a hybrid radial basis functionpseudospectral method for...

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Volume 11, Number 7

8 July 2010

Q07003, doi:10.1029/2009GC002985

ISSN: 1525‐2027

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A hybrid radial basis function–pseudospectral methodfor thermal convection in a 3‐D spherical shell

G. B. WrightDepartment of Mathematics, Boise State University, Boise, Idaho 83725, USA([email protected])

N. FlyerInstitute for Mathematics Applied to Geosciences, National Center for Atmospheric Research, Boulder,Colorado 80305, USA ([email protected])

D. A. YuenDepartment of Geology and Geophysics, University of Minnesota, Minneapolis, Minnesota 55455,USA ([email protected])

[1] A novel hybrid spectral method that combines radial basis function (RBF) and Chebyshev pseudospec-tral methods in a “2 + 1” approach is presented for numerically simulating thermal convection in a 3‐Dspherical shell. This is the first study to apply RBFs to a full 3‐D physical model in spherical geometry.In addition to being spectrally accurate, RBFs are not defined in terms of any surface‐based coordinatesystem such as spherical coordinates. As a result, when used in the lateral directions, as in this study, theycompletely circumvent the pole issue with the further advantage that nodes can be “scattered” over the sur-face of a sphere. In the radial direction, Chebyshev polynomials are used, which are also spectrally accurateand provide the necessary clustering near the boundaries to resolve boundary layers. Applications of thisnew hybrid methodology are given to the problem of convection in the Earth’s mantle, which is modeledby a Boussinesq fluid at infinite Prandtl number. To see whether this numerical technique warrants furtherinvestigation, the study limits itself to an isoviscous mantle. Benchmark comparisons are presented withother currently used mantle convection codes for Rayleigh number (Ra) 7 × 103 and 105. Results froma Ra = 106 simulation are also given. The algorithmic simplicity of the code (mostly due to RBFs) allowsit to be written in less than 400 lines of MATLAB and run on a single workstation. We find that our methodis very competitive with those currently used in the literature.

Components: 10,700 words, 7 figures, 6 tables.

Keywords: mantle convection; radial basis function; pseudospectral; isoviscous; spherical geometry.

Index Terms: 0560 Computational Geophysics: Numerical solutions (4255); 8121 Tectonophysics: Dynamics: convectioncurrents, and mantle plumes; 0550 Computational Geophysics: Model verification and validation.

Received 1 December 2009; Revised 3 May 2010; Accepted 19 May 2010; Published 8 July 2010.

Wright, G. B., N. Flyer, and D. A. Yuen (2010), A hybrid radial basis function–pseudospectral method for thermal convectionin a 3‐D spherical shell, Geochem. Geophys. Geosyst., 11, Q07003, doi:10.1029/2009GC002985.

Copyright 2010 by the American Geophysical Union 1 of 18

http://dx.doi.org/10.1029/2009GC002985

1. Introduction

[2] Mantle convection models in spherical geome-try have seen a variety of numerical methodimplementations, from finite element and finitevolume methods on a variety of grids such asicosahedral, cubed sphere, Yin‐Yang, spiral, andhexahedral [Baumgardner, 1985; Hernlund andTackley, 2003; Yoshida and Kageyama, 2004;Harder and Hansen, 2005; Stemmer et al., 2006;Hüttig and Stemmer, 2008; Zhong et al., 2000,2008], to pseudospectral (PS) methods usingspherical harmonics [Bercovici et al., 1989;Harder, 1998]. The former methods can be cum-bersome and tedious to program due to grid gen-eration and treatment of the equations near elementboundaries and are generally low order. The latterrequires more nodes than basis functions (espe-cially when dealiasing filters are used), since thereare 2N + 1 longitudinal Fourier modes for each lat-itudinal associated Legendre function of degree N,and does not easily allow for local refinement.

[3] A novel approach that is in its infancy ofdevelopment is radial basis functions (RBFs), amesh‐less method that has the advantage of beingspectrally accurate for arbitrary node layouts inmultidimensions. Former studies, using thismethod on spherical surfaces, have shown it to bevery competitive in comparison to numericalmethods that are currently used in the geosciences,algorithmically simpler, and naturally permittinglocal node refinement [Flyer and Wright, 2009;Flyer and Lehto, 2010; Flyer and Wright, 2007;Fornberg and Piret, 2008]. However, given thisearly stage of development, numerical modelingexperiments with RBFs are warranted before full‐blown mantle convection models using RBFs aredeveloped that can handle everything from variableviscosity to thermochemical convection. As aresult, this paper is of an exploratory nature fromthe perspective of numerics. Since no 3‐D modelusing RBF spatial discretization of partial differ-ential equations (PDEs) in spherical geometryexists in the math or science literature, it followsthat taking the simplest formulation for mantleconvection, isoviscous flows at various Rayleighnumbers (as is done by Bercovici et al. [1989] andHarder [1998]) would be a good starting point.

[4] The paper is organized as follows: Section 2describes the physical model; section 3 gives anintroduction to RBFs; section 4 shows how thespatial operators are discretized using RBFs;section 5 reviews the concept of influence matrices

that must be used for solving the coupled Poissonequations which result from writing the velocity interms of a poloidal potential [see Chandrasekhar,1961]; section 6 describes the time discretization;section 7 provides numerical results from two testcases with comparisons to those in the literatureand results from a Ra = 106 simulation; section 8gives timing results for the benchmark cases andsection 9 discusses extensions of the method to highRa, variable viscosity, and local node refinement.Appendices A and B give the steps for implement-ing the RBF‐PS algorithm.

2. Physical Model

[5] We consider a thermal convection model of aBoussinesq fluid at infinite Prandtl number in aspherical shell that is heated from below. Thegoverning equations are

r � u ¼ 0 ðcontinuityÞ; ð1Þ

r � � ruþ frugT� �h i

þ Ra T r̂ ¼ rp ðmomentumÞ; ð2Þ

@T

@tþ u � rT ¼ r2T ðenergyÞ; ð3Þ

where u = (ur, u�, ul) is the velocity field inspherical coordinates (� = latitude, l = longitude),p is pressure, T is temperature, r̂ is the unit vectorin the radial direction, h is the viscosity, and Ra isthe Rayleigh number. The boundary conditions onthe velocity of the fluid at the inner and outersurfaces of the spherical shell are

urjr¼Ri;Ro¼ 0|fflfflfflfflfflfflfflffl{zfflfflfflfflfflfflfflffl}

impermeable

and

r@

@r

u�r

� ��r¼Ri;Ro

¼ r@

@r

u�r

� ��r¼Ri;Ro

¼ 0|fflfflfflfflfflfflfflfflfflfflfflfflfflfflfflfflfflfflfflfflfflfflfflfflfflfflfflfflfflfflfflfflfflffl{zfflfflfflfflfflfflfflfflfflfflfflfflfflfflfflfflfflfflfflfflfflfflfflfflfflfflfflfflfflfflfflfflfflffl}shear stress free

; ð4Þ

where Ri is the radius of the inner surface of theshell and Ro is the radius of the outer surface asmeasured from the center of the earth. Theboundary conditions on the temperature are

TðRi; �; �Þ ¼ 1 and TðRo; �; �Þ ¼ 0:

Equations (1)–(3) have been nondimensionalizedwith the length scale chosen as the thickness of theshell, DR = Ro − Ri, the time scale chosen as thethermal diffusion time, t = (DR)2/� (� = thermaldiffusivity), and the temperature scale chosen as the

GeochemistryGeophysicsGeosystems G3G3 WRIGHT ET AL.: RBF-PS METHOD FOR 3-D THERMAL CONVECTION 10.1029/2009GC002985

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difference between the temperature at the inner andouter boundaries, DT.

[6] In this study, we treat the fluid as isoviscous, h =const. Thus, the dynamics of the fluid are governedby the Ra, which can be interpreted as a ratio of thedestabilizing force due to the buoyancy of theheated fluid to the stabilizing force due to the vis-cosity of the fluid. It takes the specific form of

Ra ¼ �g�DTðDRÞ3��

;

where r is the density of the fluid, g is the accel-eration due to gravity, and a is the coefficient ofthermal expansion.

[7] Chandrasekhar [1961] [see also Backus, 1966]shows that any divergence‐free field can beexpressed in terms of a poloidal and toroidalpotential, u = r × r × ((Fr)r̂) + r × (Yr̂). If thefluid is isoviscous (or the viscosity stress tensor isspherically symmetric) and satisfies (4) then thefield is purely poloidal (i.e., Y ≡ 0). As a result, thethree‐dimensional continuity and momentumequations (1) and (2) can be alternatively written asa system of two coupled Poisson equations. Thenonlinear thermal convection model can then bewritten as

DsWþ @

@rr2@W@r

� �¼ Ra r T ; ð5Þ

DsFþ @

@rr2@F@r

� �¼ r2W; ð6Þ

@T

@t¼ � ur

@T

@rþ u�

1

r

@T

@�þ u�

1

r cos �

@T

@�

� �þ 1

r2DsT þ 1

r2@

@rr2@T

@r

� �; ð7Þ

where � 2 [−p/2, p/2], l 2 (−p, p], and Ds is thesurface Laplacian operator. The velocity boundaryconditions (4) in terms of F are

Fjr¼Ri;Ro¼ 0 and

@2F@r2

��r¼Ri;Ro

¼ 0: ð8Þ

The components of the velocity u = (ur, u�, ul) aregiven by

u ¼ r�r� ðFrÞr̂½ �

¼ 1

rDsF;

1

r

@2

@r@�ðFrÞ; 1

r cos �

@2

@r@�ðFrÞ

� �: ð9Þ

We separate the angular and radial directions of theoperators as will be discussed in section 4 on

spatial discretization. Sections 5 and 6 describe thevarious steps of the algorithm.

3. Introduction to RBFs

[8] We only intend to give a brief introduction toRBFs. For a good, in depth discussion see Fasshauer[2007]. The strength of RBFs lie in approximationproblems in multidimensional space with scatterednode layouts [Fornberg et al., 2010]. In the contextof solving partial differential equations (PDEs), theglobal RBF approach can be viewed as a majorgeneralization of pseudospectral methods [Fornberget al., 2002, 2004]. The concept behind RBFs is thatby abandoning the orthogonality of the basis func-tions, the nodes can be arbitrarily scattered over thedomain, maintaining spectral accuracy with the abil-ity to node refine in a completely grid‐independentenvironment [Flyer and Lehto, 2010]. This allowsfor geometric flexibility with regard to the shapeof the domain, as well as flexibility in allowing thenodes to be concentrated where greater resolution isneeded. In addition, studies have shown that RBFscan take unusually long time steps in comparison toother methods, such as pseudospectral, spectral ele-ment and finite volume, for solving purely hyperbolicsystems [Flyer and Wright, 2007, 2009; Flyer andLehto, 2010].

[9] RBF spatial discretization is based on linearcombinations of translates of a single radiallysymmetric function that collocates the data, as isillustrated in Figure 1. The argument of the RBF, d,is the Euclidean distance between where the RBF iscentered xj 2 Rn and where it is evaluated x 2 Rn

with n being the dimension of the space, i.e., d =kx − xjk2 (from now on for simplicity we dropthe subscript 2). Since its argument only dependson a scalar distance, independent of coordinates,dimension or geometry, RBFs are exceptionallysimple to program with the algorithmic complexityof the code not increasing with dimension. Forexample, for two points on the surface of the unitsphere, x1 = (x1, y1, z1) and x2 = (x2, y2, z2) (or inspherical coordinates (�1, l1) and (�2, l2)), wherex1 is the center of the RBF and x2 is where it is tobe evaluated, the argument of the RBF is

d ¼ffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiðx2 � x1Þ2 þ ðy2 � y1Þ2 þ ðz2 � z1Þ2

q¼

ffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffi2ð1� xT2 x1Þ

q¼

ffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffi2ð1� cos �2 cos �1 cosð�2 � �1Þ � sin �2 sin �1Þ

p:

Notice that the distance is not measured as greatarcs along the sphere but rather as a straight linethrough the sphere. Thus, the RBF has no “sense”


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that it exists on a spherical manifold. It should beemphasized that the coordinate system is only usedto identify the location of the nodes and not arepresentation of any grid or manifold (i.e., geom-etry) in n‐dimensional space. Thus, if we choose torepresent the node locations in spherical coordinates,a latitude‐longitude grid is never used, but rather thenodes are placed as the user desires.

[10] Common RBFs are listed in Table 1. There aretwo distinct kinds, piecewise smooth and infinitelysmooth. Piecewise smooth RBFs lead to algebraicconvergence as they contain a jump in somederivative, e.g., ∣d∣3 jumps in the third derivative.Infinitely smooth RBFs lead to spectral conver-gence as they do not jump in any derivative and thuswill be used in this paper. This latter group features aparameter " which determines the shape of the RBFand plays an important role in both the conditioningand accuracy of RBF matrices [Fornberg and Flyer,2005; Buhmann, 2003]. How the error of the solu-tion varies as a function of the shape parameter " forsolving different classes of PDEs and what are theoptimal choices for it has been studied by Iske[2004], Wright and Fornberg [2006], Wertz et al.[2006], Driscoll and Heryundono [2007], Flyerand Wright [2007], Fasshauer and Zhang [2007],Fornberg and Zuev [2007], Fornberg and Piret[2008], and Flyer and Wright [2009].

[11] The above studies have shown that best resultsare achieved with roughly evenly distributed nodes.Since only a maximum of 20 nodes can be evenlydistributed on a sphere, there are a multitude ofalgorithms to define “even” distribution for largernumbers of nodes, such as equal partitioned area,convex hull approaches, Voronoi cells, electrostaticrepulsion [Hardin and Saff, 2004]. Although any ofthese will suffice, we have decided to use an elec-

Figure 1. (a) Data values { fj}j=1N , (b) the RBF colloca-

tion functions, and (c) the resulting RBF interpolant.

Table 1. Commonly Used RBFs

Abbreviation Name Definition

Piecewise SmoothMN monomial ∣d∣2m+1TPS thin plate spline ∣d∣2m ln ∣d∣

Infinitely SmoothMQa multiquadric

ffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffi1þ ð"dÞ2

qIMQ inverse MQ 1ffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffi

1þ ð"dÞ2q

IQ inverse quadratic 1

1þ ð"dÞ2GA Gaussian e−("d)

2

aThe MQ is used for all results in this study.


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trostatic repulsion or minimal energy (ME) approachsince the nodes do not line up along any verticesor lines, emphasizing the arbitrary node layout andcoordinate‐free nature of a RBF methodology, ascan be seen by the node layout in Figure 2a. A veryimportant consequence of this is that although thePDEs of the physical model are posed and solved inspherical coordinates, there are no pole singularities.

4. Spatial Discretization

[12] To numerically solve (5)–(7) a “2 + 1” layer-ing approach is used, where the lateral directions

(�, l) are discretized separately from the radialdirection. Using collocation, the approximate solu-tion is calculated at the nodes shown in Figure 2b.We use M + 2 Chebyshev nodes in the radialdirection (corresponding to M interior points and2 boundary points) and N “scattered” nodes oneach of the resulting M spherical surfaces. Asshown in Figure 2b, this gives a tensor productstructure between the radial and lateral directions,which allows the spatial operators to be computedin O(M2N) + O(MN2) operations instead of O(M2N2) as discussed below. While all radial deriva-tives are discretized using Chebyshev polynomials,differential operators in the latitudinal direction �and longitudinal direction l are approximated dis-cretely on each spherical surface using RBFs. Insections 4.1 and 4.2, we will discuss how to dis-cretize the lateral advection and surface Laplacianoperators using RBFs, with a novel RBF formula-tion of the latter. The radial discretization by collo-cation with Chebyshev polynomials is standard andis therefore omitted (see, for example, Fornberg[1995], Trefethen [2000] or Weideman and Reddy[2000] for details).

4.1. RBF Discretization of the LateralAdvection Operator

[13] Given a velocity field tangent to the unit spherethat is a function of time and space, u = {u�(�, l, t),ul(�, l, t)}, the lateral advection operator is givenby

u � r ¼ u�@

@�þ u�cos �

@

@�: ð10Þ

which is singular at � = ±�2, the north and southpoles, unless @

@� also vanishes there. We will nextsee that this is exactly what happens when the oper-ator is applied to an RBF.

[14] Setting " =1 (for simplicity of notation), letj(d) =(

ffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffi2ð1� cos � cos �j cosð�� jÞ � sin � sin �jÞ

p) be

an RBF centered at the node (�j, lj). Using thechain rule, the partial derivatives of the RBF j(d)with respect to l and � are given by

@

@�jðdÞ ¼ @d

@�

@

@d¼ cos � cos �j sinð�� jÞ 1

d

@j

@d

� �; ð11Þ

@

@�jðdÞ ¼ @d

@�

@

@d

¼ ðsin � cos �j cosð�� jÞ � cos � sin �jÞ 1

d

@j

@d

� �:

ð12Þ

Figure 2. (a) RBF node layout on the surface of asphere and (b) 3‐D view of the discretization of thespherical shell used in the hybrid RBF‐PS calculation.Blue is the outer boundary, and red is the inner bound-ary; black dots display the computational nodes, whichare distributed in the radial direction along the extremaof the Chebyshev polynomials. Note that the sphericalshell has been opened up in Figure 2b to show the detail.


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Inserting (11) and (12) into (10), we have

u � r ¼ u�ðcos �j sin � cosð�� jÞ � sin �j cos �Þ 1

d

@j

@d

� �þ u� cos �j sinð�� jÞ 1

d

@j

@d

� �: ð13Þ

Given that the velocities are smooth, notice thatnowhere on the sphere is (13) singular.

[15] Now, we have all the components that arenecessary to build the action of the advectionoperator on an RBF representation of the temper-ature field. We first represent T(l, �) as an RBFexpansion given by

Tð�; �Þ ¼XNj¼1

cjjðdð�; �ÞÞ: ð14Þ

where cj are the unknown expansion coefficients.We then apply the exact differential operatoru · r to (14) and evaluate it at the node locations,{(li, �i)}i=1

N , where T(l, �) is known. Note thatbecause ul and u� are time dependent we will needto create two separate differentiation matrices, oneto represent 1

cos �@@�j(d) and another to represent

@@�j(d), otherwise (13) could be written as a singledifferentiation matrix. Since they are created in thesame way, we will only demonstrate how to for-mulate Dl, the differentiation matrix for the lon-gitudinal direction:

[16] 1. Take 1cos �

@@� of (14):

1

cos �

@Tð�; �Þ@�

¼XNj¼1

cj1

cos �

@jðdÞ@�

¼XNj¼1

cj cos �j sinð�� jÞ 1

d

@j

@d

� �:

[17] 2. Evaluate step 1 at the node locations:

XNj¼1

cj cos �j sinð�i � �jÞ 1d@j

@d

� ��ð�;�Þ¼ð�i;�iÞNi¼1|fflfflfflfflfflfflfflfflfflfflfflfflfflfflfflfflfflfflfflfflfflfflfflfflfflfflfflfflfflfflfflfflfflfflffl{zfflfflfflfflfflfflfflfflfflfflfflfflfflfflfflfflfflfflfflfflfflfflfflfflfflfflfflfflfflfflfflfflfflfflffl}

Components of matrix B�

¼ B�c;

where c contains the N unknown discrete expan-sion coefficients. If we evaluate the RBFs in (14) atthe node locations {(li, �i)}i=1

N then we have thecollocation problem

ðkx1 � x1kÞ � � � ðkx1 � xNkÞ... . .

. ...

ðkxN � x1kÞ � � � ðkxN � xNkÞ

26643775

|fflfflfflfflfflfflfflfflfflfflfflfflfflfflfflfflfflfflfflfflfflfflfflfflfflfflfflfflfflfflfflfflfflfflffl{zfflfflfflfflfflfflfflfflfflfflfflfflfflfflfflfflfflfflfflfflfflfflfflfflfflfflfflfflfflfflfflfflfflfflffl}A

c1

..

.

cN

26643775

|fflfflffl{zfflfflffl}c

¼T1

..

.

TN

26643775

|fflfflffl{zfflfflffl}T

;

ð15Þ

where A is the RBF interpolation matrix for thenode set. Thus, c = A−1T and substituting this intostep 2 above gives DlT = Bl(A

−1T), in other wordsDl = BlA−1. Put verbally, the differentiation matri-ces are obtained by applying the exact differentialoperator to the interpolant and then evaluating itat the data locations. Although the computationof Dl and D� requires O(N

3) operations, it is a pre-processing step that needs to be done only once.

4.2. A Novel RBF Surface LaplacianFormulation

[18] Since RBFs do not require the nodes on aspherical surface to have any directionality andsince RBFs are not defined in terms of any surface‐based coordinate system (as discussed in section 3),then for simplicity let us center an RBF at the northpole xnp = (0, 0, 1) or (�, l) = (p/2, 0). The distancefrom this point to any point on the sphere is thengiven by

dðxÞ ¼ kx� xnpk ¼ffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffix2 þ y2 þ ðz� 1Þ2

q¼

ffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffi2ð1� sinð�ÞÞ

p:

ð16ÞNow, the surface Laplacian in spherical coordinatesin given by

Ds ¼ @2

@�2� tan �

@

@�þ 1

cos2 �

@2

@�2 : ð17Þ

However, for a radial function centered at thenorth pole there will be no l dependence. So, (17)reduces to

Ds ¼ @2

@�2� tan �

@

@�: ð18Þ

Applying (18) to an RBF (d) gives

DsðdÞ ¼ @2d

@�2@

@dþ @d

@�

� �2@2

@d2� tan �

@d

@�

@

@d: ð19Þ

With the use of (16) and after some algebra, (19)reduces to

DsðdÞ ¼ 1

4ð4� d2Þ @

2

@d2þ 4� 3d2

d

@

@d

: ð20Þ

Although we derived this formula by centering theRBF at the north pole, any node could have servedas the north pole since an RBF is invariant tocoordinate rotations. The beauty of (20) is that itexpresses the action of the surface Laplacian on anRBF simply in terms of the distances betweennodes without the coordinate system ever cominginto play. The RBF surface Laplacian differen-tiation matrix is then defined as Ls = BsA

−1,where Bs is now a matrix, evaluating (20) at


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d =ffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffi2ð1�cos�icos�icosð�i��jÞ� sin�i sin �jÞ

p,

1 ≤ i, j ≤ N, j indexing the RBF centers and i thenode locations.

5. Momentum Equation Solverand Influence Matrix Method

[19] We cannot directly solve (5) and (6) since wehave 4 boundary conditions on F, given by (8), andnone on W. We therefore use the influence matrixmethod [Peyret, 2002] to find the unknown bound-ary values on W such that all 4 boundary conditionson F are satisfied. Since (5) and (6) are linear, thesolution to each Poisson equation can be representedas a superposition of two solutions; the first, Wh andFh, satisfies the right‐hand side of the equationswith homogeneous Dirichlet boundary conditions;the second, Wj and Fj, couples the unknown bound-ary values (abbreviated bd), Wbd,j

Ri and Wbd,jRo , with Fj

Ri

andFjRo at each RBF collocation node, {�j, lj}j=1

N , onthe inner (Ri) and outer (Ro) boundary sphericalsurfaces, respectively. In Table 2, the variables aredefined in terms of the Poisson equations they solvewith the overall solution written as

W ¼ Wh þXNj¼1

WRibd; jW

Rij þ WRo

bd; jWRoj

h iand

F ¼ Fh þXNj¼1

WRibd; jF

Rij þ WRi

bd; jFRoj

h i: ð21Þ

The method is reminiscent of a Green’s functiontype approach, but instead of expanding the sourceterm of the PDEs in Dirac delta functions, weexpand the unknown boundary conditions in thisbasis, solve Laplace’s equation (as the right‐handside is taken care of by the solutions Wh and Fh)and superpose the solutions as is done by thesummations in (21). Thus, for each boundary, we

are building a table of N particular solutions,{Wj

Ri}j=1N and {Wj

Ro}j=1N , whose boundary value is 1

at the jth boundary node and 0 at all others. ThesePDEs are solved N times, corresponding to thenumber of boundary nodes we have on eachboundary surface. It is important to note thatsolving for Wj

Ri and WjRo is a preprocessing step,

since the equation is temperature independent andthus time independent. Also, once we solve for Fj

Ri

and FjRo, Wj

Ri and WjRo can be deleted as they are no

longer needed for any computations. As discussedin Appendix A, the approximate solutions to thePDEs listed in Table 2 are computed in spectralspace via a matrix diagonalization (or eigenvectordecomposition) technique which requires O(MN2) +O(M2N) operations per PDE and O(M2) + O(N2)storage. This is significant savings over a direct solveof the equations, which would require O(M2N2)operations and O(M2N2) storage.

[20] Once Fh has been computed, the unknowncoefficients Wbd, j

Ri and Wbd, jRo are determined by

requiring the linear combination of Fh, FjRi, and

FjRo in (21) satisfy the boundary conditions (8). Since

each of these variables satisfy the homogeneousDirichlet boundary conditions by construction, theunknown coefficients are determined by the secondNeumann‐type boundary condition. Inserting theexpression for F given in (21) into this boundarycondition leads to the following set of linearequations which need to be enforced at each bound-ary node j = 1,…, N:

@2

@r2FRij

��r¼Ri

WRibd; j þ

@2

@r2FRoj

��r¼Ri

WRobd; j ¼ � @2

@r2Fh

��r¼Ri

; ð22Þ

@2

@r2FRij

��r¼Ro

WRibd; j þ

@2

@r2FRoj

��r¼Ro

WRobd; j ¼ � @2

@r2Fh

��r¼Ro

: ð23Þ

Table 2. Variables Composing the Influence Matrix Method With the PDEs and Boundary Conditions They Solve and If TheyAre Time Dependenta

Variable PDE Boundary Conditions at r = Ri, Ro Time Dependent

Wh DWh = Ra r T Wh∣Ri,Ro = 0 YesFh DFh = Wh Fh∣Ri,Ro = 0 Yes

WjRi DWj

Ri = 0 WjRi∣Ro = 0, Wj

Ri∣Ri =1 if ð�; �Þ ¼ ð�j; �jÞ;0 otherwise

�No

FjRi DFj

Ri = WjRi Fj

Ri∣Ri,Ro = 0 No

WjRo DWj

Ro = 0 WjRo∣Ri = 0, Wj

Ro∣Ro =1 if ð�; �Þ ¼ ð�j; �jÞ;0 otherwise

�No

FjRo DFj

Ro = WjRo Fj

Ro∣Ri,Ro = 0 No

Wbd, jRi , Wbd, j

Ro see (22)–(23) not applicable YesaThat is, if they need to be solved at every time step.


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The coefficient matrix that arises from this 2N × 2Nlinear systems is called the influence matrix and canbe precalculated, LU decomposed and stored as itis time independent. However, the solution to thelinear system must be computed every time stepas Fh is time dependent. Once Wbd, j

Ri and Wbd, jRo are

found, F can be determined from the second equa-tion in (21) and then the velocity field can be cal-culated according to (9).

[21] While the presentation above is the moststraightforward way to describe the influencematrix technique for solving equations (5) and (6)subject to the boundary conditions (8), it is notthe most computationally efficient given how thesolutions to Wh and Fh are computed in theoverall algorithm. In Appendix A, we discuss howthe computation can be done in the spectral spaceof the discrete operators to reduce the computa-tional cost. This description is the one used in thecode.

6. Time Discretization

[22] The Chebyshev discretization of the radialcomponent of the diffusion operator has a Courant‐Friedrichs‐Lewy (CFL) condition on the timestep that is proportional to O(1/M4), which makesan explicit scheme implausible. As a result, weimplement a semi‐implicit time stepping schemewhich treats this component implicitly and theremaining terms of the energy equation explicitly.We note that implicitly time stepping the entirediffusion term, that is also the RBF discretization ofthe Ds operator, would make no difference in termsof the overall CFL condition on the energy equa-tion. This is because the CFL condition on thisoperator results in a time step restriction that scaleslike O(1/N), with N = O(1000) typically, and thetime step restriction due to the Chebyshev dis-cretization on the radial components of the non-linear advection term scales as O(1/M2), with M =O(10) typically.

[23] We separate the terms in the energy equation (7)as follows:

@T

@t¼ � ur

@T

@rþ u�

1

r

@T

@�þ u�

1

r sin �

@T

@�

� �þ 1

r2DsT|fflfflfflfflfflfflfflfflfflfflfflfflfflfflfflfflfflfflfflfflfflfflfflfflfflfflfflfflfflfflfflfflfflfflfflfflfflfflfflfflfflffl{zfflfflfflfflfflfflfflfflfflfflfflfflfflfflfflfflfflfflfflfflfflfflfflfflfflfflfflfflfflfflfflfflfflfflfflfflfflfflfflfflfflffl}

f ðT ;tÞ

þ 1

r2@

@rr2@T

@r

� �|fflfflfflfflfflfflfflfflfflffl{zfflfflfflfflfflfflfflfflfflffl}

gðT ;tÞ

: ð24Þ

Using a third‐order Adams‐Bashforth (AB3) methodcombined with a Crank‐Nicolson (CN) method,(24) can be discretized by

Tkþ1 ¼ Tk þDt

12ð23Fk � 16Fk�1 þ 5Fk�2Þ|fflfflfflfflfflfflfflfflfflfflfflfflfflfflfflfflfflfflfflfflfflfflfflfflffl{zfflfflfflfflfflfflfflfflfflfflfflfflfflfflfflfflfflfflfflfflfflfflfflfflffl}

AB3

þDt

2ðGkþ1 þ GkÞ|fflfflfflfflfflfflfflfflfflfflfflffl{zfflfflfflfflfflfflfflfflfflfflfflffl}

CN

;

ð25Þ

where all the terms are matrices of size N × M,corresponding to the values in the interior of thespherical shell, and Fk and Gk are the respectiveapproximations to f(T, t) and g(T, t) at the kth timestep and are explicitly given by

Fk ¼ �ðukr � ðTkDrÞ þ uk� � ðD�TkR�1Þ þ uk� � ðD�T

kR�1ÞÞþ LsT

kR�2 þ Bf

Gk ¼ TkLrR�2 þ Bg; ð26Þ

where � denotes element‐wise matrix multiplica-tion, and Bf and Bg contain the appropriate termsfrom the boundary conditions on T. The abbrevia-tions for the differentiation matrices are given inTable 3. The matrices ur

k, u�k, ul

k are the approxima-tions to the respective components of the velocity atthe kth time step, while the diagonal matrix R con-tains the M interior Chebyshev nodes, defined by

Rj; j ¼ 1

2ðRi þ RoÞþ 1

2ðRo� RiÞ cos j

M þ 1�

� �; j ¼ 1; . . . ;M :

ð27Þ

Equation (25) can be rewritten as

Tkþ1 ¼ Tk þDt

12ð23Fk � 16Fk�1 þ 5Fk�2Þ þDt

2Gk

� I �Dt

2LrR

�2

� ��1

: ð28Þ

As a preprocessing step (I − Dt2 LrR

−2) is LU decom-posed and stored. The computational cost of com-puting (28) is then O(M2N). However, since N istypically two orders of magnitude larger thanM, thetotal cost per time step will be dominated by thesolving the momentum equations (see Appendix A),calculating the velocity, and computing the valuesof Fk and Gk, all of which require O(M N2) opera-tions. Exact details on each step of the algorithm aregiven in Appendix B.

7. Validation on the RBF‐PS Method

[24] In this section, we first consider three bench-mark studies for 3‐D spherical shell models ofmantle convection with constant viscosity and


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report the first results in the literature from a purelyspectral method run at Ra = 106. Although there aremany numerical methods in the literature formantle convection in spherical geometry [Bercoviciet al., 1989; Zhang and Christensen, 1993; Ratcliffet al., 1996; Richards et al., 2001; Hernlund andTackley, 2003; Yoshida and Kageyama, 2004;Harder and Hansen, 2005; Stemmer et al., 2006;Choblet et al., 2007; Kameyama et al., 2008; Zhonget al., 2000, 2008], the obstacle we encountered isthat there is not a set of standardized test cases forcomparison with regard to Ra number and theviscosity profile. However, there are a number ofpublished results for Ra = 7000 with constantviscosity and we compare our method with these.Above this Ra, there does not seem to be anyconsistency in the specifications of the physicalmodel for testing the numerical methods publishedin the literature. Thus, for higher Ra, we havedecided to use Ra = 105 results from the model formantle convection, CitcomS, recently reported byZhong et al. [2008] as a benchmark comparison.The only other study in the literature that givesisoviscous results for this Ra number is byRatcliff et al. [1996], which is also included in ourcomparison.

7.1. Ra = 7000

[25] The two most common benchmarks for com-putational models of mantle convection in a spher-ical shell are the steady state tetrahedral and cubictest cases. For both of these benchmarks the fluid istreated as isoviscous andRa is set to 7000. The initialcondition for the temperature is specified as

Tðr; �; �Þ ¼ Riðr � RoÞrðRi � RoÞ þ 0:01Y 2

3 ð�; �Þ sin �r � Ri

Ro � Ri

� �ð29Þ

for the tetrahedral test case and

Tðr; �; �Þ ¼ Riðr � RoÞrðRi � RoÞ

þ 0:01 Y 04 ð�; �Þ þ

5

7Y 44 ð�; �Þ

sin �

r � Ri

Ro � Ri

� �ð30Þ

for the cubic test case, where Y‘m denotes the nor-

malized spherical harmonic of degree ‘ and orderm.The first term in each of the initial conditions rep-resents a purely conductive temperature profile,while the second terms are perturbations to thisprofile and determine the final steady state solution.The � − l temperature dependence of (29) and (30)on a spherical shell surface can be seen in Figures 3aand 3b, respectively.

[26] For this test case, two RBF‐PS simulations arereported: (1) a higher‐resolution case, in which N =1600 nodes were used on each spherical surface(i.e., in the lateral directions) and 23 total Cheby-shev nodes were used in the radial direction (i.e.,M = 21 total interior nodes), giving a total of36,800 nodes, and 2) a lower‐resolution case ofN = 900 and 19 nodes radially (17 interior nodes),giving a total of 17,100 nodes. A time step of 10−4

was used or 10,000 time steps were taken to reachsteady state at the nondimensionalized time of t =1,corresponding to roughly 58 times the age of theEarth. Figures 4a and 4b display the final RBF‐PSsteady state solutions for the tetrahedral and cubictest cases, respectively, in terms of the residualtemperature dT = T(r, �, l) − hT(r)i, where h i

Table 3. Notation for the Various Differentiation MatricesUsed in the Time‐Differencing Scheme

Matrix Operator Discretization Dimension

Dl1

cos �@@� RBF N × N

D�@@� RBF N × N

Ls Ds RBF N × NDr

@@r Chebyshev M × M

Lr @@r r2 @

@r

� Chebyshev M × M

Figure 3. The � − l dependence of the initial conditionfor the (a) tetrahedral and (b) cubic mantle convectiontest cases.


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denotes averaging over a spherical surface. Sinceno analytical solutions exist, validation is done viacomparison to other published results in the liter-ature with respect to scalar global quantities, suchas Nusselt number at the inner and outer bound-aries (Nui and Nuo), and the averaged root meansquare velocity and temperature over the volume.Table 4 contains such a comparison for the RBF‐PS method with respect to popular methods used inthe mantle convection literature. The followingobservations can be can be made:

[27] 1. The only method that is spectral in at leastone direction is the spherical harmonic–finite dif-ference method of Harder [1998]. In the work byStemmer et al. [2006], Harder’s method was usedwith Romberg extrapolation to obtain the results to

at least four digits of accuracy. With regard toalmost all quantities for both test cases, the results ofthe RBF‐PS method match exactly with Harder’sextrapolated results.

[28] 2. The number of nodes (degrees of freedom)needed to accomplish the results in point 1 is anorder of magnitude lower than what was used withthe CitcomS model reported by Zhong et al. [2008],approximately one and a half orders of magnitudelower than either the finite volume method byStemmer et al. [2006] or the method by Harder[1998] and three orders of magnitude less than theYin‐Yang, multigrid method by Kameyama et al.[2008]. It should be noted, however, that withexception to the Stemmer et al. [2006] and Harder[1998] results, a detailed convergence study wasnot performed for these methods to determine theminimal degrees of freedom needed to achieve theirreported results.

[29] 3. For the scheme to conserve energy Nui =Nuo, notice that this is the case for both tests evenwith such a low number of nodes. This results fromthe spectral accuracy of the RBF‐PS method,which by its shear high‐order convergence, willinherently dissipate physical quantities less.

[30] 4. Even though we are using Chebyshevpolynomials in the radial direction and an explicittime‐stepping scheme for the advection term in thetemperature equation, we still can take the samenumber of total time steps as Zhong et al. [2008](i.e., 10,000).

[31] 5. Even if we decrease the number of nodes byover 50%, there is only very minor changes in theresults. From our calculations, it seems that at least17 interior nodes (19 total nodes) are needed in theradial direction to resolve the flow at Ra = 7000.Although not reported, we found that in the lateraldirection the number of RBF nodes could bereduced by another 20% and the values reportedchanged only slightly (in the third decimal place).

7.2. Ra = 105

[32] For the isoviscous Ra = 105 case, there areonly two published studies for comparison, theCitcomS study of Zhong et al. [2008] and Ratcliffet al. [1996]. Both use the cubic test case initialcondition given by (30) and the model is integratedto approximately t = 0.3. Since Ra = 105 is a moreconvective regime, resulting in thinner plumesas seen in Figures 5a and 5b, larger resolutionis needed. Thus, we use 43 Chebyshev nodes

Figure 4. Steady state isosurfaces of the residual tem-perature, dT, at t = 1 for the isoviscous (a) tetrahedraland (b) cubic mantle convection test cases at Ra =7000 computed with the RBF‐PS model. Yellow corre-sponds to dT = 0.15 and denotes upwelling relative tothe average temperature at each radial level, while bluecorresponds to dT = −0.15 and denotes downwelling.The red solid sphere shows the inner boundary of the3‐D shell corresponding to the core.


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(41 interior nodes) in the radial direction and 4096nodes on each spherical surface. Since the time stepis purely restricted by the Chebyshev discretization,the increase in Chebyshev nodes results in a moresevere CFL criterion that causes a necessarydecrease in the time step. The time step for thiscase is 6 × 10−5 for stability or 50,000 time steps toreach t = 0.3 as opposed to the 35,000 time stepsused by Zhong et al. [2008].

[33] Comparative results are given in Table 5. Thefollowing 5 points are notable:

[34] 1. The difference between Nuo and Nui is0.14%, showing that we are close to absoluteconservation of energy.

[35] 2. The results of the RBF‐PS method are muchcloser to that of Zhong et al. [2008] than to those ofRatcliff et al. [1996]. We attribute this difference,most likely, to the underresolution of the runs ofthe latter model.

[36] 3. The RBF‐PS uses approximately an order ofmagnitude less degrees of freedom than the studyby Zhong et al. [2008].

[37] 4. Once steady state has been reached, differ-ences between our results and those of Zhong et al.[2008] are within 0.4% for Nuo and Nui, 0.2% forhVrmsi, and 0.9% for hTi.[38] 5. Even during the startup of the model, thecurves for Nuo, hVrmsi, and hTi as a function oftime, are almost indistinguishable from the resultsof Zhong et al. [2008], as seen in Figures 6a–6c.

7.3. Ra = 106

[39] Since no results from a purely spectral methodhave ever been reported in 3‐D spherical geometryat Ra = 106, we have decided to include this result.The common practice at this and larger Ra is tostart the simulation with an initial condition takenfrom a simulation run at a lower Ra. The primary

Table 4. Comparison Between Computational Methods for the Isoviscous Tetrahedral and Cubic Mantle Convection Test CasesWith Ra = 7000a

Model Type Nodes r × (� × l) Nuo Nui hVrmsi hTiCubic Test Case, Ra = 7000

Zhong et al. [2008] FE 393216 32 × (12 × 32 × 32) 3.6254 3.6016 31.09 0.2176Yoshida and Kageyama [2004] FD 2122416 102 × (102 × 204) 3.5554 – 30.5197 –Kameyama et al. [2008] FD 12582912 128 × (2 × 128 × 384) 3.6083 – 31.0741 0.21639Ratcliff et al. [1996] FV 200000 40 × (50 × 100) 3.5806 – 30.87 –Stemmer et al. [2006] FV 663552 48 × (6 × 48 × 48) 3.5983 3.5984 31.0226 0.21594Stemmer et al. [2006] FV Extrap. Extrap. 3.6090 – 31.0709 0.21583Harder [1998] andStemmer et al. [2006]

SP‐FD 552960 120 × (48 × 96) 3.6086 – 31.0765 0.21582

Harder [1998] andStemmer et al. [2006]

SP‐FD Extrap. Extrap. 3.6096 – 31.0821 0.21578

RBF‐PS SP 36800 23 × (1600) 3.6096 3.6096 31.0820 0.21577RBF‐PS SP 17100 19 × (900) 3.6098 3.6098 31.0834 0.21579

Tetrahedral Test Case, Ra = 7000Zhong et al. [2008] FE 393216 32 × (12 × 32 × 32) 3.5126 3.4919 32.66 0.2171Yoshida and Kageyama [2004] FD 2122416 102 × (102 × 204) 3.4430 – 32.0481 –Kameyama et al. [2008] FD 12582912 128 × (2 × 128 × 384) 3.4945 – 32.6308 0.21597Ratcliff et al. [1996] FV 200000 40 × (50 × 100) 3.4423 – 32.19 –Stemmer et al. [2006] FV 663552 48 × (6 × 48 × 48) 3.4864 3.4864 32.5894 0.21564Stemmer et al. [2006] FV Extrap. Extrap. 3.4949 – 32.6234 0.21560Harder [1998] andStemmer et al. [2006]

SP‐FD 552960 120 × (48 × 96) 3.4955 – 32.6375 0.21561

Harder [1998] andStemmer et al. [2006]

SP‐FD Extrap. Extrap. 3.4962 – 32.6424 0.21556

RBF‐PS SP 36800 23 × (1600) 3.4962 3.4962 32.6424 0.21556RBF‐PS SP 17100 19 × (900) 3.4964 3.4963 32.6433 0.21557

aNuo and Nui denote the Nusselt number at the outer and inner spherical surfaces, respectively; hVrmsi denotes the volume‐averaged RMSvelocity over the 3‐D shell; and hT i denotes the mean temperature of the 3‐D shell. Extrap. indicates that the results were obtained usingRomberg extrapolation. Dashes indicate that numbers were not reported. Abbreviations are as follows: FE, finite element; FD, finite difference;FV, finite volume; SP‐FD, hybrid spectral and finite difference; SP, purely spectral. For the RBF‐PS method, the standard deviation of all thequantities from the last 1000 time steps was less than 5 × 10−5, which is a standard measure for indicating the model has reached numericalsteady state.


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reason for this is to avoid the extremely highvelocity values that occur at higher Ra during theinitial redistribution of the temperature from aconductive profile to a convective profile, whichseverely restricts the time steps that can be taken.The initial condition used at Ra = 106 was taken

from a simulation which was started at Ra = 105

with an initial condition consisting of two terms, apurely conductive temperature profile plus a smallperturbation in the lateral direction that randomlycombined all spherical harmonics up to degree 10.This latter term was multiplied by the same sineterm in the radial direction as used in the previouscases. For the discretization, we used 81 Cheby-shev nodes in the radial direction and 6561 nodeson each spherical surface, for a total of 531,441nodes. The Ra = 105 simulation was run until thelarge spike in the radial velocity subsided, the Rawas then increased to 5 × 105 and the simulationwas restarted with the Ra = 105 solution as theinitial condition. This process was repeated oncemore and the Ra was then increased to 106 and thetime was reset to 0. The Ra = 106 simulation resultsfrom t = 0 to t = 0.08 (approximately 4 and half timesthe age of the Earth), are displayed in Figure 7.Figure 7a displays the isosurfaces of the residualtemperature at t = 0.08 and clearly shows the mantlein a purely convective regime. Figures 7b–7d showthe time traces of hTi, hVrmsi, and Nuo and Nui,respectively. Since this is a purely convectiveregime the choice of ending time is somewhatarbitrary. We chose to stop the simulation at t = 0.08since by this time the average temperature haddecreased to an acceptable level from its initialstarting value and the influence of the initial con-dition had diminished.

8. Timing Results

[40] In this short section, runtime results are pre-sented in Table 6 in order to give the reader a feelfor how long it takes to run the code. All test caseswere conducted on a workstation with one Intel i7940 2.93 GHz processor, which is a quad core pro-cessor. The code was written in MATLAB and rununder version 2009b with BLAS multithreadingenabled. 8 GB of memory was also available forthe calculations, although only a fraction of thiswas actually used. The results under “total runtime”in Table 6 include the preprocessing steps in

Figure 5. (a) Steady state isosurface temperature T =0.5 at time t = 0.3. (b) Steady state isosurface of theresidual temperature, dT = T(r, l, �) − hT(r)i, at t =0.3 for the isoviscous cubic mantle convection test caseat Ra = 105 computed with the RBF‐PS model. Thesame color scheme as Figure 4 has been applied.

Table 5. Comparison Between Computational Methods for the Isoviscous Cubic Mantle Convection Test Case With Ra = 105a

Model Type Nodes r × (� × l) Nuo Nui hVrmsi hTiZhong et al.[2008]

FE 1,327,104 48 × (12 × 48 × 48) 7.8495 (0.0054) 7.7701 (0.001) 154.8 (0.04) 0.1728 (0.0002)

Ratcliff et al.[1996]

FV 200,000 40 × (50 × 100) 7.5669 – 157.5 –

RBF‐PS SP 176,128 43 × (4096) 7.8120 7.8005 154.490 0.17123aNumbers in parentheses for Zhong et al. [2008] represent standard deviations. The standard deviation of the results from the last 1000 time steps

for the RBF‐PS model were all less than 5 × 10−5.


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Appendix B, such as setting up the differentiationmatrices and diagonalizing them.

9. Extensions of the Method: High Ra,Variable Viscosity, and Local NodeRefinement

[41] Each of these extensions requires differentalterations of the method. We discuss them below inorder of increasing complexity in terms of alteringthe method.

9.1. High Ra[42] For simulations at high Ra, the resolution ofthe model must of course be increased further. Themost expensive steps in the method are the ellipticsolver and the time step due to the severe CFLrestriction of Chebyshev discretization. With onlymoderate losses in accuracy, the time step restric-tion would be alleviated if sixth‐order implicit (orcompact) finite differences were used instead todiscretize the radial direction. This would be aminor change to the implementation of the method.The advantage of a sixth‐order implicit schemeover a sixth‐order explicit scheme is that (1) thestencil size is almost 50% less for the same accuracy;(2) they have smaller error constants; (3) informa-tion only needs to be extrapolated to one pointoutside the boundary; and (4) most importantly,they have spectral‐like resolution in the sense thatthey resolve higher wave numbers [Lele, 1992]. Todate, the authors are not aware of a study that hasemployed such a scheme. With regard to theelliptic solver, global RBFs do not scale well ashigh Ra convection is reached (i.e., Ra ∼ O(109)).However, RBF‐based finite difference schemes(see section 9.3), already show exceptionallypromising results on spherical surfaces, in termscomputational scaling and accuracy (B. Fornbergand E. Lehto, A filter approach for stabilizingRBF‐generated finite difference methods for con-vective PDEs, manuscript in preparation, 2010;N. Flyer et al., A radial basis function generatedfinite difference method for the unsteady shallowwater equations on a sphere, manuscript in prepa-ration, 2010).

9.2. Variable Viscosity

[43] To handle variable viscosity (depth and/orhorizontal dependent), the PDEs in their primitivevariables, i.e., (1) and (2), would need to be dis-cretized. The main difficulty here is not in the

Figure 6. Time plots of the (a) hT i, (b) hVrmsi, and(c) Nuo, comparing the results (obtained through Com-putational Infrastructure for Geodynamics (http://www.geodynamics.org/cig/workinggroups/mc/workarea/benchmark/3dconvention/)) of the CitcomS and RBF‐PSfor isoviscous mantle convection at Ra = 105.


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discretization of the variable coefficients terms thatappear in (2), which can be handled straightfor-wardly by the method, but from solving theresulting coupled equations to ensure that the flowremains incompressible. This system will be ofsaddle point type for which many potential meth-ods are available (see, for example, the review byBenzi et al. [2005]). The most promising approachfor the RBF‐PS system, currently being exploredby authors, would use a block factorization of thediscretized momentum and continuity equations,giving an upper block triangular system andresulting in a Schur complement problem. To avoidthe excessive computational cost in solving thissystem directly at every time step, we wouldinstead employ a preconditioned Krylov subspacemethod (see Benzi et al. [2005, section 10] for adiscussion). A good choice for the preconditionermight be to use a coarse solution to the isoviscousproblem, as presented by the method in section 5,since the velocities should be sufficiently smoothgiven that they are solutions to Poisson‐type PDEs.

9.3. Local Refinement

[44] Local refinement will take the greatest alter-ation to the code. Once variables are separated, inthis case (�, l) from r, the result is a tensor‐likegrid and local refinement becomes difficult. ForRBFs, refinement is simple in the sense that thenodes can be easily clustered [Flyer and Lehto,2010]. Thus, the next developmental step, whichis currently in progress, is to use 3‐D RBF‐basedfinite difference stencils [Wright and Fornberg,2006] for discretizing the equations. Here, all dif-ferentiation matrices are based on local high‐orderfinite difference‐type stencils that are generated bymeans of RBFs from nodes scattered in 3‐D space,which results in exceptionally sparse matrices tosolve.

10. Summary

[45] This paper develops the first spectral RBFmethod for 3‐D spherical geometries in the math/science literature. Applications of this new hybrid

Figure 7. Results for Ra = 106 test case: (a) Residualtemperature, dT, at t = 0.08, where yellow correspondsto dT = 0.1, blue corresponds to dT = −0.1, and thered solid sphere shows the inner boundary of the 3‐Dshell corresponding to the core. (b) Average tempera-ture, hTi, versus time. (c) Average RMS velocity, hVrmsi,versus time. (d) Inner and outer Nusselt numbers, Nuiand Nuo, versus time.


14 of 18

purely spectral methodology are given to theproblem of thermal convection in the Earth’smantle, which is modeled by a Boussinesq fluidat infinite Prandtl number. To see whether thisnumerical technique warrants further investigation,the study limits itself to an isoviscous mantle. TwoRa number regimes are tested, the classical case inthe literature of Ra = 7000 and the latest resultsfrom the CitcomS model [Zhong et al., 2008] atRa = 105. Also, a Ra = 106 simulation is run, asthere are no results in the literature from a purelyspectral model at this Ra. For the Ra = 7000 case,the method perfectly conserved energy, matchedthe extrapolated results of the only other partiallyspectral method (reported by Stemmer et al. [2006])to four or 5 significant digits and used anywherebetween 1 to 3 orders of magnitude less nodes thanother methods. For the Ra = 105 case, the methodalmost perfectly conserved energy, and gave resultsthat differed from CitcomS by 0.2% to 0.9%(depending on the scalar quantity being measured)yet required approximately an order of magnitudeless nodes. All calculations were run on a work-station using a single quad core processor with Ra =7000 case taking about 8 min and the Ra = 105 casetaking about 6.5 h. Given this outcome, the meth-odology warrants more extensive testing. It will bealtered to accommodate the PDEs in primitive formso that variable viscosity and thermochemicalconvection can be considered in future test runs.

[46] We conclude by noting that the spectral RBFmethod presented here may also be a potentiallypromising technique for simulating the geodynamo,which is well approximated by an isoviscous andBoussinesq fluid. For the geodynamo, the Prandtlnumber is finite and the (now time‐dependent)momentum and energy equations are coupled to atime‐dependent induction equation for the magneticfield [see, e.g., Glatzmaier and Roberts, 1995]. Forthe hybrid RBF method, a natural approach wouldbe to decompose the velocity and magnetic fieldsinto toroidal and poloidal potentials as is com-monly done in other models [see, e.g., Christensenet al., 2001; Glatzmaier and Roberts, 1995; Oishiet al., 2007]. This decomposition results in a cou-pled set of time‐dependent parabolic equations anda set of time‐independent elliptic equations. For the

former, a similar approach to the discretization forthe energy equation discussed in section 6 could beused, while the latter could be solved using thetechnique discussed in Appendix A.

Appendix A: Solution of the MomentumEquations via Matrix Diagonalization

[47] The two coupled Poisson equations represent-ing the momentum equations are solved throughmatrix diagonalization (i.e., spectral decomposi-tion) of the RBF surface Laplacian and Chebyshevradial operators and the influence matrix methodbriefly discussed in section 5. In this appendix, wefirst show how the two coupled equations in rows 1and 2 of Table 2 are solved for Wh and Fh. We thendescribe the influence matrix method in the spectralspace of the discrete operators We conclude withdetails on how the full solution for F is obtained.

[48] If N is the number of nodes on a sphericalsurface and M is the number of interior Chebyshevnodes in the radial direction, then let Wh, Fh 2RN×M

be the respective matrices for the unknown valuesof the two potentials at all the interior node pointsat time t. Also let T 2 RN×M be the known values ofthe temperature at time t. Using the notation fromTable 3 for the RBF differentiation matrix for thesurface Laplacian and the Chebyshev differentiationmatrix for the radial component of the 3‐DLaplacian,the discrete form of the first two equations in Table 2is written as

LsWh þ WhLr ¼ Ra T R; ðA1Þ

LsFh þ FhLr ¼ Wh R2; ðA2Þ

where R is the diagonal matrix given in (27). Now,letting Ls = VsLsVs

−1 and Lr = VrLrVr−1 be the spectral

decompositions of the operators Ls and Lr, respec-tively, (A1) and (A2) can be written, after somemanipulations as

LsðV�1s WhVrÞ þ ðV�1

s WhVrÞLr ¼ Ra ðV�1s T RVrÞ;

LsðV�1s FhVrÞ þ ðV�1

s FhVrÞLr ¼ ðV�1s WhVrÞðV�1

r R2VrÞ:

Table 6. Runtime Results for the RBF‐PS Method for the Ra = 7000 and Ra = 100,000 Cases on a Single 2.93 GHz Intel i7 940Quad Core Processor

Test CaseTotal Numberof Nodes

Runtime perTime Step Total Runtime Total Time Steps

Ra = 7000 36,800 0.0516 s 8 min 16 s 10,000 (to t = 1)Ra = 100,000 176,128 0.44 s 6 h 27 min 50,000 (to t = 0.3)


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By defining bWh = Vs−1WhVr, bFh = Vs

−1FhVr, bT = Vs−1T

RVr, and eR2 = Vr−1R2Vr, the above equations can be

written as the diagonal system of equations

LsbWh þ bWhLr ¼ Ra bT ; ðA3Þ

LsbFh þ bFhLr ¼ bWheR2: ðA4Þ

The solutions to these equations are given explicitlyas

ðbWhÞi; j ¼ RaðbTÞi; j

ðLsÞi; i þ ðLrÞj; jand ðbFhÞi; j ¼

ðbWheR2Þi; jðLsÞi; i þ ðLrÞj; j

;

ðA5Þ

for i = 1,…, N, and j = 1,…, M.

[49] The operators Ls and Lr are time independentso that their spectral decompositions can be com-puted as a preprocessing step. The solution to (A3)and (A4) thus requires O(MN) operations per timestep. However, the total cost in computing bFh pertime step is dominated by the cost of computing bTand bWh

eR2, which requires O(MN2) and O(M2N)operations, respectively. Since N is typically twoorders of magnitude greater than M, the formercomputation will dominate everything.

[50] We also apply the influence matrix techniquein spectral space as it allows some reduction in thestorage and computational cost of computing thefinal value of F. The setup of this technique issimilar to that described in section 5 in that we lookfor a superposition of solutions to the equations.However, by working in the spectral space of thelateral operator Ls, the lateral directions can bedecoupled so that the superposition consists only ofthree N × M linear systems instead of N:

W ¼ Wh þ WRi

bdWRi þ W

Ro

bdWRo and

F ¼ Fh þ WRi

bdFRi þ W

Ro

bdFRo; ðA6Þ

where bars indicate the variables are in the spectralspace of the lateral operator Ls only, and Wbd

Ri andWbdRo are N × N diagonal matrices with the unknown

values for enforcing the boundary conditions. Thevalues of Wh and Fh, can be obtained by multi-plying bWh and bFh (computed from (A3) and (A4))on the right by Vr

−1. The remaining values aredetermined from the solution of N independenttwo‐point boundary value systems in the radial

direction which correspond to the transform of thetwo sets of coupled PDEs in rows 3–6 of Table 2into the spectral space of Ls. The complete set ofequations can be written in discrete form as

LsWRi þ W

RiLr ¼ BRi ;

LsFRi þ F

RiLr ¼ WRieR2;

(and

LsWRo þ W

RoLr ¼ BRo ;

LsFRo þ F

RoLr ¼ WRoeR2;

(ðA7Þ

where BRi andBRo contain themodifications from theboundary conditions (WRi)j = 1 at r =Ri and (W

Ro)j = 1at r = Ro (the remaining boundary conditions on thevariables are homogeneous). The solutions to bothof these systems can be computed by a transfor-mation into the spectral space of Lr as done in (A3)and (A4) and then solving a diagonal system. Thetotal operation for the solution of FRi and FRo canthen be computed in O(M2N) operations. However,this can be done as a preprocessing step as thesevalues are time independent.

[51] To find the unknown values on the diagonalsof Wbd

Ri and WbdRo we apply the second set of

boundary conditions in (8) to the right‐hand side ofthe F equation in (A6). The discrete set of equa-tions that result are given by

FRiDRi

rr

� �W

Ri

bd þ FRoDRi

rr

� �W

Ro

bd ¼ �FhDRirr ; ðA8Þ

FRiDRo

rr

� �W

Ri

bd þ FRoDRo

rr

� �W

Ro

bd ¼ �FhDRorr ; ðA9Þ

where DrrRi, Drr

Ro 2 RM×1 are the discrete Chebyshevsecond derivative operators in the radial directionat the inner and outer boundaries, respectively.Equations (A8) and (A9) correspond to N decoupled2 × 2 linear system for finding the unknowns.Thus, all these systems can be solved in O(N)operations. Furthermore, since FRi and FRo aretime independent the values in parentheses on theleft‐hand side of these equations can be com-puted as a preprocessing step. The values on theright‐hand side are time dependent and need tobe computed every time step, which requiresO(MN)operations.

[52] Once WbdRi and Wbd

Ro are determined, F is com-puted according to (A6) and then transformed backinto physical space by computing F = VsF. Thecomputation of F requires O(MN) operations andthe computation of F requires O(MN2) operations.


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Combining this cost with the cost of computingbFh from (A3) and (A4), we see the total compu-tational cost per time step for computing F will beO(MN2) operations.

Appendix B: Overview of RBF‐PSAlgorithmB1. Setup (Preprocessing)

[53] 1. Discretize @@�,

1cos �

@@�, and Ds using colloca-

tion with N RBFs. This will result in 3 N × Nmatrices, D�, Dl, and Ls (see sections 4.1 and 4.2).

[54] 2. Discretize @@r ;

@@r r2 @

@r

� ; @2

@r2

��r¼Ri

; and @2

@r2

��r¼Ro

using collocation with M + 2 Chebyshev poly-nomials. Since these are only applied on the inte-rior of the shell, this will result in 2M ×M matricesDr and Lr for the first two operators, and 2 M × 1vectors for the last two.

[55] 3. Form the M × M diagonal matrices R(see (27)) and R2.

[56] 4. LU decompose the M × M matrix (I − Dt/2LrR

−2) for the AB3‐CN time stepping method (seesection 6), where I is the M × M identity matrix.

[57] 5. Momentum equations:i. Compute the spectral decomposition of Ls and

Lr (see Appendix A). This results in the followingmatrices: 1 full N × N (Vs), 1 diagonal N × N (Ls),1 full M × M (Vr), and 1 diagonal M × M (Lr).

ii. Compute the inverses of Vr and Vs.iii. Compute the fullM ×Mmatrix eR2 = Vr

−1R2Vr.iv. Solve (A7) for FRi and FRo, which results in 2

N × M matrices.v. Compute the influence matrix entries FRiDrr

Ri,FRiDrr

Ro, FRoDrrRi, and FRoDrr

Ro from (A8) and (A8).This results in 4 vectors of length N.

B2. Execution

[58] 1. Momentum equations:i. With the temperature Tk at the kth time step,

calculate bFh (i.e., Fh in the spectral space of boththe operators Ls and Lr) according to (A5).

ii. Calculate Fh = bFhVr−1 (i.e., Fh in the spectral

space of Ls only).iii. Solve for the influence matrix system (A7)

for the unknowns WbdRi and Wbd

Ro.iv. Compute F by updating Fh with the influ-

ence matrix terms according to (A6).v. Compute F = Vs F to find the poloidal

potential in physical space.

vi. Compute the velocity field (9), which can bewritten

u ¼ ðLsFR�1|fflfflfflffl{zfflfflfflffl}ur

;D�FRDrR�1|fflfflfflfflfflfflfflffl{zfflfflfflfflfflfflfflffl}

u�

;D�FRDrR�1|fflfflfflfflfflfflfflffl{zfflfflfflfflfflfflfflffl}

u�

Þ:

[59] 2. Compute the Fk and Gk in (26) with thevelocity field from the previous step and the tem-perature Tk.

[60] 3. Solve (28) for the temperature at the nexttime step, Tk+1, and repeat step 1 with this newtemperature.

Acknowledgments

[61] G. B.Wright was supported byNSF grants ATM‐0801309and DMS‐0934581. N. Flyer was supported by NSF grantsATM‐0620100 and DMS‐0934317. N. Flyer’s work was in partcarried out while she was a Visiting Fellow at Oxford Centre forCollaborative Applied Mathematics (OCCAM) under supportprovided by award KUK‐C1‐013‐04 to the University ofOxford, UK, by King Abdullah University of Science andTechnology (KAUST). D. A. Yuen was supported by NSFgrant DMS‐0934564.

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