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Radial Basis Function-Generated Finite Differences: A Mesh-Free Method for Computational Geosciences

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Handbook of Geomathematics

Abstract

Radial basis function-generated finite differences (RBF-FD) is a mesh-free method for numerically solving partial differential equations that emerged in the last decade and have shown rapid growth in the last few years. From a practical standpoint, RBF-FD sprouted out of global RBF methods, which have shown exceptional numerical qualities in terms of accuracy and time stability for numerically solving PDEs, but are not practical when scaled to very large problem sizes because of their computational cost and memory requirements. RBF-FD bypass these issues by using local approximations for derivatives instead of global ones. Matrices in the RBF-FD methodology go from being completely full to 99 % empty. Of course, the sacrifice is the exchange of spectral accuracy from the global RBF methods for high-order algebraic convergence of RBF-FD, assuming smooth data. However, since natural processes are almost never infinitely differentiable, little is lost and much gained in terms of memory and runtime. This chapter provides a survey of a group of topics relevant to using RBF-FD for a variety of problems that arise in the geosciences. Particular emphasis is given to problems in spherical geometries, both on surfaces and within a volume. Applications discussed include nonlinear shallow water equations on a sphere, reaction–diffusion equations, global electric circuit, and mantle convection in a spherical shell. The results from the last three of these applications are new and have not been presented before for RBF-FD.

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Author information

Authors and Affiliations

  1. Institute for Mathematics Applied to Geosciences, National Center for Atmospheric Research, Boulder, CO, USA

    Natasha Flyer

  2. Department of Mathematics, Boise State University, Boise, ID, USA

    Grady B. Wright

  3. Department of Applied Mathematics, University of Colorado, Boulder, CO, USA

    Bengt Fornberg

Authors
  1. Natasha Flyer
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  2. Grady B. Wright
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  3. Bengt Fornberg
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Corresponding author

Correspondence to Natasha Flyer .

Editor information

Editors and Affiliations

  1. Geomathematische Gruppe, TU Kaiserslautern FB Mathematik, Kaiserslautern, Germany

    Willi Freeden

  2. University of Central Florida Dept. Mathematics, Orlando, Florida, USA

    M. Zuhair Nashed

  3. Inst. Analysis, TU Braunschweig FB 1 Mathematik und Informatik, Braunschweig, Germany

    Thomas Sonar

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Flyer, N., Wright, G.B., Fornberg, B. (2013). Radial Basis Function-Generated Finite Differences: A Mesh-Free Method for Computational Geosciences. In: Freeden, W., Nashed, M., Sonar, T. (eds) Handbook of Geomathematics. Springer, Berlin, Heidelberg. https://doi.org/10.1007/978-3-642-27793-1_61-1

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  • DOI: https://doi.org/10.1007/978-3-642-27793-1_61-1

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  • Online ISBN: 978-3-642-27793-1

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