RUNGE-KUTTA PAIRS FOR PERIODIC INITIAL-VALUE PROBLEMS

We study the relative merits of the phase-lag property of Runge-Kutta pairs and we propose new explicit embedded pairs for the numerical solution of first order differential systems with periodical solution. We analyze two families of 5(4) pairs and one family of 6(5) pairs with respect to the attainable phase-lag order. From each family we choose a pair with the highest achievable phase-lag order, optimized with respect to a measure of the magnitude of its truncation error coefficients. The new 5(4) algebraic order pairs are of phase-lag order 8(4) and 8(6) and they are both non-dissipative, while the 6(5) pair is dissipative and of phase-lag order 10(6). The new pairs exhibit an improved performance, in comparison with other currently known general and special purpose methods, when they are applied to semidiscretized hyperbolic equations and problems describing free and weakly forced oscillations.