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An extension and analysis of the Shu-Osher representation of Runge-Kutta methods

Authors: L. Ferracina and M. N. Spijker
Journal: Math. Comp. 74 (2005), 201-219
MSC (2000): Primary 65M20; Secondary 65L05, 65L06
Published electronically: June 11, 2004
MathSciNet review: 2085408
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Abstract: In the context of solving nonlinear partial differential equations, Shu and Osher introduced representations of explicit Runge-Kutta methods, which lead to stepsize conditions under which the numerical process is total-variation-diminishing (TVD). Much attention has been paid to these representations in the literature.

In general, a Shu-Osher representation of a given Runge-Kutta method is not unique. Therefore, of special importance are representations of a given method which are best possible with regard to the stepsize condition that can be derived from them.

Several basic questions are still open, notably regarding the following issues: (1) the formulation of a simple and general strategy for finding a best possible Shu-Osher representation for any given Runge-Kutta method; (2) the question of whether the TVD property of a given Runge-Kutta method can still be guaranteed when the stepsize condition, corresponding to a best possible Shu-Osher representation of the method, is violated; (3) the generalization of the Shu-Osher approach to general (possibly implicit) Runge-Kutta methods.

In this paper we give an extension and analysis of the original Shu-Osher representation, by means of which the above questions can be settled. Moreover, we clarify analogous questions regarding properties which are referred to, in the literature, by the terms monotonicity and strong-stability-preserving (SSP).

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Additional Information

L. Ferracina
Affiliation: Mathematical Institute, Leiden University, Niels Bohrweg 1, 2333 CA Leiden, The Netherlands

M. N. Spijker
Affiliation: Mathematical Institute, Leiden University, Niels Bohrweg 1, 2333 CA Leiden, The Netherlands

Keywords: Initial value problem, conservation law, method of lines (MOL), Runge-Kutta formula, Shu-Osher representation, total-variation-diminishing (TVD), strong-stability-preserving (SSP), monotonicity.
Received by editor(s): May 7, 2003
Received by editor(s) in revised form: August 3, 2003
Published electronically: June 11, 2004
Article copyright: © Copyright 2004 American Mathematical Society