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arXiv · 1008.0197

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

Aurora-Mihaela Marica·Enrique Zuazua
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Abstract

We study the propagation properties of the solutions of the finite-difference space semi-discrete wave equation on an 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.

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Aurora-Mihaela Marica, Enrique Zuazua. 2010-08-01. Localized solutions for the finite difference semi-discretization of the wave equation. https://doi.org/10.1016/j.crma.2010.03.020

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