Spectral radial basis functions for full sphere computations

  • Authors:

  • Philip W. Livermore;Chris A. Jones;Steven J. Worland

  • Affiliations:

  • School of Mathematics, Leeds University, Leeds, LS2 9JT, UK;School of Mathematics, Leeds University, Leeds, LS2 9JT, UK;Formerly at School of Engineering, Computing and Mathematics, University of Exeter, EX4 4QF, UK

  • Venue:
  • Journal of Computational Physics
  • Year:
  • 2007

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Abstract

The singularity of cylindrical or spherical coordinate systems at the origin imposes certain regularity conditions on the spectral expansion of any infinitely differentiable function. There are two efficient choices of a set of radial basis functions suitable for discretising the solution of a partial differential equation posed in either such geometry. One choice is methods based on standard Chebyshev polynomials; although these may be efficiently computed using fast transforms, differentiability to all orders of the obtained solution at the origin is not guaranteed. The second is the so-called one-sided Jacobi polynomials that explicitly satisfy the required behavioural conditions. In this paper, we compare these two approaches in their accuracy, differentiability and computational speed. We find that the most accurate and concise representation is in terms of one-sided Jacobi polynomials. However, due to the lack of a competitive fast transform, Chebyshev methods may be a better choice for some computationally intensive timestepping problems and indeed will yield sufficiently (although not infinitely) differentiable solutions provided they are adequately converged.