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