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

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Abstract

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 \({\mathbb {R}}^d\). 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.

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Acknowledgments

We would like to acknowledge useful discussions concerning this work within the CLOT group at the University of Utah. The first, third and fourth authors acknowledge funding support under NIGMS Grant R01-GM090203. The second author acknowledges funding support under NSF-DMS Grant 1160379 and NSF-DMS Grant 0934581.

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Authors and Affiliations

  1. Department of Mathematics, University of Utah, Salt Lake City, UT, 84112, USA

    Varun Shankar & Aaron L. Fogelson

  2. Department of Mathematics, Boise State University, Boise, ID, 83725-1555, USA

    Grady B. Wright

  3. School of Computing, University of Utah, Salt Lake City, UT, 84112, USA

    Robert M. Kirby

Authors
  1. Varun Shankar
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  2. Grady B. Wright
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  3. Robert M. Kirby
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  4. Aaron L. Fogelson
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Correspondence to Varun Shankar.

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Shankar, V., Wright, G.B., Kirby, R.M. et al. A Radial Basis Function (RBF)-Finite Difference (FD) Method for Diffusion and Reaction–Diffusion Equations on Surfaces. J Sci Comput 63, 745–768 (2015). https://doi.org/10.1007/s10915-014-9914-1

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