Global error estimation with adaptive explicit Runge-Kutta methods

M.C. Calvo, D.J. Higham, J.M. Montijano, L. Rández

Research output: Contribution to journalArticle

8 Citations (Scopus)

Abstract

Users of locally-adaptive software for initial value ordinary differential equations are likely to be concerned with global errors. At the cost of extra computation, global error estimation is possible. Zadunaisky's method and 'solving for the error estimate' are two techniques that have been successfully incorporated into Runge-Kutta algorithms. The standard error analysis for these techniques, however, does not take account of the stepsize selection mechanism. In this paper, some new results are presented which, under suitable assumptions show that these techniques are asymptotically valid when used with an adaptive, variable stepsize algorithm - the global error estimate reproduces the leading term of the global error in the limit as the error tolerance tends to zero. The analysis is also applied to Richardson extrapolation (step halving). Numerical results are provided for the technique of solving for the error estimate with several Runge-Kutta methods of Dormand, Lockyer, McGorrigan and Prince.
LanguageEnglish
Pages47-63
Number of pages16
JournalIMA Journal of Numerical Analysis
Volume16
Issue number1
DOIs
Publication statusPublished - 1996

Fingerprint

Runge Kutta methods
Explicit Methods
Error Estimation
Runge-Kutta Methods
Error analysis
Error Estimates
Richardson Extrapolation
Variable Step Size
Runge-Kutta
Standard error
Error Analysis
Tolerance
Ordinary differential equation
Likely
Tend
Valid
Extrapolation
Ordinary differential equations
Numerical Results
Software

Keywords

  • differential equations
  • mathematics
  • Runge-Kutta algorithms
  • stepsize selection mechanism
  • Richardson extrapolation

Cite this

Calvo, M.C. ; Higham, D.J. ; Montijano, J.M. ; Rández, L. / Global error estimation with adaptive explicit Runge-Kutta methods. In: IMA Journal of Numerical Analysis. 1996 ; Vol. 16, No. 1. pp. 47-63.
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Global error estimation with adaptive explicit Runge-Kutta methods. / Calvo, M.C.; Higham, D.J.; Montijano, J.M.; Rández, L.

In: IMA Journal of Numerical Analysis, Vol. 16, No. 1, 1996, p. 47-63.

Research output: Contribution to journalArticle

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T1 - Global error estimation with adaptive explicit Runge-Kutta methods

AU - Calvo, M.C.

AU - Higham, D.J.

AU - Montijano, J.M.

AU - Rández, L.

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