Stable Two-Step Runge-Kutta Collocation Methods for Oscillatory Systems of Initial Value Problems in ODEs

摘要

We present stable two-step Runge-Kutta collocation methods for solution of highly oscillatory systems of first order initial value problems in ordinary differential equations. These methods are obtained based on multistep collocation at Gaussian points, which are shown to be self-starting, convergent, with large regions of absolute stability. They provide excellent approximations of solutions of oscillatory systems of ordinary differential equations, over the entire interval of integration. This approach has several fascinating advantages, for example, the numerical tests indicate that the methods compare favourably with the standard integrators, both in the quality of the numerical solutions and the computational effort. The results obtained from the preliminary experiments also coincide well with the theoretical values and demonstrate the effectiveness and reliability of this approach.