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

  • Natasha Flyer
  • Grady B. Wright
  • Bengt Fornberg
Reference work entry


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.


Nusselt Number Spherical Shell Discontinuous Galerkin Shallow Water Equation Discontinuous Galerkin Method 
These keywords were added by machine and not by the authors. This process is experimental and the keywords may be updated as the learning algorithm improves.


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Copyright information

© Springer-Verlag Berlin Heidelberg 2015

Authors and Affiliations

  • Natasha Flyer
    • 1
  • Grady B. Wright
    • 2
  • Bengt Fornberg
    • 3
  1. 1.Institute for Mathematics Applied to GeosciencesNational Center for Atmospheric ResearchBoulderUSA
  2. 2.Department of MathematicsBoise State UniversityBoiseUSA
  3. 3.Department of Applied MathematicsUniversity of ColoradoBoulderUSA

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