable resolution requirements, but without having to form any local trianglesor tetrahedra, etc. Furthermore, with a derivative being a local property of a function, it makessense to also rely on spatially localized approximations. While global approximations can have highformal orders of accuracy, their cost is typically high. This is due both to high operation countsand to costly data flow on modern computers with hierarchical memory structures.
Historically, one can recognize an evolutionary path FD ⇒ PS ⇒ RBF ⇒ RBF-FD that starts byextending from FD methods (first applied to PDEs just over a century ago [66]) to PS methods.It transpires that each PS method can be seen as a special case of an RBF approximation in acertain limit. With the RBF representation, geometric flexibility has been achieved. When thenRBFs are used to create weights for scattered node FD-like stencils (i.e. RBF-FD approximations),approximations have again become ‘local’, with associated high computational speeds and excellentscaling properties for massively large problem sizes.
Concerning interpolation over scattered nodes, using standard basis functions, the following theo-rem may at first appear discouraging:
Mairhuber-Curtis Theorem [13, 57]: Given any set of basis functions Fk(x), k = 1, 2, . . . , Nwith x ∈ Rd, d ≥ 2, the problem of determining an interpolant
s(x) =
N∑k=1
λkFk(x), (1)
satisfying s(xk) = fk, is singular for infinitely many configurations of distinct nodes xk, k =1, 2, . . . , N .
Proof: The interpolation requirement s(xk) = fk implies that the coefficients λk in (1) will satisfy
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Figure 1: Graphical illustration of the RBF concept. (a) Example of 2-D scattered data, (b) Basisfunction set - one rotated Gaussian is located at each data point, (c) The unique linear combinationof the Gaussians which agrees with all the provided data.
the linear system F1(x1) F2(x1) · · · FN (x1)F1(x2) F2(x2) · · · FN (x2)...
......
F1(xN ) F2(xN ) · · · FN (xN )
λ1λ2...λN
=
f1f2...fN
. (2)
In more than 1-D, it is possible to move the nodes continuously so that two nodes end up inter-changed, without them having coincided at any time. The effect on the coefficient matrix in (2) isthat two rows have become interchanged, i.e. its determinant has changed sign. By continuity, thedeterminant must therefore have been zero somewhere along the way.
The consequence of the theorem above is that vast numbers of seemingly ‘innocent’ node configura-tions will give rise to singular systems. The RBF idea for overcoming this issue is sketched in Figure1. The basis functions are here radially symmetric, typically with one centered at each node pointxk, i.e. of the form φ(||x− xk||) . Here φ is a radial function (such as φ(r = ||x− xk||) = e−(εr)
2),
ε is a shape parameter, and the norm is the standard Euclidean distance function. Again, lettingthe data value be fk at node xk, k = 1, 2, . . . , N , the coefficients in the RBF interpolant of f(x)
s(x) =
N∑k=1
λkφ(||x− xk||) (3)
can be found by solving a system very similar to (2)φ(||x1 − x1||) φ(||x1 − x2||) · · · φ(||x1 − xN ||)φ(||x2 − x1||) φ(||x2 − x2||) · · · φ(||x2 − xN ||)...
Moving two nodes so that they change places again interchanges two rows but now also two columns,leaving the sign of the determinant unaffected. Therefore, the singularity argument above no longerapplies. The key difference to the assumptions in the Mairhuber-Curtis Theorem is that the basisfunctions φ(||x− xk||) depend on the node locations.
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Type of basis function Radial function φ(r)
Piecewise smooth RBFs
Polyharmonic spline (PHS) rm, m = 1, 3, 5, . . .rmlog(r), m = 2, 4, 6, . . .
Compact support (‘Wendland’) (1− εr)m+p(εr), p certain polynomials
Table 1: Some common choices for radial functions.
2.1 Different RBF types
Table 1 lists a number of RBF types. For most of these, we will show in Section 2.2 that thesystem (4) can never be singular, no matter how any number of (distinct) nodes are scattered inany number of dimensions.
2.1.1 Piecewise smooth RBFs
The listed ‘piecewise smooth’ radial functions will cause a singularity at the origin of the associatedRBF and, in the compactly supported ‘Wendland’ case, also at r = 1/ε. This is entirely acceptablein many applications, but puts them at a disadvantage in other cases, such as when seeking accuratesolutions to convection-type PDEs over long times [34]. The property of compactly supported RBFsto produce sparse rather than full linear systems is advantageous in some contexts such as imagerendering, but less so when approximating PDEs, since the differentiation matrices that result fromthem nevertheless become full matrices.
PHS-type RBFs are associated with several optimality results, such as interpolating scattered datawith the least possible amounts of overall curvature [15, 63]. They are also of particular interest inthe context of RBF-FD.
It can be noted that φ(r) = r3 in 1-D reproduces cubic splines, albeit with highly unusual endconditions. With slight modifications in the form of (3), one can however obtain either ‘natural’or ‘Not-a-Knot’ splines. Similar modifications can be applied also to other RBF types and forscattered nodes in higher dimensions, offering easy-to-apply approaches for enhancing the accuracyat domain boundaries [26].
2.1.2 Infinitely smooth RBFs
As noted above, φ(r) = r3 in 1-D leads to a cubic spline, featuring a jump in the 3rd derivative ateach node. Disregarding possible boundary effects, its accuracy is well known to be O(h4) on a gridwith spacing h. Similarly, φ(r) = r5 leads to O(h6) errors, etc. This raises the obvious question whyone would use radial functions that cause jumps in any derivative. For the infinitely smooth ones,there are no such jumps, and that suffices to obtain spectral accuracy - better than any algebraicorder O(hp), p ∈ N (assuming that no counterpart to the polynomial Runge phenomenon arises)
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[56].
All smooth radial functions (of which there are many more options than what are listed in Table1) feature a shape parameter, denoted by ε. While one could apply an ε-scaling also to thePHS functions, such as φ(r) = rm, i.e. use φ(r) = (ε r)m, this would serve no purpose since(ε r)m = εmrm, and the scale factor would then vanish analytically by the time the interpolants(x) is obtained.
2.2 Non-singularity theorems
Following Bochner [5], we will first show that the RBF matrix
is guaranteed to be non-singular for GA RBFs, no matter how the nodes (assumed to be distinct)are scattered in any number of dimensions. This result will then be generalized to several otherRBF types.
2.2.1 Gaussian RBFs
A symmetric (real-valued) matrix A is positive definite if and only if αTAα 6= 0 for every vectorα 6= 0. All eigenvalues are then positive, and the matrix will be non-singular. The proof that theA-matrix for GA RBFs is positive definite can be carried out in three steps:
Step 1: Recall the Fourier transform of Gaussians. We define the 1-D Fourier transform(FT) as
u(x) =1√2π
∞
−∞u(ω) e iωxdω
u(ω) =1√2π
∞
−∞u(x) e−iωxdx
.
Applying the 1-D result u(x) = e−ε2x2 ⇔ u(ω) = 1√
2εe−ω
2/(4ε2) d times, gives for d-D
u(x) = e−ε2||x||2 ⇔ u(ω) =
1
2d/2εde−||ω||
2/(4ε2). (6)
Inverting u(ω) back to physical space produces the identity
e−ε2||x||2 =
1
(2π)d/2
ˆRd
1
2d/2εde−||ω||
2/(4ε2) ei x·ω dω. (7)
It may at first seem that this way to rewrite the GA radial function e−ε2||x||2 has introduced a lot
of extra complexity. However, the key point will turn out to be that x, appearing quadraticallyas ||x||2 in the exponent in the left hand side (LHS), appears only linearly, as x, in one of theexponents in the RHS.
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Step 2: Proof that A is positive semidefinite. Let α = [α1, . . . , αN ]T 6= 0. Then
αTA α =∑N
j=1
∑Nk=1 αjαke
−ε2||xj−xk||2 apply (7)=∑N
j=1
∑Nk=1 αjαk
1(2π)d/2
´Rd
1(2ε2)d/2
e−||ω||2/(4ε2) ei (xj−xk)·ωdω
= 1(2ε)dπd/2
´Rd e
−||ω||2/(4ε2)(∑N
j=1
∑Nk=1 αjαk e
i (xj−xk)·ω)dω
The double sum inside the integral can be written as N∑j=1
αj eixj ·ω
( N∑k=1
αk eixk·ω
)=
∥∥∥∥∥N∑m=1
αm eixm·ω
∥∥∥∥∥2
≥ 0.
Thus αTA α ≥ 0 , and we have shown that the matrix A is positive semidefinite.
Step 3: Proof that A is positive definite. Based on the result above, it only remains to showthat
∑Nm=1 αm eixm·ω cannot be identically zero (as a function of ω) unless all the coefficients αm
are zero. Several different short proofs for this are available [17, 28, 63].
2.2.2 Some other RBF types
If (6) is replaced by u(x) = f(ε||x||) ⇔ u(ω) = g(||ω||/ε) with g(||ω||) > 0, the replacement forthe leading factor (e−||ω||
2/(4ε2)) inside the integral in (7) will again be positive, and the positivedefiniteness proof will carry through just as in the GA case. This situation arises for ex. for manytypes of compactly supported RBFs.
Another variation of the nonsingularity proof (related to the theory of completely monotone func-tions [71]), proceeds as follows: Taking the inverse Laplace transform of φ(
√r) for different radial
functions φ(r) gives formulas such as:
IQ:1
1 + (εr)2=∞
0
e−s e−s(εr)2ds
IMQ:1√
1 + (εr)2=∞
0
e−s√πs
e−s(εr)2ds
.
In all cases when the factor in front of e−s(εr)2
inside the integral is positive, we observe (using hereIQ as an illustration)
αTA α =∑N
j=1
∑Nk=1 αjαk
11+ε2||xj−xk||2
=´∞0 e−s
(∑Nj=1
∑Nk=1 αjαke
− s ε2||xj−xk||2)ds .
From the non-singularity proof for GA RBFs, we know that the double sum is positive wheneverthe vector α = [α1, α2, . . . , αN ]T is not identically zero. Therefore, the integral and, with that,the quantity αTA α will also be positive, i.e. A is a positive definite matrix.
The proofs above do not directly apply to the commonly used MQ case. It transpires however thatnon-singularity again is assured, with the (symmetric) A-matrix now having one positive eigenvalueand all the remaining ones negative. The original proof by Micchelli [60] was later much simplified[64].
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Figure 2: (a) A set of 41 scattered nodes in the unit circle, (b) The error in max norm when thetest function f(x, y) = 59/(67 + (x+ 1
7)2 + (y − 111)2) is interpolated using these nodes, displayed
as a function of the shape parameter ε.
3 Near-flat RBFs
With ε available as a free parameter, it is natural to explore how the choice of ε influences theaccuracy that is obtained. A typical test is shown in Figure 2. As first noted in [78], it oftenhappens that the error decreases rapidly with ε until the calculation suddenly breaks down due tothe increasing ill-conditioning of the linear system (4). This may suggest that a tradeoff will berequired between accuracy and numerical conditioning (described as an ‘uncertainty principle’ in[68]). It was however soon realized that the RBF interpolation problem actually does not becomeill-conditioned in this flat basis function limit, and that the apparent problem was particular tothe RBF-Direct procedure: Solution of (4) followed by evaluation of (3). RBF-Direct uses ill-conditioned expansion coefficients λk as intermediate quantities for arriving at what should be a wellconditioned result [14, 36]. Several well-conditioned stable numerical algorithms were subsequentlydeveloped, cf. Section 3.2, giving results as seen in one typical case in Figure 3. Sometimes, themost accurate ε-range can be reached already with RBF-Direct. In other cases, such as the oneillustrated here, this requires a stable algorithm.
If the nodes are lattice-based, it can happen that the RBF interpolant diverges when ε→ 0 [30, 35],although never in the GA case [69], a fact contributing to making GA a popular RBF choice. Fornode sets with some irregularity, the interpolant will in the flat ε → 0 limit take the form of amultivariate polynomial [14, 36]. One reason that ε small often is better than ε → 0 is that, withRBF interpolants converging to polynomials, the boundary accuracy often deteriorates due to theRunge phenomenon [37]. In the high degree polynomial case, Chebyshev-style node clustering nearthe boundaries is the most often used remedy (in spite of disadvantages, such as causing adversestability conditions in the context of explicit time stepping of PDEs). As was noted in Section2.1.1, a number of additional options are available for RBFs.
3.1 The ill-conditioning of the A-matrix
Sideways translates of near-flat basis functions all look the same, and it is intuitively obvious thatthey must form a very ill-conditioned base to expand in. Just how bad it is can readily be quantified[37]. For example, when using infinitely smooth RBFs on scattered nodes in 2-D, the eigenvalues
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Figure 3: (a) Test function f(x) = e−7(x+ 1
2)2−8(y+ 1
2)2−9(z− 1√
2)2
, (b) N = 1849 ME (minimal energy)nodes on the surface of the unit sphere, (c) MQ interpolation errors (in max norm), as functions ofε, when using RBF-Direct vs. using the stable RBF-QR algorithm. The RBF-QR error level seenhere for small ε is unrelated to the nearby machine rounding level of 10−16.
of the A-matrix form distinct groups, following the specific pattern
until the last eigenvalue is reached, causing the last group to possibly contain fewer eigenvaluesthan the general pattern would suggest. Different choices of scattered node locations or of RBFtypes (IQ, MQ, or GA), make no difference in this regard. However, use of lattice based nodes orBessel-type RBFs result in exceptions (with smaller groups, implying worse conditioning). Moreconcisely, we can write the eigenvalue pattern above as
1, 2, 3, 4, . . . , (9)
indicating how many eigenvalues there are of orders ε0, ε2, ε4, ε6, etc. Table 2 shows some moresuch sequences. The patterns are readily recognizable; for ex. in the d-D non-periodic case, the kth
entry is(d+k−2k−1
). Given these patterns, one can immediately calculate the orders of both cond(A)
and det(A) =∏nk=1 λk as functions of n (here λk denote the eigenvalues of A). For the examples
in Figures 2 and 3, cond(A) becomes equal to O(ε−16) and O(ε−84), respectively.
3.2 Overview of some stable algorithms
The most straightforward approach for calculating in the small ε regime is to use extended precisionarithmetic. The main drawback is that the cost usually becomes excessive. Given the results quotedin Section 3.1, one can determine in advance just how many digits of precision would be needed asfunction of N and ε in various geometrical settings. For example, in the case shown in Figure 3,lowering the ε-value for onset of ill-conditioning by a factor of 100 (about what is needed in thiscase to ‘safely’ reach the optimal accuracy range) increases cond(A) by a factor of 10084 = 10168,showing that the arithmetic precision would have to be increased from 16 to about 180 digits.
Some types of preconditionings and SVD enhancements have been suggested for the RBF-Directapproach. While preconditioning can speed up certain iterative procedures, cf. [17] Ch. 34, this
Table 2: Numbers of eigenvalues of different sizes (powers of ε) for different geometries and typesof shape parameter.
does not address the issue that significant information becomes lost already when the coefficientmatrix A is formed (with all its entries virtually the same when ε is small). Recovery of suchmissing information is challenging or impossible.
Stable algorithms produce the same interpolant s(x) as mathematically defined by (3), (4), butwithout involving the ill-conditioned expansion coefficients λk. By using only computational stepsthat remain well conditioned even when ε → 0, standard double precision arithmetic suffices. Sofar, two main classes of stable algorithms have been developed. The first realizations of these weredenoted Contour-Pade [35] and RBF-QR [33], respectively. Related to the latter is the recentRBF-GA algorithm [32].
3.2.1 Contour-Pade algorithm
Although ε typically is a real-valued quantity, it can be extended to complex values. Focusing onthe GA case, it can be shown that the interpolant s(x, ε), for any fixed evaluation point x, thenbecomes a meromorphic function of ε (i.e. with poles as its only singularities across the finitecomplex ε-plane). Furthermore, it is known that s(x, 0) is finite even as ε → 0. The origin ε = 0must therefore be a removable singularity of s(x, ε). The actual algorithm requires a number oftechnicalities to be addressed, but its key principle is that Cauchy’s integral theorem allows theevaluation of an analytic function at a point (such as ε = 0) using an integration path that doesnot need to anywhere come close to it - i.e. the path can follow such a large circle around theorigin in the ε-plane that RBF-Direct can safely be used along it. In its original form, the Contour-Pade algorithm is nowadays mostly of historical interest, having established the feasibility of stablealgorithms.
3.2.2 RBF-QR algorithm
As we have noted repeatedly, translates of near-flat RBFs form a basis that is ill-suited for im-mediate numerical use. This naturally raises the question whether the underlying approximationspace also is bad, or if the conditioning issue can be resolved by finding an alternate good basis inexactly the same space. The latter turns out to be the case, leading to the follow-up issue of howone can carry out the basis conversion by analytic means also in scattered node cases, i.e. so thatno numerical cancellations will arise anywhere in the process.
One can here draw a parallel to the set of monomials P = 1, x, x2, . . . , x100 versus Chebyshevpolynomials T = T0, T1, T2, . . . , T100 over x ∈ [−1, 1]. Both sets span exactly the same functionspace, yet the monomials are an ill-conditioned base. For numerical work, it is critical that the
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Chebyshev polynomials are available in some type of closed form, e.g. as Tn(x) = cos(n arccos x),or through a 3-term recursion, and need not be obtained by numerically forming different linearcombinations of the monomials.
The RBF-QR method offers a systematic approach for converting a set of near-flat basis functionswith scattered centers to a well conditioned base for exactly the same space, in a numerically stablemanner. It was first implemented for nodes on the surface of a sphere [33], and more recently (inthe special case of GA RBFs) for arbitrary node sets in 1-D, 2-D, and 3-D [29, 51].
3.2.3 RBF-GA algorithm
The RBF-QR algorithm involves extensive manipulations of power series expansions. Rather thanexpanding to the extent that remainders can be ignored, the RBF-GA algorithm utilizes shorterexpansions combined with exact remainder formulas, for GA RBFs expressible in terms of incom-plete gamma functions. This leads to a stable algorithm that is free from both infinite expansionsand inexact truncations. It applies to GA RBFs in any number of dimensions, and is presentlyboth the algebraically simplest and the computationally fastest stable option available (at around10 times the cost of RBF-Direct, in either 2-D or 3-D). Although it may be slightly less accuratethan RBF-QR in some cases (such as for large lattice-like node sets), it is nevertheless well suitedfor generating RBF-FD approximations.
4 Three examples of solving PDEs using global RBFs
The three examples illustrate implementation issues and resulting accuracies, as well as how PDEcomplexity has been increased over the last decade - from Poisson’s equation in a simple 2-D domainto a nonlinear time dependent PDE system describing mantle convection in a 3-D spherical shell.In the former case, perfectly well understood solutions were reproduced, whereas in the latter case,it provided physical insights not previously reached by any other investigative method.
4.1 Poisson’s equation
We consider as test problem Poisson’s equation on a domain Ω, with a Dirichlet condition on theboundary ∂Ω:
u(x) = g(x) on boundary ∂Ω∆u(x) = f(x) in interior of Ω
. (10)
This is discretized at node locations x1, . . . ,xNB
on ∂Ω and xNB+1, . . . ,xN within Ω.
4.1.1 Two strategies for RBF discretization
The two main discretization approaches can be summarized as follows (for simplicity described inthe 2-D Poisson case):
Kansa’s formulation: Let the solution to (10) be of the form
u(x) =N∑j=1
λjφ(||x− xj ||) . (11)
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Enforcing this at all nodes gives a linear system for the λj of the following structure:φ(||x− xj ||)|x=xi
−−−−−−−−−4φ(||x− xj ||)|x=xi
λ
=
g
−f
, (12)
where i = 1, . . . , NB for the upper matrix block, and i = NB + 1, . . . , N for the lower block.This straightforward approach has proven to be widely successful, even if rare possibilities forsingularities have been noted [43].
Symmetric formulation: The assumed form of the solution is now changed from (11) to
u(x) =
NB∑j=1
λjφ(||x− xj ||) +
N∑j=NB+1
λj4φ(||x− xj ||) ,
i.e. we use 4φ(||x− xj ||) rather than φ(||x− xj ||) as RBF at the interior nodes. The counterpartto (12) becomes (in abbreviated notation)
|φ | 4φ
|− − −−− + −−−4φ | 42φ
|
λ
=
g
−f
,
with (for the standard RBF choices) an assured symmetric and positive definite coefficient matrix[16, 89]. Although this is an obvious advantage, actual numerical performance of the two approachesseems relatively comparable (with different studies suggesting slight advantages either way, e.g.[50, 65]).
Generalizations to other linear or nonlinear operators is straightforward. If Newton’s method isused, the cost per iteration becomes comparable to that of solving a linear case, as either willrequire the solution of a full N ×N linear system.
4.1.2 Test calculation: Circular domain
Naturally, the earliest implementations of RBFs for PDEs were focused on showing that the ap-proach is viable for very simple test problems. We summarize here the study [50], since this alsocompared RBF-Direct against Contour-Pade (the only stable choice in 2003). In order to alloweasy comparisons of RBFs against FD2 (second order FD) and PS methods, the domain was chosenas the unit circle. All the node sets had NB = 16 nodes on the boundary ∂Ω and NI = 48 nodes inthe interior of Ω. For FD2, the nodes were equispaced in both angle and radius, and for PS againequispaced in angle, but of Chebyshev-type radially (across −1 ≤ r ≤ 1, with angle 0 ≤ θ < π).For RBF, the nodes were somewhat irregularly scattered, cf. Figure 4. Figure 5 shows a typicalresult. Kansa’s approach is here applied to (10) with g(x) and f(x) selected in such way that theequation has as its solution u(x) = 100/(100 + (x − 0.2)2 + 2y2). Even when using RBF-Direct,
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Figure 4: Illustration of how polar type grids create highly non-uniform resolutions in different di-rections near the origin, (a) Polar: equispaced in radius, (b) Polar: Chebyshev along each diameter,(c) Irregular (but avoiding clustering) - as typically used in RBF contexts.
Figure 5: Max norm errors, as functions of ε, when solving a 2-D Poisson test problem using threechoices of RBFs: GA, IQ, MQ (a) Using RBF-Direct and (b) Using Contour-Pade. The dashedlines across both subplots compares the accuracies reached by FD2, PS (both independent of ε).
the RBF approach is seen to be the most accurate option (if the optimal ε is used). The use of astable algorithm not only improves the accuracy further still, but also makes the choice of ‘optimal’ε very much less critical.
It can be noted that a second order method (such as FD2, or second order finite elements) gainsa factor of 4 in accuracy when step sizes are halved, i.e. in 2-D when 4 times as many nodes areused. The error is then inversely proportional to the number of nodes. In the present test case,the errors for MQ and IQ RBFs are roughly 10−6 times those for FD2 implying that, in order tomatch the RBF accuracy, FD2 would need the node count N = 64 to be increased by a factor ofabout one million.
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Figure 6: Top row: Patterns produced by RBF solutions of the Brusselator reaction-diffusionequations for two different parameter settings, Bottom row: The skin patterns on two frog species(Tabasara rain frog, and Poison dart frog, respectively). Illustration provided by Cecile Piret.
4.2 Reaction-Diffusion equations on curved surfaces
Solving PDEs over curved surfaces has a substantial history, both in terms of application areasand with regard to numerical approaches. Some different methods (including RBFs) are discussedin [72]. RBFs are particularly well suited to the task, since they avoid the singularities that areintrinsic to any surface-bound coordinate system, exemplified for a sphere with the two poles if usingspherical coordinates. Another key advantage is that spectral accuracy readily becomes available(in contrast to for ex. surface triangularization-based finite element discretizations).
The solution of PDEs over biological surfaces was pioneered in 1952 by A.M. Turing [82] in thecontext of pattern formation on animals. Both this topic, and also other processes occurring on cellsurfaces and on other types of biological membranes, have since received extensive mathematicaland numerical attention. The solutions presented in [62] use global RBFs, in combination with theOrthogonal Gradient method (OGr), allowing a single ‘cloud’ of nodes to be used both for definingthe surface and for discretizing the PDE. Figure 6 illustrates an N = 560 node set in the shapeof a frog, and two RBF-generated solutions to the brusselator equations over this surface. Thisnonlinear reaction-diffusion system closely models actual formation of skin patterns on animals (forwhich the time evolution gets frozen at some embryonic stage). The very high accuracy of the RBFapproach is evident in Figure 6, as the finest resolved features have about 4 points per wavelength,to be compared to the theoretical limits of 2 for Fourier-PS and π for Chebyshev-PS.
The article [39] describes solutions to another convection-diffusion type PDE (the Barkley model),again over surfaces of biological objects. The global RBF approach was in this case somewhatdifferent (a ‘projection’ approach, for which the surfaces were given in the form of level surfaces ofspecified 3-D functions).
14
Method No. of nodes Nucrust Nucore 〈VRMS〉 〈T 〉 Ref.
Table 3: Comparison between methods in the literature for the standard Ra = 7, 000 case.
4.3 Mantle flow in a spherical shell
A number of increasingly large geoscience-oriented test cases were solved using global RBF-basedspatial discretization between 2007 and 2011. The geometries were at first confined to the surfaceof a sphere [19, 21, 22], and then followed by a 3-D mantle convection simulation [87]. Theseworks were all summarized in [18]. We highlight here the mantle flow simulation, since it decisivelybrought RBF based simulations from ‘just another approach that can work on toy problems’ to (i)confirming a physical prediction previously outside numerical reach, and (ii) doing so using a PC,against supercomputer calculations employing the full range of traditional methodology approaches(see Table 3, abbreviating a more extensive table in [87]).
The physical scenario is as follows: the flow is incompressible; the temperature (T ) is governedby a mixed convective-diffusive PDE; the momentum is governed by Stokes flow, an elliptic PDE;the impermeable boundaries are slip-free (Neumann boundary conditions in the angular direction),with T = 1 at the core and T = 0 at the crust. The coupled system of three PDEs is approximatedby RBF discretization on each of many concentric spherical shells, together with Chebyshev-PSdiscretization radially (see Fig. 7 a). Since no analytic solutions are available, isoviscous flowat low Ra = 7, 000 (within the steady-state regime) has become a commonly used benchmark.The standard initial condition in this case is a combination of fourth-order spherical harmonicstimes linear decay in the radial direction. The summary in Table 3 compares results for the globalvariables, Nucrust, Nucore, 〈VRMS〉, 〈T 〉 (Nu is the Nusselt Number, VRMS is the root mean squarevelocity, and 〈·〉 indicates globally averaged quantities). For this test, energy conservation impliesthat solutions should satisfy Nucrust = Nucore. The RBF-CH method, using a much lower levelof discretization, achieves near perfection in terms of accuracy compared to the previously mostaccurate method, the Romberg extrapolated SPH-FD method. The RBF-CH simulation was theonly one that was run on standard PC hardware.
Fig. 7c shows a snapshot from a Ra = 106 simulation, dominated by turbulent convection. Thisis a much more physically realistic case, since Ra ≈ 107 for the current Earth. This RBF-CHsimulation is the only spectral model in the literature to be run in spherical geometry at sucha high Ra. It showed an instability at Ra = 70, 000 that had been theorized [3] but remainedcontroversial, as it had not been seen in any previous numerical simulations. These mantle flowsimulations demonstrate strikingly that global RBFs can be very competitive already on standardPCs.
15
Figure 7: Mantle convection: (a) RBF-CH discretization, (b) steady state solution at Ra = 7, 000;yellow = upwelling, blue = downwelling, red = core, (c) snapshot of a Ra = 106 solution after atime corresponding to about 4.5 times the age of the Earth.
5 Basic properties of RBF-FD approximations
RBF-FD combines many key strengths of RBFs with those of traditional FD approximations. Theprimary factor behind their development was the high computational cost of global RBFs. Whenusing RBF-Direct, finding an interpolant or calculating a differentiation matrix (DM) each costO(N3) operations for N nodes, with an additional O(N2) operations each time a DM is applied(for ex. during time stepping). In parallel with the successful application usage of global RBFs,as described above, efforts were under way on several fronts to dramatically reduce these costs. Ofseveral potentially viable approaches (such as ’fast algorithms’ based on multipole ideas, innovativepreconditioners, etc.), RBF-FD is at present the leading option. The rest of this article will bedevoted to this.
The RBF-FD concept was first outlined in a conference presentation by A.I. Tolstykh in 2000[79]. It was shortly afterwards independently introduced a number of times, e.g. [73, 83, 86].Since this approach still is in rapid development, the present discussion will not attempt to becomprehensive, but only to highlight how it already has proven to be highly competitive againstprevious alternatives. Active application areas not discussed here include elasticity [48, 80], flamepropagation [2, 49], and mechanics [11, 67].
5.1 RBF-FD weights
Traditional FD approximations are grid based and, when above 1-D, typically combine 1-D approx-imations. FD weights are determined so that the approximations become exact for polynomialsof as high degrees as possible. Some effective algorithms for generating FD weights are given in[25]. The polynomial approach does not generalize well to scattered nodes in more than 1-D, withthe Mairhuber-Curtis theorem being just one reason. Instead of relying on multivariate polyno-mials, one can enforce the exact result for all the RBFs that are centered at the nodes of thestencil of size n. Straightforward algebra will then show that the weights wk at the stencil nodesxk, k = 1, 2, . . . , n can be obtained by solving the linear system
16
A
w1
w2...wn
=
Lφ(||x− x1||)|x=xc
Lφ(||x− x2||)|x=xc...
Lφ(||x− xn||)|x=xc
. (13)
The matrix A is the same as the one in (4), and xc is the location at which the stencil is ap-proximating the L-operator (typically chosen as a node point near the stencil center). A commongeneralization of (3) is to add multivariate polynomials to the RBF basis, and then impose match-ing constraints. For instance, if one includes linear terms in a 2-D case, (13) should be replacedby
p 1 x1 y1
A p...
......
p 1 xn yn− − − + − − −1 · · · 1 px1 · · · xn p 0y1 · · · yn p
w1...wn−
wn+1
wn+1
wn+3
=
Lφ(||x− x1||)|x=xc...
Lφ(||x− xn||)|x=xc
−L 1 |x=xc
L x |x=xc
L y |x=xc
, (14)
where only the weights w1, w2, . . . , wn should be used. For a derivation, see [28]. The pattern in(14) generalizes directly to higher dimensions and different polynomial orders.
5.2 Node distributions
While Cartesian lattices are commonly used for FD and PS methods, hexagonal lattices (in caseof 2-D) generally allow for more cost-effective discretizations (cf. Section 6.2). Such lattices havebeen used only rarely in the past because of algebraic complexity, and difficulties with both localrefinements and with generalizations to higher dimension. When using RBFs, and especially RBF-FD, all these concerns vanish. Later in this article, Figure 16 will illustrate other advantages withdeviating from Cartesian grids. Quasi-uniform scattered node sets are often highly effective as wellas easily generated. The Delaunay-based algorithm in [61] offers one convenient option. In the caseof 2-D, the algorithm described in [27] is particularly fast.
On Cartesian lattices, nodes are usually sequentially ordered by the lattice directions. For scatterednodes, the ordering is in principle arbitrary. However, both for achieving fast memory access (withnode sets that do not fit into high speed cache memory) and for optimal convergence rate withcertain iterative linear solvers, the node ordering needs to be optimized. Re-orderings based onreverse Cuthill-McGee or ‘locality sensitive hashing’ can be highly beneficial [6].
5.3 Time stabilization - hyperviscosity
For a purely convective PDE, there should not be any solution modes that feature long-term growthor decay. In the case of linear spatial operators, this can be studied via eigenvalue analysis. If themethod of lines (MOL) discretization for ∂u
∂t = Lx takes the form dudt = D u, the eigenvalues of
the ‘differentiation matrix (DM)’ D should be purely imaginary. RBF-FD approximations for Lintroduce a low level of ‘jitter’ on the eigenvalues, typically scattering them small distances to eachside of the imaginary axis, with the physically relevant ones typically scattered the least. While asmall distance to the left of the axis generally is harmless (causing spurious modes to decay slowly),
17
small scatter to the right causes exponential growth in time. What is needed is an approach thatleaves the physically relevant (smooth) eigenvalues/modes intact, but ‘nudges’ spurious oscillatoryones from the right half-plane over into the left one. This can be achieved by hyper-viscosity [31],adapted from turbulence simulations. As an additional benefit, this permits the use of larger (andtherefore more accurate) RBF-FD stencils. Without this enhancement, stencils in 2-D can rarelyexceeded around n = 8− 12 nodes, whereas with it, n-values up to around 100 were instrumentalfor obtaining the high accuracies reported in [20, 31]. There are at present two main hyperviscosityapproaches, best suited for global RBFs and for RBF-FD approximations, respectively.
5.3.1 The A−1 method
This approach applies to RBF types with the A-matrix positive definite (e.g. GA, IQ, IMQ, etc.,but not MQ or PHS). As noted in Section 3.1, the A-matrix eigenvalues will decrease very rapidly tozero if ε is small. The corresponding eigenvectors at the same time become increasingly oscillatory.The matrix A−1 will have the same eigenvectors, but its eigenvalues are the inverses for those ofA, i.e. they will start out O(1) and then rapidly become very large (and again all positive). Hence,adding a term −γ A−1u with a very small constant γ > 0 to the RHS of a MOL discretizationof a convective PDE d
dtu = Lu will leave all the physically relevant (reasonably smooth) modesessentially intact, but will rapidly damp out all highly oscillatory (spurious noise) modes.
5.3.2 Powers of the Laplacian
The concept is again to leave smooth modes intact, but to quickly damp out rapidly oscillatinghigh ones. Adding a small multiple of the Laplacian operator ∆ to the PDE’s RHS would damphigh modes, but also interfere with low ones (which represent physical information). The analysisand test results in [20, 31] show that using relatively high powers of ∆ achieves what is needed.These references discuss implementation issues, e.g. guidelines for powers and multiplying factorsto use, and convenient formulas for GA-type RBFs.
A standard test case for studying how well a solution is advected intact, i.e. without trailing wavestrains or diffusion, is known as solid body rotation [85]. An initial condition, such as a C1 cosinebell, is advected around the unit sphere at an angle α tilted relative to the polar axis. The governingequation in spherical coordinates is given by
∂h
∂t+ (cosα− tan θ sinϕ sinα)
∂h
∂ϕ+ cosϕ sinα
∂h
∂θ= 0 (15)
Using N = 25, 600 MD nodes, a stencil size n = 74, GA RBFs with ε = 8, and ∆8-type hypervis-cosity, the long-term evolution is illustrated in Figure 8. In spite of the very long integration time(1, 000 revolutions around the sphere), there are no visible hints of instabilities or even of loss inpeak height (here less than 1%). The main errors remain right at the base of the cosine bell, wherethere is a jump in the second derivative.
5.4 Compact (implicit) approximations to elliptic PDEs
Since a derivative is a ‘local’ property of a function, there is something intuitively contradictoryabout enhancing the order of a FD approximation by invoking data located increasingly far away.When the task is to solve a PDE (rather than just to approximate an operator), compact approx-imations offer a different opportunity for improving the order of accuracy. For finite differences,
18
Figure 8: The numerical solution and the magnitude of the errors for the solid body rotationtest case, using the stabilized RBF-FD approach. The displays are over the (ϕ, θ)-plane, withϕ ∈ [−π, π], θ ∈ [−π
2 ,π2 ].
19
the concept has a long history [12, 38] with several more recent enhancements available (such as tononlinear PDEs in 2-D and 3-D, etc.) [40, 52, 53, 91].
Before considering compact approximations in scattered node RBF-FD cases, we illustrate the basicidea in the case of approximating 4u = ∂2u
∂x2+ ∂2u∂y2
on a 2-D lattice, with spacing h in each direction.The most obvious FD approximation can be written as 1
1 −4 11
u/h2 = 4u+O(h2) . (16)
Using only a 3 × 3 stencil size, it is impossible to find weights that improve the accuracy abovesecond order. Extending the stencil to 5 nodes in both directions permits fourth order accuracy,but causes problems when solving the PDE 4u = f :
(i) The center weight becomes smaller in magnitude than the sum of magnitudes of the remainingweights, i.e. diagonal dominance is lost. This damages the convergence rate of many iterativeschemes, and it also opens up the possibility of system singularities.
(ii) Wider stencils need more boundary information than what is readily available.
Taylor expansions will however reveal that, if the task is not to approximate 4u but to solve4u = f , then 1 4 1
4 −20 41 4 1
u/(6h2) =
11 8 1
1
f/12 +O(h4), (17)
and for the special case of solving 4u = 0, 1 4 14 −20 41 4 1
u/(6h2) = 0 +O(h6) .
The latter approximations suffer neither of the two problems noted above, but achieve neverthelesssignificantly improved levels of accuracy.
Equation (17) can be re-cast as a compact approximation to 4u as
[1] ∆u =
14 1 1
41 −5 114 1 1
4
u /h2 +
− 18
− 18 − 1
8− 1
8
∆u+O(h4). (18)
RBF-FD counterparts to (18) for scattered nodes can readily be generated, as described in [28, 88].The latter reference provides several test examples, showing that the advantages noted above forcompact formulas carry over from lattice based FD cases to scattered node RBF-FD cases.
20
6 Three examples of solving PDEs with RBF-FD
6.1 The shallow water equations on a sphere
The equations in a 3-D Cartesian coordinate system for a rotating fluid are
∂u
∂t=− (u · ∇)u− f(x× u)− g∇h, (19)
∂h
∂t=−∇ · (hu) , (20)
where f is the Coriolis force, ∇ = ∂xi + ∂y j + ∂zk, u = ui + vj + wk is the velocity vector, h isthe geopotential height and x = x, y, zT represents the position vector. Working in Cartesiancoordinates requires a projection operator that confines the motion to the surface of the sphere,that is ∇ → P∇ = [px · ∇,py · ∇,pz · ∇], where
P∇ =
(1− x2) −xy −xz−xy (1− y2) −yz−xz −yz (1− z2)
∂∂x∂∂y∂∂z
=
px · ∇py · ∇pz · ∇
Notice that each component of the projected gradient for a given direction is a linear combinationof the other three. In addition, the right hand side of (19) needs to be projected, with the modifieddifferential operators, in the corresponding i, j, and k directions. For example, in the case of the umomentum equation (corresponding to the velocity in the x direction), this results in
Notice that the only differential operator L that needs to be discretized is P∇ and can be calculatedas done in Section 5.1. It was noticed that the addition of any polynomials beyond a constant didnot affect the results. For further details see [20], which also provides a greatly simplified way tocalculate the projected gradient for the sphere.
By adding a forcing term hmtn to the right hand side of the geopotential height h equation in(20), flow over a mountain can be simulated [77, 85]. Two mountain profiles, one where hmtn isa C1 cone and the other a C∞ mountain, are considered to illustrate the sensitivity of high-ordermethods to Gibbs phenomena. This is important because topographical features are rarely evenC1. Figure 9 shows the results for a 15 day run using Runge-Kutta 4th-order time-stepping, withthe reference solution given by a Discontinuous Galerkin (DG) shallow water model [4], where eachelement contains 12x12 Legendre quadrature nodes to represent the solution, which results in atotal of 884, 736 degrees of freedom and an average resolution around 26 km. The results are givenin Figure 9. The key differences between the two columns of panels is that 1) even though the C∞
Gaussian mountain is slightly steeper than the C0 mountain, there are no high-frequency wavesemanating throughout the domain, and 2) after n = 31, stencil size has no bearing on convergenceor accuracy with the C1 cone forcing. This latter fact is that with non-smooth forcing the only wayto increase accuracy is to increase resolution about the base of the mountain and not the order ofthe method.
21
Figure 9: Left column, Cone mountain results: (1) profile of mountain; (2) Plotted in meteres,RBF-FD solution for h at day 15, N = 25, 600 and n = 31 with contour intervals at 50 m; (3)magnitude in the error between the RBF-FD solution and DG reference solution in meters; Contourinterval is 0.5m with white denotes errors less than 0.1m (4) `2 error as function of the resolutionN for varying stencil sizes. Right column, same as left but for the Gaussian mountain forcing.Dashed circle in all plots is the base of the mountain.
22
Figure 10: Left and middle panel: Convergence plots in the `2norm with regard to the geopotentialheight fields h for the RBF-FD and global RBF, respectively, against the 3 reference solutions.Notice the RBF-FD and DG solutions perfectly agree. Right panel: The error as a function ofcomputer runtime for the RBF-FD and DG methods.
We consider three reference solutions: 1) the DG reference solution mentioned above, 2) a sphericalharmonic solution from the DWD (Deutscher Wetterdienst, German National Weather Service)that has a spectral truncation of T426, that is it uses 182,329 spherical harmonic bases, and 3)an RBF-FD based on N = 163, 824 icosahedral-type nodes on the sphere, representing a 60kmresolution, and a stencil size of n = 31. The left panel of Figure 10 shows that the `2 error in theRBF-FD method, whether using the RBF-FD reference solution or DG one, are almost identical.This same trend is also seen in the middle panel of Figure 10 with global RBFs. In contrast, theerror from the SH reference solution is an order of magnitude larger. Given that DG, RBF-FD, andglobal RBFs are vastly different numerical methods, this strongly indicates that the SH spectralmodel is providing a less accurate solution, while DG and RBF are in line with one another.
The next consideration is time benchmarking of RBF-FD against DG. The present benchmarkingwas done on a MacBook Pro laptop with an Intel i7 2.2 GHz quad-core processor, using only asingle core, and 8 GB of memory. The RBF-FD code was written in MATLAB and the DG codein C++. The RBF-FD reference solution of N = 163, 842 and n = 31 (i.e. 60 km resolution) wasused for calculating the `2 error versus run-time (i.e. wall-clock time) for both methods as shownin the right panel of Figure 10. The RBF-FD method was computationally faster than the DGmethod, from about an order and a half of magnitude for coarser resolutions to 4 times faster forthe finest resolutions.
6.2 The compressible Navier-Stokes equations on a limited area domain
The compressible Navier-Stokes equations in a 2-D Cartesian coordinate system, x, z, for strati-fied fluid flow (important in atmospheric processes) are given by
23
∂u
∂t=− (u · ∇) u− cpθ∇P − gk + µ∆u, momentum
∂θ
∂t=− (u · ∇) θ + µ∆θ, energy (22)
∂P
∂t=− (u · ∇)P − R
cv(∇ · u)P, mass
where P = (p/P0)R/cp is the non-dimensional Exner pressure
(P0 = 1× 105 Pa
), and θ = T/P is
the potential temperature. The constants cp = 1004 and cv = 717 are the specific heat at constantpressure and constant volume, respectively, R = cp−cv = 287, and µ, the dynamic viscosity. Theseequations are often used for testing novel numerical methods in atmospheric modeling, as will bedone here.
A commonly used test case is known as the Straka density current [75]. A bubble of cold air fallsto the ground and develops three smooth and distinct rotors due to shear instability, as it spreadssideways. The computational domain is [−25.6, 25.6]km in x with periodic boundary conditions,and [0, 6.4]km in z with no-flux and free-slip boundary conditions on the velocity and Neumannon the temperature and pressure. The dynamic viscosity is µ = 75 m2/s. PHS RBFs, r7, togetherwith polynomials up to third degree are used to approximate all spatial derivatives locally by theRBF-FD approach with a stencil size of n = 37. The remaining system of first order ODEs is time-stepped with RK4. Figure 11 shows the behavior of the numerical solution in time from t = 0suntil the final time t = 900s.
The RBF-FD approach makes it particularly easy to test how different node distributions (allwith the same total number of nodes) influence the extent to which the physics is captured. InFigure 12, three different node layouts are examined: Cartesian, hexagonal, and quasi-uniformlyscattered. Convergence under refinement leads in all cases to the same solution, as seen in thehighest resolution displays (bottom row of subplots). However, in numerical weather prediction,the ability to work at such fine resolutions as 100m is a luxury rather than a reality. The fact thatobservational data that initializes models is observed on the order of kilometers, makes the degreeto which the physics is captured at coarser resolutions more important. Three key features to benoticed are: 1) formation of the rotors, 2) how much cold air they have entrenched (larger negativevalues of θ- black) and 3) where the front location is. In the coarsest case shown, using only 720nodes in the domain (about 700m resolution), the hexagonal and scattered nodes calculations givemore clear evidence of the first rotor being formed. At the next higher resolution (2,700 nodesin the entire domain; about 350m resolution), they provide a better picture of the formation ofsubsequent rotors, as well as more accurate entrenchment of cold air (black) near the front, lookingmore similar to the high resolution 90m case. Cartesian nodes furthermore give solutions moreprone to Gibbs phenomenon oscillations (overshoots in white of 2.4oK as opposed to 1oK).
The calculations for Figure 12 all used RBF-FD stencils of size n = 37, generated from φ(r) = r7,supported with polynomials up through degree 3. The differences between the columns of sub-plots reflect only the intrinsic resolution capabilities of the different node layouts. The traditionalCartesian choice is the least effective one. If using a fixed node separation, a hexagonal layout can‘pack’ more nodes into a fixed region than can a Caresian one. Conversely, in the present casewith fixed node numbers, their separation becomes somewhat larger. Even so, at every resolutionlevel, the hexagonal choice gives better accuracy than the Cartesian one. The big advantage of
24
Figure 11: The time evolution of the potential temperature θ using a hexagonal node layout at100m resolution
25
Figure 12: The potential temperature θ for the Straka density current test case at the final timet = 900s, shown as a function of the total number of nodes when using the RBF-FD method ondifferent node sets. For plot clarity, only half of the solution is displayed.
generalizing further, from hexagonal to quasi-uniformly scattered nodes, is that it then becomestrivial to implement spatially variable node densities, i.e. to do local refinement in select criticalareas. It is very important to note that this major increase in gemetric flexibility (from hexagonalto quasi-uniformly scattered) hardly has any negative effect at all on the accuracy that is achieved,nor on the algorithmic complexity of the code.
To place Figure 12 in context with the results of other numerical methods, a comparison is donewith DG, spectral element (SE), finite volume (FV), and upwinding schemes in Figure 13. As canbe seen, when no filtering is used in the RBF-FD method, there is a trade-off between capturingfeatures at low resolutions and preserving monotonicity. Only the FV and upwind schemes do notexhibit Gibbs’ oscillations and have solutions with monotonic properties. However, the price tobe paid is that the solution is smoothed out both with regard to rotor formation and the amountof cold air that has been entrenched. The DG and SE solutions have more structure, but thebeginning formation of the second rotor is still not seen as well as in the RBF-FD model.
Without any explicit viscosity, the solution enters the turbulent regime with the dynamics nowmodeled by the Euler equations. In such regimes, there is no convergence to any solution as energycascades to smaller and smaller scales, eventually entering the sub-grid scale domain. Nevertheless,it is interesting to observe whether the model remains stable in this regime. Figure 14 shows thesolution at 50m and 25m resolutions on a hexagonal layout (optimal in 2-D and easily implementedwith RBFs). The fact that now there is no explicit viscosity, i.e. µ = 0 in (22), does not affectthe time stability and the time step did not have to be altered between the two cases. Stability isgoverned solely by the fact that the time step could not exceed the speed of sound in air.
26
Figure 13: Comparison at 400m between 4 different numerical methods. Contour intervals: RBF-FD 1K [1]; DG & SE 0.25oK , FV 1oK , 5th-order Upwind 1oK .
Figure 14: The potential temperature θ for the Straka density current test case with the dynamicviscosity µ = 0 on a hexagonal node layout of 50m and 25m
27
6.3 Forward seismic modeling
6.3.1 Background
Seismic exploration is the primary tool used for finding and then mapping out hydrocarbon de-posits. In forward modeling, subsurface structures are assumed to be known, and the task is tosimulate elastic wave propagation through the medium. Inversion programs then update subsur-face assumptions to reconcile the model response with actual measurements. There are typicallyhundreds of irregularly curved interfaces present, often interrupted by fracture lines with associatedtranslations between the strata on the two sides. During the history of the earth, the vast majorityof all hydrocarbons (such as natural gas and oil), being lighter than water, have migrated up to thesurface and then biodegraded. What is left are mostly small pockets where hard layers somehowhave formed traps for this upward migration, due to their curvature or the presence of cornersresulting from fractures. With drilling being far more expensive (and environmentally damaging)than seismic exploration, the latter is constantly pushed to its limits, leading to some of the largestcomputational tasks in any field.
Figure 15 shows a extremely simplified model for the Marmousi test case, itself a highly simplified2-D vertical slice off the coast of Madagascar (as shown in [59], Figure 2). The governing elasticwave equations are in 2-D
The dependent variables are u, v (horizontal and vertical velocities) and f, g, h (components of thesymmetric stress tensor), and the material is specified by ρ (density) and λ, µ (Lame parametersfor compression and shear). Away from interfaces, these equations support two types of waves:P-waves (pressure or primary) with speed cp =
√(λ+ 2µ)/ρ and S-waves (shear or secondary)
with speed cs =√λ/ρ. Each incoming wave to an interface results in four main outgoing waves
- reflected and transmitted both P- and S-waves (as well as waves following interfaces). Withtypically hundreds of interfaces, wave patterns become extremely complicated. Simulated returnsignals at the surface need to accurately represent both wave propagation long distances throughregions with smoothly varying material properties as well as reflection-transmissions (with respectto amplitudes, phase angles, and directions).
In the smoothly varying regions, the dominant error source is numerical dispersion. The onlypractical remedy against this is to use high order approximations [23]. Industry standard movedfrom second to fourth order in the 1980’s, and FD approximations of extremely high (around20th) order are nowadays in common use. It has proven much more difficult to achieve accurateinterface treatments [54, 55, 76]. While closed form expressions are available in simplified cases(such as straight interfaces between constant media), incorporating these in full production codeshas so far not been cost effective. Industry standard has surprisingly remained at foregoing specialtreatments (beyond mild smoothing of interfaces), accepting typically first order convergence forreflected waves. The present RBF-FD/AC method (with AC standing for Analytic Correction)achieves third order accuracy both in smooth regions and across smoothly curved interfaces, makingit very competitive. For simplicity, we describe here its concept only in 1-D and show only a 2-Dtest result. It has however already been tested equally favorably in 3-D.
28
Figure 15: Subsurface acoustic velocities in a ‘micro-Marmousi’ test case.
6.3.2 RBF-FD/AC approach
Figure 16 illustrates a typical node layout and the different stencil types used in a still moresimplified test case, with just one curved interface in 2-D. The nodes are distributed to straddle theinterface but then smoothly transition to become lattice-based a short distance away from it. Threestencil types are used: (a) regular FD when the whole stencil is lattice based, (b) standard RBF-FDwhere some nodes are irregularly placed, while away from the interface, and (c) RBF-FD/AC whenan interface intersects a stencil.
Although the RBF-FD/AC discretization has been successfully tested in 1-D, 2-D and 3-D, welimit ourselves here to describing it in 1-D (for a 2-D description, see [58]). The governing equationreduces in 1-D to
∂
∂t
[uf
]=
[0 1
ρ∂∂x
ρc2 ∂∂x 0
] [uf
]. (24)
While ρ and c typically both jump at an interface, continuity of motion and traction requires uand f to be continuous (in the 2-D case, there will similarly be 4 continuity relations linking the 5variables in (23)) . Denoting left and right sides of the interface by subscripts L and R respectively,it will then hold that [
uLfL
]−[uRfR
]=
[00
],
which implies∂k
∂tk
[uLfL
]−[uRfR
]=
[00
], k = 0, 1, 2, . . .
With use of (24), these time identities will translate to relations between spatial derivatives for uand f on the two sides. The idea is to embed these relations in the supplementing polynomials forthe RBF-FD approximation (but not in the RBFs themselves; cf. [90] where a similar approachwas considered in the context of Maxwell’s equations). Figure 17 illustrates how one thus arrivesat ‘interface aware’ supplementary polynomials (with their changes across the interface dependenton the material properties on the two sides).
6.3.3 2-D test case
In the geometry shown in Figure 15, part (a) of Figure 18 shows the vertical velocity v associatedwith an underground explosive source, and part (b) a very accurate calculation of the solution ata certain later time. Parts (c) and (d) display the error at this same later time for RBF-FD/ACsolutions when using N = 38, 400 and N = 153, 600 nodes, respectively (using IMQ type RBFs;stencils of type (b) (as shown in Figure 16): n = 19, polynomials degree 3; stencils of type (c):
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Figure 16: The three stencil types (a), (b), (c) used in a hybrid approach combining FD withRBF-FD/AC.
Figure 17: The naıve supporting monomials up through degree 4 compared to the interface specificones in the special case of cL = 1, cR = 2, ρL = ρR = 1.
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Figure 18: Test calculation for the ‘micro-Marmousi’ example using the RBF-FD/AC approach,showing better than a factor of 10 reduction in error when the number of nodes is doubled.
n = 38, polynomials degree 2). Since the color bars for these latter two cases ((c) and (d) in Figure18) are identical, one can readily note that halving the typical node separation h has reduced theerror by more than a factor of ten. For a scheme that is third order accurate everywhere, theexpected error reduction would have been a factor of eight.
7 Conclusions
Ever since RBFs were first introduced for multivariate interpolation, their range of applicationshas grown tremendously. In some sense, computational experiences with RBFs for PDEs are nowwell ahead of the more strict analysis of these methods. With so many free parameters associatedwith irregularly scattered nodes, strict numerical analysis obviously becomes far more difficultthan for lattice-based methods, such as FD or PS. This lack of rigorous theory, for example withregard to stability during time stepping, might have somewhat delayed the broad adoption of RBFbased methods for large applications. However, the number of successful large-scale benchmarkcomparisons against alternative PDE approaches is now steadily increasing. The present authorshope that this will stimulate new advances on all fronts of the topic of RBFs for PDEs: theoretical,computational, as well as still further extending the range of application areas.
Acknowledgments:
The presented research was supported by the NSF grants DMS-0914647, DMS-0934317, OCI-0904599 and by Shell International Exploration and Production, Inc.
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