Spectral deferred correction methods for ordinary differential equations

Alok Dutt, Leslie Greengard, Vladimir Rokhlin

Research output: Contribution to journalArticle

Abstract

We introduce a new class of methods for the Cauchy problem for ordinary differential equations (ODEs). We begin by converting the original ODE into the corresponding Picard equation and apply a deferred correction procedure in the integral formulation, driven by either the explicit or the implicit Euler marching scheme. The approach results in algorithms of essentially arbitrary order accuracy for both non-stiff and stiff problems; their performance is illustrated with several numerical examples. For non-stiff problems, the stability behavior of the obtained explicit schemes is very satisfactory and algorithms with orders between 8 and 20 should be competitive with the best existing ones. In our preliminary experiments with stiff problems, a simple adaptive implementation of the method demonstrates performance comparable to that of a state-of-the-art extrapolation code (at least, at moderate to high precision). Deferred correction methods based on the Picard equation appear to be promising candidates for further investigation.

Original languageEnglish (US)
Pages (from-to)241-266
Number of pages26
JournalBIT Numerical Mathematics
Volume40
Issue number2
DOIs
StatePublished - Jun 2000

Keywords

  • Deferred correction
  • Initial value problems
  • Spectral methods
  • Stiffness

ASJC Scopus subject areas

  • Software
  • Computer Networks and Communications
  • Computational Mathematics
  • Applied Mathematics

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