HOMOGENIZATION OF LOCALLY PERIODIC WAVE SYSTEMS FOR LONG TIMES
Grégoire Allaire, Agnes Lamacz-Keymling, Jeffrey B. Rauch · HAL (Le Centre pour la Communication Scientifique Directe) · 2024
This paper is concerned with the homogenization of wave equations in periodic media for long times. We extend our previous results, which were limited to the scalar wave equation with purely periodic coefficients, in two directions. First, for a system of equations in purely periodic media we derive a high-order homogenized system and prove that its solution is an approximation of the exact solution with error O(ϵ N1 ) for times ϵ -N2 with N 1 and N 2 as large as one likes, where ϵ denotes the period. Second, we generalize this analysis for a class of systems with locally periodic coefficients of the type a(x, x/ϵ) with a being periodic with respect to its second argument. The admissible systems for our analysis are those where the density tensor is constant, although the diffusion or rigidity tensor is truly locally periodic. The strategy in the system case follows that of scalar case, (i) derive a high-order homogenized equation by a criminal two-scale asymptotic expansion, (ii) eliminate higher order time derivatives, (iii) filter, (iv) prove stability of the resulting high-order homogenized equation, (v) derive an error estimate. Steps (ii), (iii), (iv) for the system case are entirely new. The new elimination and filtering respect the symmetry of the system, and they preserve its divergence form, which is not obvious for coefficients a(x, x/ϵ).