Numerical simulation of high frequency waves in highly heterogeneous media is a challenging problem. Resolving the fine structure of the wave field typically requires extremely small time steps and spatial meshes. We show that capturing macroscopic quantities of the wave field, such as the wave energy density, is achievable with much coarser discretizations. We obtain such a result using a time splitting algorithm that solves separately and successively propagation and scattering in the simplified regime of the parabolic wave equation in a random medium. The mathematical theory of the convergence and statistical properties of the algorithm is based on the analysis of the Wigner transforms in random media. Our results provide a step toward understanding time and space discretizations that are needed in order for the numerical algorithm to capture the correct macroscopic statistics of the wave energy density in a random medium.
Mots clés : high frequency waves in random media, time splitting, multiscale analysis
@article{M2AN_2004__38_6_961_0,
author = {Bal, Guillaume and Ryzhik, Lenya},
title = {Time splitting for wave equations in random media},
journal = {ESAIM: Mod\'elisation math\'ematique et analyse num\'erique},
pages = {961--987},
publisher = {EDP-Sciences},
volume = {38},
number = {6},
year = {2004},
doi = {10.1051/m2an:2004046},
mrnumber = {2108940},
zbl = {1130.74393},
language = {en},
url = {http://www.numdam.org/articles/10.1051/m2an:2004046/}
}
TY - JOUR AU - Bal, Guillaume AU - Ryzhik, Lenya TI - Time splitting for wave equations in random media JO - ESAIM: Modélisation mathématique et analyse numérique PY - 2004 SP - 961 EP - 987 VL - 38 IS - 6 PB - EDP-Sciences UR - http://www.numdam.org/articles/10.1051/m2an:2004046/ DO - 10.1051/m2an:2004046 LA - en ID - M2AN_2004__38_6_961_0 ER -
%0 Journal Article %A Bal, Guillaume %A Ryzhik, Lenya %T Time splitting for wave equations in random media %J ESAIM: Modélisation mathématique et analyse numérique %D 2004 %P 961-987 %V 38 %N 6 %I EDP-Sciences %U http://www.numdam.org/articles/10.1051/m2an:2004046/ %R 10.1051/m2an:2004046 %G en %F M2AN_2004__38_6_961_0
Bal, Guillaume; Ryzhik, Lenya. Time splitting for wave equations in random media. ESAIM: Modélisation mathématique et analyse numérique, Tome 38 (2004) no. 6, pp. 961-987. doi : 10.1051/m2an:2004046. http://www.numdam.org/articles/10.1051/m2an:2004046/
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