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Global error estimation with adaptive explicit Runge-Kutta methods

Calvo, M.C. and Higham, D.J. and Montijano, J.M. and Rández, L. (1996) Global error estimation with adaptive explicit Runge-Kutta methods. IMA Journal of Numerical Analysis, 16 (1). pp. 47-63. ISSN 0272-4979

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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.

Item type: Article
ID code: 193
Keywords: differential equations, mathematics, Runge-Kutta algorithms, stepsize selection mechanism, Richardson extrapolation, Mathematics, Computational Mathematics, Applied Mathematics, Mathematics(all)
Subjects: Science > Mathematics
Department: Faculty of Science > Mathematics and Statistics
Related URLs:
    Depositing user: Ms Sarah Scott
    Date Deposited: 01 Mar 2006
    Last modified: 04 Sep 2014 09:56
    URI: http://strathprints.strath.ac.uk/id/eprint/193

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