The Neumann boundary condition for the two-dimensional Lax–Wendroff scheme

We study the stability of the two-dimensional Lax–Wendroff scheme with a stabilizer that approximates solutions to the transport equation. The problem is first analyzed in the whole space in order to show that the so-called energy method yields an optimal stability criterion for this finite difference scheme. We then deal with the case of a half-space when the transport operator is outgoing. At the numerical level, we enforce the Neumann extrapolation boundary condition and show that the corresponding scheme is stable. Eventually we analyze the case of a quarter-space when the transport operator is outgoing with respect to both sides. We then enforce the Neumann extrapolation boundary condition on each side of the boundary and propose an extrapolation boundary condition at the numerical corner in order to maintain stability for the whole numerical scheme.

Keywords

transport equation, numerical scheme, domain with corners, boundary condition, stability

2010 Mathematics Subject Classification

65M06, 65M12, 65M20

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Research of the authors was supported by Agence Nationale de la Recherche project NABUCO, ANR-17-CE40-0025.

This article was completed while the first author was visiting the Institut de Mathématiques de Toulouse as a CNRS researcher. A. B. warmly thanks CNRS and the Institut de Mathématiques de Toulouse for its hospitality during this visit.

Received 11 October 2022

Accepted 30 January 2023

Published 15 November 2023