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

TL;DR

This paper investigates the propagation of solutions in finite-difference semi-discrete wave equations, constructing high-frequency wave packets and analyzing their behavior, especially regarding control properties and dispersive effects introduced by numerical schemes.

Contribution

It provides a rigorous construction of high-frequency wave packets and analyzes their behavior near characteristic rays, highlighting dispersive effects affecting observability in semi-discrete schemes.

Findings

Wave packets propagate along bi-characteristic rays with near-zero group velocity.

Observability constants blow up polynomially as mesh size decreases.

Dispersive effects significantly influence control properties of semi-discrete wave equations.

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.