Methods and Algorithms for Scientific Computing

A Radial Basis Function (RBF) Compact Finite Difference (FD) Scheme for Reaction-Diffusion Equations on Surfaces

We present a new high-order, local meshfree method for numerically solving reaction diffusion equations on smooth surfaces of codimension 1 embedded in $\mathbb{R}^d$. The novelty of the method is in the approximation of the Laplace--Beltrami operator for a given surface using Hermite radial basis function (RBF) interpolation over local node sets on the surface. This leads to compact (or implicit) RBF generated finite difference (RBF-FD) formulas for the Laplace--Beltrami operator, which gives rise to sparse differentiation matrices. The method only requires a set of (scattered) nodes on the surface and an approximation to the surface normal vectors at these nodes. Additionally, the method is based on Cartesian coordinates and thus does not suffer from any coordinate singularities. We also present an algorithm for selecting the nodes used to construct the compact RBF-FD formulas that can guarantee the resulting differentiation matrices have desirable stability properties. The improved accuracy and computational cost that can be achieved with this method over the standard (explicit) RBF-FD method are demonstrated with a series of numerical examples. We also illustrate the flexibility and general applicability of the method by solving two different reaction-diffusion equations on surfaces that are defined implicitly and only by point clouds.

  • 1.  Advanpix Multiprecision Computing Toolbox for MATLAB, http://www.advanpix.com/, accessed 2016-06-09.Google Scholar

  • 2.  U. M. Ascher S. J. Ruuth and  B. T. . Wetton, Implicit-explicit methods for time-dependent PDEs , SIAM J. Numer. Anal , 32 ( 1997 ), pp. 797 -- 823 . LinkISIGoogle Scholar

  • 3.  V. Bayona N. Flyer B. Fornberg and  G. A. Barnett , On the role of polynomials in RBF-FD approximations: II. Numerical solution of elliptic PDEs , J. Comput. Phys. , 332 ( 2017 ), pp. 257 -- 273 . CrossrefISIGoogle Scholar

  • 4.  V. Bayona M. Moscoso M. Carretero and  M. Kindelan , RBF-FD formulas and convergence properties , J. Comput. Phys. , 229 ( 2010 ), pp. 8281 -- 8295 . CrossrefISIGoogle Scholar

  • 5.  R. L. Bishop and  S. I. Goldberg , Tensor Analysis on Manifolds , Macmillan , New York , 1968 . Google Scholar

  • 6.  T. Cecil J. Qian and  S. Osher , Numerical methods for high dimensional Hamilton--Jacobi equations using radial basis functions , J. Comput. Phys. , 196 ( 2004 ), pp. 327 -- 347 . CrossrefISIGoogle Scholar

  • 7.  G. Chandhini and  Y. Sanyasiraju , Local RBF-FD solutions for steady convection--diffusion problems , Int. J. Numer. Meth. , 72 ( 2007 ), pp. 352 -- 378 . CrossrefISIGoogle Scholar

  • 8.  P. Cignoni M. Corsini and  G. Ranzuglia , MESHLAB: An open-source $3$D mesh processing system , ERCIM News , ( 2008 ), pp. 45 -- 46 . Google Scholar

  • 9.  L. Collatz , The Numerical Treatment of Differential Equations , 3 rd ed., Springer , Berlin , 1966 . Google Scholar

  • 10.  O. Davydov and  D. T. Oanh , Adaptive meshless centres and RBF stencils for Poisson equation , J. Comput. Phys. , 230 ( 2011 ), pp. 287 -- 304 . CrossrefISIGoogle Scholar

  • 11.  T. A. Driscoll and  B. Fornberg , Interpolation in the limit of increasingly flat radial basis functions , Comput. Math. Appl. , 43 ( 2002 ), pp. 413 -- 422 . CrossrefISIGoogle Scholar

  • 12.  G. E. Fasshauer , Meshfree Approximation Methods with MATLAB , Interdisciplinary Mathematical Sciences 6 , World Scientific , Singapore , 2007 . CrossrefGoogle Scholar

  • 13.  G. E. Fasshauer and  M. J. McCourt , Stable evaluation of Gaussian radial basis function interpolants , SIAM J. Sci. Comput. , 34 ( 2012 ), pp. A737 -- A762 . LinkISIGoogle Scholar

  • 14.  N. Flyer B. Fornberg V. Bayona and  G. A. Barnett , On the role of polynomials in RBF-FD approximations I. Interpolation and accuracy , J. Comput. Phys. , 321 ( 2016 ), pp. 21 -- 38 . CrossrefISIGoogle Scholar

  • 15.  N. Flyer E. Lehto S. Blaise G. B. Wright and  A. St-Cyr , A guide to RBF-generated finite differences for nonlinear transport: Shallow water simulations on a sphere , J. Comput. Phys. , 231 ( 2012 ), pp. 4078 -- 4095 . CrossrefISIGoogle Scholar

  • 16.  N. Flyer and  G. B. Wright , A radial basis function method for the shallow water equations on a sphere , Proc. R. Soc. A , 465 ( 2009 ), pp. 1949 -- 1976 . CrossrefGoogle Scholar

  • 17.  B. Fornberg and  N. Flyer , A Primer on Radial Basis Functions with Applications to the Geosciences , SIAM , Philadelphia , 2014 . Google Scholar

  • 18.  B. Fornberg E. Larsson and  N. Flyer , Stable computations with Gaussian radial basis functions , SIAM J. Sci. Comput. , 33 ( 2011 ), pp. 869 -- 892 . LinkISIGoogle Scholar

  • 19.  B. Fornberg and  E. Lehto , Stabilization of RBF-generated finite difference methods for convective PDEs , J. Comput. Phys. , 230 ( 2011 ), pp. 2270 -- 2285 . CrossrefISIGoogle Scholar

  • 20.  B. Fornberg E. Lehto and  C. Powell , Stable calculation of Gaussian-based RBF-FD stencils , Comput. Math. Appl. , 65 ( 2013 ), pp. 627 -- 637 . CrossrefISIGoogle Scholar

  • 21.  B. Fornberg and  C. Piret , A stable algorithm for flat radial basis functions on a sphere , SIAM J. Sci. Comput. , 30 ( 2007 ), pp. 60 -- 80 . LinkISIGoogle Scholar

  • 22.  B. Fornberg and  G. Wright , Stable computation of multiquadric interpolants for all values of the shape parameter , Comput. Math. Appl. , 48 ( 2004 ), pp. 853 -- 867 . CrossrefISIGoogle Scholar

  • 23.  B. Fornberg and  J. Zuev , The Runge phenomenon and spatially variable shape parameters in RBF interpolation , Comput. Math. Appl. , 54 ( 2007 ), pp. 379 -- 398 . CrossrefISIGoogle Scholar

  • 24.  E. Fuselier and  G. Wright , Scattered data interpolation on embedded submanifolds with restricted positive definite kernels: Sobolev error estimates , SIAM J. Numer. Anal. , 50 ( 2012 ), pp. 1753 -- 1776 . LinkISIGoogle Scholar

  • 25.  E. Fuselier and  G. B. Wright , Order-preserving derivative approximation with periodic radial basis functions , SIAM J. Numer. Anal. , 41 ( 2015 ), pp. 23 -- 53 . Google Scholar

  • 26.  E. J. Fuselier and  G. B. Wright , A high-order kernel method for diffusion and reaction-diffusion equations on surfaces , J. Sci. Comput., ( 2013 ), pp. 1 -- 31 . Google Scholar

  • 27.  Q. T. . Gia, Approximation of parabolic pdes on spheres using spherical basis functions , Adv. Comput. Math. , 22 ( 2005 ), pp. 377 -- 397 . CrossrefISIGoogle Scholar

  • 28.  V. Kostić R. S. Varga and  L. Cvetković , Localization of generalized eigenvalues by Cartesian ovals , Numer. Linear Algebra Appl. , 19 ( 2012 ), pp. 728 -- 741 . CrossrefISIGoogle Scholar

  • 29.  E. Larsson and  B. Fornberg , Theoretical and computational aspects of multivariate interpolation with increasingly flat radial basis functions , Comput. Math. Appl. , 49 ( 2005 ), pp. 103 -- 130 . CrossrefISIGoogle Scholar

  • 30.  E. Larsson E. Lehto A. Heryudono and  B. Fornberg , Stable computation of differentiation matrices and scattered node stencils based on Gaussian radial basis functions , SIAM J. Sci. Comput. , 35 ( 2013 ), pp. A2096 -- A2119 . LinkISIGoogle Scholar

  • 31.  S. K. Lele , Compact finite difference schemes with spectral-like resolution , J. Comput. Phys. , 103 ( 1992 ), pp. 16 -- 42 . CrossrefISIGoogle Scholar

  • 32.  C. Macdonald and  S. Ruuth , The implicit closest point method for the numerical solution of partial differential equations on surfaces , SIAM J. Sci. Comput. , 31 ( 2010 ), pp. 4330 -- 4350 . LinkISIGoogle Scholar

  • 33.  F. Narcowich and  J. Ward , Generalized hermite interpolation via matrix-valued conditionally positive definite functions , Math. Comput. , 63 ( 1994 ), pp. 661 -- 688 . CrossrefISIGoogle Scholar

  • 34.  P.-O. Persson , DistMesh -- a simple mesh generator in MATLAB . http://persson.berkeley.edu/distmesh/, accessed 2016-06-09. , http://persson.berkeley.edu/distmesh/. Google Scholar

  • 35.  P.-O. Persson and  G. Strang , A simple mesh generator in MATLAB , SIAM Rev. , 46 ( 2004 ), pp. 329 -- 345 . LinkISIGoogle Scholar

  • 36.  C. Piret , The orthogonal gradients method: A radial basis functions method for solving partial differential equations on arbitrary surfaces , J. Comput. Phys. , 231 ( 2012 ), pp. 4662 -- 4675 . CrossrefISIGoogle Scholar

  • 37.  R. Schaback , Multivariate interpolation by polynomials and radial basis functions , Constr. Approx. , 21 ( 2005 ), pp. 293 -- 317 . CrossrefISIGoogle Scholar

  • 38.  V. Shankar G. B. Wright R. M. Kirby and  A. L. Fogelson , A radial basis function (RBF)-finite difference (FD) method for diffusion and reaction--diffusion equations on surfaces , J. Sci. Comput. , 63 ( 2014 ), pp. 745 -- 768 . CrossrefISIGoogle Scholar

  • 39.  C. Shu H. Ding and  K. S. Yeo , Local radial basis function-based differential quadrature method and its application to solve two-dimensional incompressible Navier--Stokes equations , Comput. Methods Appl. Mech. Eng. , 192 ( 2003 ), pp. 941 -- 954 . CrossrefISIGoogle Scholar

  • 40.  D. Stevens H. Power M. Lees and  H. Morvan , The use of PDE centers in the local RBF Hermitean method for $3$D convective-diffusion problems , J. Comput. Phys. , 228 ( 2009 ), pp. 4606 -- 4624 . CrossrefISIGoogle Scholar

  • 41.  G. W. Stewart , Gershgorin theory for the generalized eigenvalue problem ${A}x=\lambda {B}x$ , Math. Comput. , 29 ( 1975 ), pp. 600 -- 606 . CrossrefISIGoogle Scholar

  • 42.  G. B. Wright N. Flyer and  D. A. Yuen , A hybrid radial basis function--pseudospectral method for thermal convection in a $3$-D spherical shell , Geochem. Geophys. Geosyst. , 11 ( 2010 ). ISIGoogle Scholar

  • 43.  G. B. Wright and  B. Fornberg , Scattered node compact finite difference-type formulas generated from radial basis functions , J. Comput. Phys. , 212 ( 2006 ), pp. 99 -- 123 . CrossrefISIGoogle Scholar

  • 44.  W. Zongmin , Hermite--Birkhoff interpolation of scattered data by radial basis functions , Approx. Theory Appl. , 8 ( 1992 ), pp. 1 -- 10 . Google Scholar