Reliability of local error control algorithms for initial value ordinary differential equations

Higham, Desmond J.

doi:10.1007/978-1-5041-2940-4_26

Cited by 1 publication

(2 citation statements)

References 8 publications

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“…As remarked in [12] the results of Theorem 1 are relevant to explain the global error behavior in non-stiff problems. For some stiff problems a possible order reduction on the basic method and the error estimate might occur.…”

Section: Holds Then the Adaptive Rk Methods Possess The Tp Property Wmentioning

confidence: 92%

“…In the above-mentioned TP theory developed by Higham [10][11][12]2] a continuous variation of the step-size along the integration interval was assumed in the sense that the step-size h n+1 in advancing from t n to t n+1 =t n +h n+1 satisfies an asymptotic relation h n+1 = (t n ) 1/q + o( 1/q ) with a fixed integer q 1 and a continuous problem-depending function (t) * > 0. Such a relation implies that, in general, the step-size will vary from step to step but this does not affect negatively the performance of explicit RK codes in which the computational cost is independent of the size of the step.…”

Section: Introductionmentioning

confidence: 99%

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Global error estimation based on the tolerance proportionality for some adaptive Runge–Kutta codes

Calvo

González-Pinto

Montijano

2008

Journal of Computational and Applied Mathematics

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Modern codes for the numerical solution of Initial Value Problems (IVPs) in ODEs are based in adaptive methods that, for a user supplied tolerance , attempt to advance the integration selecting the size of each step so that some measure of the local error is . Although this policy does not ensure that the global errors are under the prescribed tolerance, after the early studies of Stetter [Considerations concerning a theory for ODE-solvers, in: R. The reliability of standard local error control algorithms for initial value ordinary differential equations, in: Proceedings: The Quality of Numerical Software: Assessment and Enhancement, IFIP Series, Springer, Berlin, 1997], it has been proved that in many existing explicit Runge-Kutta codes the global errors behave asymptotically as some rational power of . This step-size policy, for a given IVP, determines at each grid point t n a new step-size h n+1 = h(t n ; ) so that h(t; ) is a continuous function of t.In this paper a study of the tolerance proportionality property under a discontinuous step-size policy that does not allow to change the size of the step if the step-size ratio between two consecutive steps is close to unity is carried out. This theory is applied to obtain global error estimations in a few problems that have been solved with the code Gauss2 [S. Gonzalez-Pinto, R. Rojas-Bello, Gauss2, a Fortran 90 code for second order initial value problems, http://www.netlib.org/ode/ ], based on an adaptive two stage Runge-Kutta-Gauss method with this discontinuous step-size policy.

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Section: Holds Then the Adaptive Rk Methods Possess The Tp Property Wmentioning

confidence: 92%

Section: Introductionmentioning

confidence: 99%

Global error estimation based on the tolerance proportionality for some adaptive Runge–Kutta codes

Calvo

González-Pinto

Montijano

2008

Journal of Computational and Applied Mathematics

View full text Add to dashboard Cite

show abstract

Reliability of local error control algorithms for initial value ordinary differential equations

Cited by 1 publication

References 8 publications

Global error estimation based on the tolerance proportionality for some adaptive Runge–Kutta codes

Global error estimation based on the tolerance proportionality for some adaptive Runge–Kutta codes

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