A Radial Basis Function (RBF)-Finite Difference (FD) Method for Diffusion and Reaction–Diffusion Equations on Surfaces

In this paper, we present a method based on radial basis function (RBF)-generated finite differences (FD) for numerically solving diffusion and reaction–diffusion equations (PDEs) on closed surfaces embedded in R d . Our method uses a method-of-lines formulation, in which surface derivatives that ap...

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Bibliographic Details
Published in:Journal of scientific computing 2015-06, Vol.63 (3), p.745-768
Main Authors: Shankar, Varun, Wright, Grady B., Kirby, Robert M., Fogelson, Aaron L.
Format: Article
Language:English
Subjects:
Algorithms
Computation
Computational Mathematics and Numerical Analysis
Cost control
Derivatives
Differential equations
Diffusion
Finite difference method
Interpolation
Mathematical analysis
Mathematical and Computational Engineering
Mathematical and Computational Physics
Mathematical models
Mathematics
Mathematics and Statistics
Method of lines
Methods
Nodes
Operators (mathematics)
Partial differential equations
Radial basis function
Reaction-diffusion equations
Singularity (mathematics)
Theoretical
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Summary:In this paper, we present a method based on radial basis function (RBF)-generated finite differences (FD) for numerically solving diffusion and reaction–diffusion equations (PDEs) on closed surfaces embedded in R d . Our method uses a method-of-lines formulation, in which surface derivatives that appear in the PDEs are approximated locally using RBF interpolation. The method requires only scattered nodes representing the surface and normal vectors at those scattered nodes. All computations use only extrinsic coordinates, thereby avoiding coordinate distortions and singularities. We also present an optimization procedure that allows for the stabilization of the discrete differential operators generated by our RBF-FD method by selecting shape parameters for each stencil that correspond to a global target condition number. We show the convergence of our method on two surfaces for different stencil sizes, and present applications to nonlinear PDEs simulated both on implicit/parametric surfaces and more general surfaces represented by point clouds.
ISSN:0885-7474
1573-7691
DOI:10.1007/s10915-014-9914-1