Abstract. In this article, we describe the propagation properties of the one-dimensional wave and transport equations with variable coefficients semi-discretized in space by finite difference schemes on non-uniform meshes obtained as diffeomorphic transformations of uniform ones. In particular, we introduce and give a rigorous meaning to notions like the principal symbol of the discrete wave operator and the corresponding bi-characteristic rays. The main mathematical tool we employ is the discrete Wigner transform, which, in the limit as the mesh size parameter tends to zero, yields the so-called Wigner (semi-classical) measure. This measure provides the dynamics of the bi-characteristic rays, i.e., the solutions of the Hamiltonian system describing the propagation, in both physical and Fourier spaces, of the energy of the solution to the wave equation. We show that, due to dispersion phenomena, the high-frequency numerical dynamics does not coincide with the continuous one. Our analysis holds for C 0,1 (R)-coefficients and non-uniform grids obtained by means of C 1,1 (R)-diffeomorphic transformations of a uniform one. We also present several numerical simulations that confirm the predicted paths of the space-time projections of the bi-characteristic rays. Based on the theoretical analysis and simulations, we describe some of the pathological phenomena that these rays might exhibit as, for example, their reflection before touching the boundary of the space domain. This leads, in particular, to the failure of the classical properties of boundary observability of continuous waves, arising in control and inverse problems theory.
We build non-uniform numerical meshes for the finite difference and finite element approximations of the 1d wave equation, ensuring that all numerical solutions reach the boundary, as continuous solutions do, in the sense that the full discrete energy can be observed by means of boundary measurements, uniformly with respect to the mesh-size. The construction of the nonuniform mesh is achieved by means of a concave diffeomorphic transformation of a uniform grid into a non-uniform one, making the mesh finer and finer when approaching the right boundary. For uniform meshes it is known that high-frequency numerical wave packets propagate very slowly without never getting to the boundary. Our results show that this pathology can be avoided by taking suitable non-uniform meshes. This also allows to build convergent numerical algorithms for the approximation of boundary controls of the wave equation.
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. RésuméOnétudie les propriétés de propagation des solutions de l'équation des ondes semi-discretisée en espace par différences finies sur une grille uniforme dans tout l'espace euclidien. On réalise une construction de paquets d'ondes concentrésà haute fréquence qui se propagent le long des rayons bicaractéristiques de l'Optique Géométriqueà une vitesse de groupe arbitrairement petite. Notre analyse est motivée par la théorie du contrôle. Plus précisement, l'équation des ondes continue vérifie la propriété d'observabilité : pour un temps suffisament grand, l'énergie totale de ses solutions peutêtre estimée en fonction de l'énergie localiséeà l'extérieur d'un ensemble compact. Cette propriété n'est pas verifiée de manière uniforme par rapport au pas de discrétisation pour le schéma semidiscret pour un temps fini quelconque, si bien que la constante d'observabilité semi-discrète diverge avec un taux polynomial arbitraire. Nous donnons une construction précise de ces paquets d'ondes et decrivons l'effet dispersif rajouté que le schéma numérique génère autour du rayon de propagation. Dans cet article, on considère le problème de Cauchy associéà l'équation des ondes d-dimensionnelle semi-discretisée en espace par un schéma centré en différences finies dans un maillage uniforme. Nous avons deux objectifs. Le premier est de montrer l'éxistence de paquets d'ondes concentrés autour d'un nombre d'onde fixé a priori,à haute fréquence, propageant avec une vitesse de groupe arbitrairement petite autour des rayons bicharactéristiques du schéma semi-discret et donc peu visibles depuis le domaine d'observation. On déduit dans [3] que la constante d'observabilité semi-discrète diverge au moins d'une manière polynomiale arbitraire, et ce pour tout temps d'observabilité fini. Le deuxième objectif est de donner une forme asymptotique précise pour ces paquets d'ondes, de façonà mettre enévidence la dispersion numérique qui n'apparaît pas dans le modèle continu.Notre résultat complète la littérature exi...
We analyze the propagation properties of the numerical versions of one and two-dimensional wave equations, semi-discretized in space by finite difference schemes. We focus on high-frequency solutions whose propagation can be described, both at the continuous and semi-discrete level, by micro-local tools. We do it both for uniform and non-uniform numerical grids and also for constant coefficients and variable ones. The energy of continuous and semi-discrete high-frequency solutions propagates along bi-characteristic rays, but their dynamics differ from the continuous to the semi-discrete setting, because of the different nature of the corresponding Hamiltonians. One of the main objectives of this paper is to illustrate through accurate numerical simulations that, in agreement with the micro-local theory, numerical high-frequency solutions can bend in an unexpected manner, as a result of the accumulation of the local effects introduced by the heterogeneity of the numerical grid. These effects are enhanced in the multi-dimensional case where the interaction and combination of such behaviors in the various space directions may produce, for instance, the rodeo effect, i. e. waves that are trapped by the numerical grid in closed loops, without ever getting to the exterior boundary. Our analysis allows explaining all such pathological behaviors. Moreover, the discussion in this paper also contributes to the existing theory about the necessity of filtering high-frequency numerical components when dealing with control and inversion problems for waves, which is based very much in the theory of rays and, in particular, on the fact that they can be observed when reaching the exterior boundary of the domain, a key property that can be lost through numerical discretization.2010 Mathematics Subject Classification. 35A21, 37C05, 65M06, 70K05.
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