Runge-Kutta-Nyström methods for hyperbolic problems with time-dependent coefficients (Q1098598)

Galerkin fully discrete approximations for hyperbolic equations with time-dependent coefficients are analyzed. The schemes are based on Runge- Kutta-Nyström methods applied to the semidiscrete (second-order in time) equations. This approach permits the analysis of arbitrary high- order methods and does not make use of derivative values. A preconditioned iterative procedure for the approximate but efficient solution of the resulting linear systems is considered. At every time step, using previously known values, an initial approximation which is within the global order of accuracy of the method is constructed via an extrapolation technique. This is then refined an additional O(k 2) by the preconditioned iterative method. If some additional conditions are satisfied, then an O(k r) refinement is shown to suffice, and the stability and the rate of convergence of the base scheme is preserved. A specific class of methods exhibit parallel features that can be exploited to reduce the final execution time to that of a low-order method.