Journal of Scientific Computing

, Volume 56, Issue 3, pp 535–565 | Cite as

A High-Order Kernel Method for Diffusion and Reaction-Diffusion Equations on Surfaces

Article

Abstract

In this paper we present a high-order kernel method for numerically solving diffusion and reaction-diffusion partial differential equations (PDEs) on smooth, closed surfaces embedded in \(\mathbb{R }^d\). For two-dimensional surfaces embedded in \(\mathbb{R }^3\), these types of problems have received growing interest in biology, chemistry, and computer graphics to model such things as diffusion of chemicals on biological cells or membranes, pattern formations in biology, nonlinear chemical oscillators in excitable media, and texture mappings. Our kernel method is based on radial basis functions and uses a semi-discrete approach (or the method-of-lines) in which the surface derivative operators that appear in the PDEs are approximated using collocation. The method only requires nodes at “scattered” locations on the surface and the corresponding normal vectors to the surface. Additionally, it does not rely on any surface-based metrics and avoids any intrinsic coordinate systems, and thus does not suffer from any coordinate distortions or singularities. We provide error estimates for the kernel-based approximate surface derivative operators and numerically study the accuracy and stability of the method. Applications to different non-linear systems of PDEs that arise in biology and chemistry are also presented.

Keywords

Radial basis functions Mesh-free Manifold Collocation Method-of-lines Pattern formation Turing patterns Spiral waves 

Mathematics Subject Classification(2000)

58J45 35K57 41A05 41A25 41A30 41A63 65D25 65M20 65M70 46E22 35B36 

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Copyright information

© Springer Science+Business Media New York 2013

Authors and Affiliations

  1. 1.Department of Mathematics and Computer ScienceHigh Point UniversityHigh PointUSA
  2. 2.Department of MathematicsBoise State UniversityBoiseUSA

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