Stable and conservative time propagators for second order hyperbolic systems

In this paper we construct a hierarchy of arbitrary high (even) order accurate explicit time propagators for semi-discrete second order hyperbolic systems. An accurate semi-discrete problem is obtained by approximating the corresponding spatial derivatives using high order accurate finite difference operators satisfying the summation by parts rule. In order to obtain a strictly stable semi-discrete problem, boundary conditions are imposed weakly using the simultaneous approximation term method. The time discretization starts with a second order central difference scheme, then using the modified equation approach (even in the presence of a first order derivative in time) we derive arbitrary high order accurate time marching schemes. For the fully discrete problem, we introduce a suitable weighted inner product and use the energy method to derive an optimal CFL condition, which provides a useful and rigorous criterion for stability. Numerical examples are also provided.