A Stable Scheme for the Numerical Computation of Long Wave Propagation in Temporal Laminates

A temporal laminate is a material whose parameters are homogeneous in space but vary periodically and discontinuously in time. In this article, we consider wave propagation through a temporal laminate where the period of the disturbance moving through the media is large relative to ε the period of the lamination. It is worth noting that the constituent materials and the mixing coefficient can be chosen so that the effective speed in a temporal laminate is greater than the individual phase speeds. We show that the analytic problem admits stable long wave modes, but shorter wave modes grow as they pass through the laminate layers. Computing wave motion through this composite medium using the standard upwind, finite-difference method under the CFL condition for numerical wave propagation in the individual media will produce growing short wave modes. Numerical results are degraded since accuracy is quickly lost due to the growth of short waves which enter into the computation through truncation and round-off error. A new CFL constraint is derived for a finite-difference numerical scheme which will allow us to compute the stable long wave motion. Numerical results are given for the direct numerical simulation of the homogenization problem (ε→0).