Adaptive approximation of solutions to problems with multiple layers by chebyshev pseudo-spectral methods

Alvin Bayliss, Andreas Class, Bernard J Matkowsky

Research output: Contribution to journalArticlepeer-review

16 Scopus citations

Abstract

We develop end analyze a family of mappings which enhance the accuracy of Chebyshev pseudo-spectral methods in approximating functions with multiple regions of localized rapid variation (layers). The mapping family depends on 3N − 1 free parameters, where N is the number of layers. N parameters depend upon the locations of the layers and on the widths of the layers, while the other N − 1 parameters depend on the resolution of each layer relative to the first layer. The parameters can be determined adaptively by minimizing a functional which measures the error of the approximation. Techniques to simplify the minimization process are developed. We further demonstrate that the appropriate choice of mappings can lead to a significant reduction in the condition number of matrices associated with Chebyshev pseudo-spectral differentation. We illustrate the effectiveness of the proposed mapping and adaptive procedure by examples in which we approximate (i) given functions exhibiting multiple layers and (ii) the solution of a system of partial differential equations modeling combustion in counterflowing jets so that two distinct flames occur.

Original languageEnglish (US)
Pages (from-to)160-172
Number of pages13
JournalJournal of Computational Physics
Volume116
Issue number1
DOIs
StatePublished - Jan 1 1995

ASJC Scopus subject areas

  • Numerical Analysis
  • Modeling and Simulation
  • Physics and Astronomy (miscellaneous)
  • Physics and Astronomy(all)
  • Computer Science Applications
  • Computational Mathematics
  • Applied Mathematics

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