Crime Pays; Homogenized Wave Equations for Long Times

TL;DR

This paper investigates the accuracy of asymptotic expansions in periodic homogenization of wave equations over large times, introducing a novel 'criminal' algorithm that extends approximation validity to very long times using high order homogenized equations.

Contribution

It introduces a new 'criminal' homogenization algorithm that improves approximation accuracy for wave equations over very long times using high order homogenized equations.

Findings

Standard two-scale expansion accurate up to time $ ext{O}( ext{}\epsilon^{-2+ ext{}\delta)$

Criminal algorithm achieves $O( ext{ }\epsilon^N)$ error for times $ ext{ }\epsilon^{-N}$

High order homogenized equations provide dispersive corrections for moderate wave numbers

Abstract

This article examines the accuracy for large times of asymptotic expansions from periodic homogenization of wave equations. As usual, $\epsilon$ denotes the small period of the coefficients in the wave equation. We first prove that the standard two scale asymptotic expansion provides an accurate approximation of the exact solution for times $t$ of order $\epsilon^{-2+\delta}$ for any $\delta>0$. Second, for longer times, we show that a different algorithm, that is called criminal because it mixes different powers of $\epsilon$, yields an approximation of the exact solution with error $O(\epsilon^N)$ for times $\epsilon^{-N}$ with $N$ as large as one likes. The criminal algorithm involves high order homogenized equations that, in the context of the wave equation, were first proposed by Santosa and Symes and analyzed by Lamacz. The high order homogenized equations yield dispersive corrections for moderate wave numbers. We give a systematic analysis for all time scales and all high order corrective terms.