Transparent numerical boundary conditions for evolution equations: Derivation and stability analysis

Coulombel, Jean-François

Annales de la Faculté des sciences de Toulouse : Mathématiques, Série 6, Tome 28 (2019) no. 2, pp. 259-327.

Le but de cet article est de proposer une étude systématique des conditions aux limites transparentes pour les approximations par différences finies d’équations d’évolution. On essaie de maintenir la discussion au plus haut niveau de généralité possible afin d’appliquer la théorie à la plus large classe de problèmes.

On aborde deux problèmes principaux. On construit en premier lieu des conditions aux limites numériques transparentes, c’est-à-dire qu’on exhibe les relations satisfaites par la solution du problème de Cauchy quand les données initiales sont nulles hors d’un certain domaine. Notre construction englobe les discrétisations d’équations de type transport, diffusion ou dispersif avec un « stencil » arbitrairement grand. Le second problème que nous abordons est celui de la stabilité du problème mixte obtenu en imposant les conditions aux limites numériques construites à la première étape. On étudie ici le cas des équations de transport discrétisées. Sous une hypothèse de bord non-caractéristique, notre résultat principal classifie les schémas numériques pour lesquels les conditions aux limites transparentes vérifient la condition dite de Kreiss–Lopatinskii uniforme. En adaptant des travaux antérieurs au cadre non-local considéré ici, notre analyse aboutit finalement à des estimations de trace et de semi-groupe pour les conditions aux limites numériques transparentes. L’article se conclut avec des exemples et de futures extensions possibles.

The aim of this article is to propose a systematic study of transparent boundary conditions for finite difference approximations of evolution equations. We try to keep the discussion at the highest level of generality in order to apply the theory to the broadest class of problems.

We deal with two main issues. We first derive transparent numerical boundary conditions, that is, we exhibit the relations satisfied by the solution to the pure Cauchy problem when the initial condition vanishes outside of some domain. Our derivation encompasses discretized transport, diffusion and dispersive equations with arbitrarily wide stencils. The second issue is to prove sharp stability estimates for the initial boundary value problem obtained by enforcing the boundary conditions derived in the first step. We focus here on discretized transport equations. Under the assumption that the numerical boundary is non-characteristic, our main result characterizes the class of numerical schemes for which the corresponding transparent boundary conditions satisfy the so-called Uniform Kreiss–Lopatinskii Condition. Adapting some previous works to the non-local boundary conditions considered here, our analysis culminates in the derivation of trace and semigroup estimates for such transparent numerical boundary conditions. Several examples and possible extensions are given.

Classification : 65M06, 65M12, 35L02, 35K05, 35Q41
Mots clés : evolution equations, difference approximations, transparent boundary conditions, stability

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