High-order explicit Runge-Kutta pairs with low stage order

doi:10.1016/S0168-9274(96)00041-4

Applied Numerical Mathematics

Volume 22, Issues 1–3, November 1996, Pages 345-357

https://doi.org/10.1016/S0168-9274(96)00041-4 Get rights and content

Abstract

To illustrate his idea for propagating an approximate solution of an initial value problem, Runge (1895) included a pair of formulas or orders 1 and 2 (a 1,2 pair) whose difference could be used to estimate the error incurred in each step. Later work by Merson (1957) stimulated the search for p − 1, p pairs (p > 3) of Runge-Kutta formulas which would be more efficient. A scheme developed by Butcher (1963) conveniently tabulated the algebraic conditions imposed by the requirements of accuracy. Yet the direct solution of these conditions to obtain explicit Runge-Kutta pairs has remained a more challenging problem. Families of different types have been discovered through continuing research, and some of these are briefly reviewed. A recent approach (Verner, 1994) to solving the order conditions is illustrated with some new parametric families of 7,8 and 8,9 pairs.

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^☆: This work was supported by the Natural Sciences and Engineering Research Council of Canada, and the Information Technology Research Centre of Ontario.

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High-order explicit Runge-Kutta pairs with low stage order☆

Abstract

Comput. Math. Appl.

J. Comput. Appl. Math.

J. Comput. Appl. Math.

Coefficients for the study of Runge-Kutta integration processes

J. Austral. Math. Soc.

Implicit Runge-Kutta processes

Math. Comp.

On the attainable order of Runge-Kutta methods

Math. Comp.

The non-existence of ten stage eighth order explicit Runge-Kutta methods

BIT

The Numerical Analysis of Ordinary Differential Equations

Some explicit Runge-Kutta methods of high order

SIAM J. Numer. Anal.

An eighth order Runge-Kutta process with eleven function evaluations per step