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Variable order and stepsize in general linear methods. (English) Zbl 1416.65187
Summary: This paper describes the implementation of a class of IRKS methods [W. M. Wright, General linear methods with Inherent Runge-Kutta stability. Auckland: The University of Auckland. (PhD Thesis) (2002)]. These GLM algorithms are practical with reliable error estimators [the second author and H. Podhaisky, Appl. Numer. Math. 56, No. 3–4, 345–357 (2006; Zbl 1089.65080)]. The current robust ODE solvers in variable stepsize as well as in variable-order mode are based upon heuristics. In this paper, we examine an optimisation approach, based on Euler-Lagrange theory [the second author, IMA J. Numer. Anal. 6, 433–438 (1986; Zbl 0615.65076); Computing 44, No. 3, 209–220 (1990; Zbl 0719.65059)] to control the stepsize as well as the order and implement the GLMs in an efficient manner. A set of nonstiff to mildly stiff problems have been used to investigate this approach in fixed-order and variable-order modes.
65L05 Numerical methods for initial value problems involving ordinary differential equations
Full Text: DOI
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