Time Compact High Order Difference Methods for Wave Propagation

We construct one-step explicit difference methods for solution of wave propagation problems with fourth order accuracy in both space and time by using a principle that can be generalized to arbitrary order. We use the first order system form and a staggered grid. The fourth order accuracy in time is obtained by transferring time derivatives in the truncation error to space derivatives. Discontinuous coefficients corresponding to interfaces between different materials are considered as a special case of variable coefficients, and the method is applied across the discontinuities. The accuracy is much improved compared to second order methods even for this type of problems. A certain norm is shown to be conserved, ensuring good accuracy even for long time integration.