Two-Step Runge-Kutta Methods and Hyperbolic Partial Differential Equations

Abstract

Abstract. The purpose of this study is the design of efficient methods for the solution of an ordinary differential system of equations arising from the semidiscretization of a hyperbolic partial differential equation. Jameson recently introduced the use of one-step Runge-Kutta methods for the numerical solution of the Euler equations. Improvements in efficiency up to 80% may be achieved by using two-step Runge-Kutta methods instead of the classical onestep methods. These two-step Runge-Kutta methods were first introduced by Byrne and Lambert in 1966. They are designed to have the same number of function evaluations as the equivalent one-step schemes, and thus they are potentially more efficient. By solving a nonlinear programming problem, which is specified by stability requirements, optimal two-step schemes are designed. The optimization technique is applicable for stability regions of any shape. INTRODUCTIONIn this paper we consider a class of pseudo-Runge-Kutta methods for the solution of an ordinary differential system of equations (0.1) y' = f(y) which arises from the semidiscretization of a hyperbolic partial differential equation. In 1982, Jameson [7] initiated interest in the use of one-step Runge-Kutta methods for the numerical solution of the Euler equations. He applied the van der Houwen [4] optimal schemes in codes for the solution of the Euler equations by central differences. These schemes are optimal because they have regions of stability enclosing a maximal interval on the imaginary axis, as is required when central differences are used for the semidiscretization. Here we demonstrate that greater efficiency is achieved by using two-rather than onestep Runge-Kutta formulae. These Runge-Kutta methods were first considered by Byrne and Lambert [1] in 1966.We define an explicit two-step w-stage Runge-Kutta method as m (0.2) yn+x = (1 -ß)yn + ßyn_x + h ][>,/(>V.) + V0Í)) » i=i