Collocation methods for boundary value problems on long intervals

Peter Markowich*, Christian A. Ringhofer

*Corresponding author for this work

Research output: Contribution to journalArticlepeer-review

16 Scopus citations

Abstract

This paper deals with the numerical solution of boundary value problems of ordinary differential equations posed on infinite intervals. We cut the infinite interval at a finite, large enough point and insert additional, so-called asymptotic boundary conditions at the far (right) end and then solve the resulting two-point boundary value problem by an-stable symmetric collocation method. Problems arise, because standard theory predicts the use of many grid points as the length of the interval increases. Using the exponential decay of the 'infinite' solution, an 'asymptotic' a priori mesh-size sequence which increases exponentially, and which therefore only employs a reasonable number of meshpoints, is developed and stability, as the length of the interval tends to infinity, is shown. We also show that the condition number of the collocation equations is asymptotically proportional to the number of meshpoints employed when using this exponentially graded mesh. Using £-stage collocation at Gaussian points and requiring an accuracy O() at the knots implies that the number of meshpoints is 0(-1/2k) as 0.

Original languageEnglish (US)
Pages (from-to)123-150
Number of pages28
JournalMathematics of Computation
Volume40
Issue number161
DOIs
StatePublished - Jan 1 1983

Keywords

  • Asymptotic properties
  • Difference equations
  • Nonlinear boundary value problems
  • Singular points
  • Stability of difference equations

ASJC Scopus subject areas

  • Algebra and Number Theory
  • Computational Mathematics
  • Applied Mathematics

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