The role of the multiquadric shape parameters in solving elliptic partial differential equations

J. Wertz, E. J. Kansa*, Leevan LING

*Corresponding author for this work

Research output: Contribution to journalArticlepeer-review

76 Citations (Scopus)

Abstract

This study examines the generalized multiquadrics (MQ), φj(x) = [(x-xj)2+cj 2] β in the numerical solutions of elliptic two-dimensional partial differential equations (PDEs) with Dirichlet boundary conditions. The exponent β as well as cj 2 can be classified as shape parameters since these affect the shape of the MQ basis function. We examined variations of β as well as cj 2 where cj 2 can be different over the interior and on the boundary. The results show that increasing,β has the most important effect on convergence, followed next by distinct sets of (cj 2)Ω∂Ω ≪ (cj 2)∂Ω. Additional convergence accelerations were obtained by permitting both (cj 2)Ω∂Ω and (cj 2)∂Ω to oscillate about its mean value with amplitude of approximately 1/2 for odd and even values of the indices. Our results show high orders of accuracy as the number of data centers increases with some simple heuristics.

Original languageEnglish
Pages (from-to)1335-1348
Number of pages14
JournalComputers and Mathematics with Applications
Volume51
Issue number8 SPEC. ISS.
DOIs
Publication statusPublished - Apr 2006

Scopus Subject Areas

  • Modelling and Simulation
  • Computational Theory and Mathematics
  • Computational Mathematics

User-Defined Keywords

  • Different shape parameters
  • Elliptic PDEs
  • Generalized multiquadrics

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