Dispersive homogenized models and coefficient formulas for waves in general periodic media

Article type: Research Article

Authors: Dohnal, T. | Lamacz, A. | Schweizer, B.^*;

Affiliations: Fakultät für Mathematik, Technische Universität Dortmund, Vogelpothsweg 87, D-44227 Dortmund, Germany. E-mail: ben.schweizer@tu-dortmund.de

Correspondence: [*] Corresponding author. E-mail: ben.schweizer@tu-dortmund.de.

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 aε that is periodic with characteristic length scale ε; no spatial symmetry properties are imposed. Classical homogenization theory allows to describe solutions uε 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 wε describe uε well on time intervals (0,Tε−2). More precisely, we provide a norm and uniform error estimates of the form ∥uε(t)−wε(t)∥⩽Cε for t∈(0,Tε−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.

Keywords: wave equation, large time homogenization, dispersive model, Bloch analysis

DOI: 10.3233/ASY-141280

Journal: Asymptotic Analysis, vol. 93, no. 1-2, pp. 21-49, 2015