On Asymptotic Global Error Estimation and Control of Finite Difference Solutions for Semilinear Parabolic Equations

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Abstract

The aim of this paper is to extend the global error estimation and control addressed in Lang and Verwer[SIAM J. Sci. Comput. 29, 2007] for initial value problems to finite difference solutions of semilinear parabolic partial differential equations. The approach presented there is combined with an estimation of the PDE spatial truncation error by Richardson extrapolation to estimate the overall error in the computed solution. Approximations of the error transport equations for spatial and temporal global errors are derived by using asymptotic estimates that neglect higher order error terms for sufficiently small step sizes in space and time. Asymptotic control in a discrete L 2-norm is achieved through tolerance proportionality and uniform or adaptive mesh refinement. Numerical examples are used to illustrate the reliability of the estimation and control strategies.

Original languageEnglish
JournalComputer Methods in Applied Mechanics and Engineering
Volume288
Pages (from-to)110-126
ISSN0045-7825
DOIs
Publication statusPublished - 2015

Keywords

  • Asymptotic global error control
  • Asymptotic global error estimation
  • Defects and local errors
  • Finite difference method
  • Method of lines
  • Numerical integration for PDEs

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