期刊
SIAM JOURNAL ON SCIENTIFIC COMPUTING
卷 39, 期 5, 页码 A2129-A2151出版社
SIAM PUBLICATIONS
DOI: 10.1137/16M1095457
关键词
RBF-FD; RBF-HFD; manifolds; reaction diffusion
资金
- NSF-DMS [1160432]
- [NSF-DMS 1160379]
- [NSF-ACI 1440638]
- Direct For Computer & Info Scie & Enginr
- Office of Advanced Cyberinfrastructure (OAC) [1440638] Funding Source: National Science Foundation
- Direct For Mathematical & Physical Scien
- Division Of Mathematical Sciences [1521748, 1160379, 1160432] Funding Source: National Science Foundation
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.
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