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A Radial Basis Function (RBF) Compact Finite Difference (FD) Scheme for Reaction-Diffusion Equations on Surfaces
KTH, School of Engineering Sciences (SCI), Mathematics (Dept.), Numerical Analysis, NA.
2017 (English)In: SIAM Journal on Scientific Computing, ISSN 1064-8275, E-ISSN 1095-7197, Vol. 39, no 5, A2129-A2151 p.Article in journal (Refereed) Published
Abstract [en]

We present a new high-order, local meshfree method for numerically solving reaction diffusion equations on smooth surfaces of codimension 1 embedded in 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.

Place, publisher, year, edition, pages
SIAM Publications , 2017. Vol. 39, no 5, A2129-A2151 p.
Keyword [en]
RBF-FD, RBF-HFD, manifolds, reaction diffusion
National Category
Mathematics
Identifiers
URN: urn:nbn:se:kth:diva-219525DOI: 10.1137/16M1095457ISI: 000415797300068OAI: oai:DiVA.org:kth-219525DiVA: diva2:1163496
Note

QC 20171207

Available from: 2017-12-07 Created: 2017-12-07 Last updated: 2017-12-07Bibliographically approved

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Lehto, Erik

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