Localized solutions for the finite difference semi-discretization of the wave equation

Marica A, Zuazua E (2010)


Publication Type: Journal article

Publication year: 2010

Journal

Book Volume: 348

Pages Range: 647-652

Journal Issue: 11-12

DOI: 10.1016/j.crma.2010.03.020

Abstract

We study the propagation properties of the solutions of the finite difference space semi-discrete wave equation on a uniform grid of the whole Euclidean space. We provide a construction of high frequency wave packets that propagate along the corresponding bi-characteristic rays of Geometric Optics with a group velocity arbitrarily close to zero. Our analysis is motivated by control theoretical issues. In particular, the continuous wave equation has the so-called observability property: for a sufficiently large time, the total energy of its solutions can be estimated in terms of the energy concentrated in the exterior of a compact set. This fails to be true, uniformly on the mesh-size parameter, for the semi-discrete schemes and the observability constant blows-up at an arbitrarily large polynomial order. Our contribution consists in providing a rigorous derivation of those wave packets and in analyzing their behavior near that ray, by taking into account the subtle added dispersive effects that the numerical scheme introduces. (C) 2010 Published by Elsevier Masson SAS on behalf of Academie des sciences.

Authors with CRIS profile

Involved external institutions

How to cite

APA:

Marica, A., & Zuazua, E. (2010). Localized solutions for the finite difference semi-discretization of the wave equation. Comptes Rendus Mathematique, 348(11-12), 647-652. https://dx.doi.org/10.1016/j.crma.2010.03.020

MLA:

Marica, Aurora, and Enrique Zuazua. "Localized solutions for the finite difference semi-discretization of the wave equation." Comptes Rendus Mathematique 348.11-12 (2010): 647-652.

BibTeX: Download