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.
展开▼
