MATHICSE Technical Report : Effective models for the multidimensional wave equation in heterogeneous media over long time and numerical homogenization

A family of effective equations that capture the long time dispersive effects of wave propagation in heterogeneous media in an arbitrary large periodic spatial domain $\Omega \subset \mathbb{R}^d$ over long time is proposed and analyzed. For a wave equation with highly oscillatory periodic spatial tensors of characteristic length $\varepsilon$, we prove that the solution of any member of our family of effective equations are $\varepsilon$-close in the $L^\infty (0; T^\varepsilon; L2(\Omega))$ norm to the true oscillatory wave over a time interval of length $T^\varepsilon = O(\varepsilon^2)$.We show that the previously derived effective equation in [Dohnal, Lamacz, Schweizer, Multiscale Model. Simul., 2014] belongs to our family of effective equation. Moreover, while Bloch waves techniques were previously used, we show that asymptotic expansion techniques give an alternative way to derive such effective equations. An algorithm to compute the tensors involved in the dispersive equation and allowing for efficient numerical homogenization methods over long time is proposed.