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A Family of Fifth-Order Runge-Kutta Pairs

Authors: S. N. Papakostas and G. Papageorgiou
Journal: Math. Comp. 65 (1996), 1165-1181
MSC (1991): Primary 65L05
MathSciNet review: 1333323
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Abstract: The construction of a Runge-Kutta pair of order $5(4)$ with the minimal number of stages requires the solution of a nonlinear system of $25$ order conditions in $27$ unknowns. We define a new family of pairs which includes pairs using $6$ function evaluations per integration step as well as pairs which additionally use the first function evaluation from the next step. This is achieved by making use of Kutta's simplifying assumption on the original system of the order conditions, i.e., that all the internal nodes of a method contributing to the estimation of the endpoint solution provide, at these nodes, cost-free second-order approximations to the true solution of any differential equation. In both cases the solution of the resulting system of nonlinear equations is completely classified and described in terms of five free parameters. Optimal Runge-Kutta pairs with respect to minimized truncation error coefficients, maximal phase-lag order and various stability characteristics are presented. These pairs were selected under the assumption that they are used in Local Extrapolation Mode (the propagated solution of a problem is the one provided by the fifth-order formula of the pair). Numerical results obtained by testing the new pairs over a standard set of test problems suggest a significant improvement in efficiency when using a specific pair of the new family with minimized truncation error coefficients, instead of some other existing pairs.

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Additional Information

S. N. Papakostas
Affiliation: Department of Mathematics, Zografou Campus, National Technical University of Athens, Athens 157 80, Greece

G. Papageorgiou
Affiliation: Department of Mathematics, Zografou Campus, National Technical University of Athens, Athens 157 80, Greece

Keywords: Initial Value Problems, Runge-Kutta, pairs of embedded methods, phase-lag
Received by editor(s): September 7, 1993
Received by editor(s) in revised form: September 5, 1994, and April 5, 1995
Article copyright: © Copyright 1996 American Mathematical Society