Dispersive homogenized models and coefficient formulas for waves in general periodic media
Tomáš Dohnal, T. Dohnal, Agnes Lamacz, Agnes Lamacz, Ben Schweizer · Asymptotic Analysis · 2015
Abstract We analyze a homogenization limit for the linear wave equation of second order. The spatial operator is assumed to be of divergence form with an oscillatory coefficient matrix a ε that is periodic with characteristic length scale ε ; no spatial symmetry properties are imposed. Classical homogenization theory allows to describe solutions u ε well by a non-dispersive wave equation on fixed time intervals ( 0 , T ) . Instead, when larger time intervals are considered, dispersive effects are observed. In this contribution we present a well-posed weakly dispersive equation with homogeneous coefficients such that its solutions w ε describe u ε well on time intervals ( 0 , T ε − 2 ) . More precisely, we provide a norm and uniform error estimates of the form ∥ u ε ( t ) − w ε ( t ) ∥ ⩽ C ε for t ∈ ( 0 , T ε − 2 ) . They are accompanied by computable formulas for all coefficients in the effective models. We additionally provide an ε -independent equation of third order that describes dispersion along rays and we present numerical examples.