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

作者：Dohnal T; Lamacz A; Schweizer B^*

来源：Asymptotic Analysis, 2015, 93(1-2): 21-49.

DOI：10.3233/ASY-141280

• wave equation
• large time homogenization
• dispersive model
• Bloch analysis

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(epsilon) that is periodic with characteristic length scale epsilon; no spatial symmetry properties are imposed. Classical homogenization theory allows to describe solutions u(epsilon) 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(epsilon) describe u(epsilon) well on time intervals (0, T epsilon(-2)). More precisely, we provide a norm and uniform error estimates of the form parallel to u(epsilon) (t) - w(epsilon) (t) parallel to <= C epsilon for t is an element of (0, T epsilon(-2)). They are accompanied by computable formulas for all coefficients in the effective models. We additionally provide an e-independent equation of third order that describes dispersion along rays and we present numerical examples.

• 出版日期2015