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

Varun Shankar1, Grady B Wright2, Robert M Kirby1

  • 1School of Computing, University of Utah, Salt Lake City, UT 84112.

Journal of Scientific Computing
|May 19, 2015
PubMed
Summary
This summary is machine-generated.

This study introduces a novel numerical method using Radial Basis Function-generated Finite Differences to solve diffusion equations on surfaces. The technique offers stable and accurate solutions for complex geometries without coordinate singularities.

Keywords:
RBF-FDfinite differencesmanifoldsmesh-freemethod-of-linesradial basis functionsreaction-diffusion

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Area of Science:

  • Numerical Analysis
  • Computational Geometry
  • Partial Differential Equations

Background:

  • Solving partial differential equations (PDEs) on surfaces is crucial for various scientific and engineering fields.
  • Existing numerical methods often struggle with complex surface geometries, coordinate singularities, and stability issues.
  • Accurate and stable numerical solvers are needed for diffusion and reaction-diffusion processes on embedded surfaces.

Purpose of the Study:

  • To present a novel numerical method for solving diffusion and reaction-diffusion equations on closed surfaces.
  • To develop a stable and accurate approach using Radial Basis Function-generated Finite Differences (RBF-FD).
  • To demonstrate the method's applicability to surfaces represented by scattered nodes and point clouds.

Main Methods:

  • Utilizes a method-of-lines formulation with RBF interpolation for approximating surface derivatives.
  • Employs extrinsic coordinates exclusively, avoiding issues with intrinsic coordinate systems.
  • Introduces an optimization procedure for stabilizing discrete differential operators by selecting shape parameters.
  • Requires only scattered surface nodes and their corresponding normal vectors.

Main Results:

  • Demonstrates the convergence of the RBF-FD method on various surfaces for different stencil sizes.
  • Successfully applies the method to nonlinear PDEs on implicit, parametric, and point cloud surfaces.
  • The RBF-FD method shows stability and accuracy, handling complex geometries effectively.

Conclusions:

  • The proposed RBF-FD method provides a robust and versatile tool for numerically solving PDEs on closed surfaces.
  • The extrinsic coordinate approach and operator stabilization enhance the method's reliability.
  • This technique is applicable to a wide range of surfaces, including those represented by scattered data.