Space–time multiscale model for wave propagation in heterogeneous media

Abstract

A space–time multiscale model for wave propagation in heterogeneous media is developed. The model builds on the authors' previous work on the higher-order mathematical homogenization theory with multiple spatial and temporal scales, and is aimed at addressing the issues of stability and mathematical consistency. Starting from the weak forms of homogenized macroscopic equations of motion, terms causing the solution secularity are identified and enforced to vanish. This condition recovers the missing boundary conditions and gives rise to two secularity constrains imposing the uniform validity of asymptotic expansions. Finite element semi-discretization in space along with an analytical solution in time are employed to incorporate the secularity constrains in the leading-order solution and account for the slow time dependence of the leading solution. Pade approximation is utilized to develop the time stepping schemes on the fast time scale. The formulation is verified for wave propagation problems in semi-infinite and finite domains.