EFFECTIVE MODELS FOR LONG TIME WAVE PROPAGATION IN LOCALLY PERIODIC MEDIA

A family of effective equations for the wave equation in locally periodic media over long time is derived. In particular, explicit formulas for the effective tensors are provided. To validate the derivation, an a priori error estimate between the effective solutions and the original wave is proved. As the dependence of the estimate on the domain is explicit, the result holds in arbitrarily large periodic hypercube. This constitutes the first analysis for the description of long time effects for the wave equation in locally periodic media. Thanks to this result, the long time a priori error analysis of the numerical homogenization method presented in [A. Abdulle and T. Pouchon, SIAM J. Numer. Anal., 54 (2016), pp. 1507-1534] is generalized to the case of a locally periodic tensor.