Eigenvalue stability of radial basis function discretizations for time-dependent problems

R. B. Platte, T. A. Driscoll

Research output: Contribution to journalArticle

53 Scopus citations

Abstract

Differentiation matrices obtained with infinitely smooth radial basis function (RBF) collocation methods have, under many conditions, eigenvalues with positive real part, preventing the use of such methods for time-dependent problems. We explore this difficulty at theoretical and practical levels. Theoretically, we prove that differentiation matrices for conditionally positive definite RBFs are stable for periodic domains. We also show that for Gaussian RBFs, special node distributions can achieve stability in 1-D and tensor-product nonperiodic domains. As a more practical approach for bounded domains, we consider differentiation matrices based on least-squares RBF approximations and show that such schemes can lead to stable methods on less regular nodes. By separating centers and nodes, least-squares techniques open the possibility of the separation of accuracy and stability characteristics.

Original languageEnglish (US)
Pages (from-to)1251-1268
Number of pages18
JournalComputers and Mathematics with Applications
Volume51
Issue number8 SPEC. ISS.
DOIs
StatePublished - Apr 2006

Keywords

  • Least squares
  • Method of lines
  • Numerical stability
  • RBF
  • Radial basis functions

ASJC Scopus subject areas

  • Modeling and Simulation
  • Computational Theory and Mathematics
  • Computational Mathematics

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