Localized solutions for the finite difference semi-discretization of the wave equation |
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Authors: | Aurora Marica Enrique Zuazua |
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Institution: | 1. Ikerbasque, Basque Foundation for Sciences, Basque Country, Spain;2. BCAM – Basque Center for Applied Mathematics, Bizkaia Technology Park, 500, 48160 Derio, Basque Country, Spain |
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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. |
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