On the stability of totally upwind schemes for the hyperbolic initial boundary value problem

Abstract In this paper, we present a numerical strategy to check the strong stability (or GKS-stability) of one-step explicit totally upwind schemes in 1D with numerical boundary conditions. The underlying approximated continuous problem is the one-dimensional advection equation. The strong stabilit...

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Bibliographic Details
Published in:IMA journal of numerical analysis 2024-04, Vol.44 (2), p.1211-1241
Main Authors: Boutin, Benjamin, Le Barbenchon, Pierre, Seguin, Nicolas
Format: Article
Language:English
Subjects:
Analysis of PDEs
Mathematics
Numerical Analysis
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Summary:Abstract In this paper, we present a numerical strategy to check the strong stability (or GKS-stability) of one-step explicit totally upwind schemes in 1D with numerical boundary conditions. The underlying approximated continuous problem is the one-dimensional advection equation. The strong stability is studied using the Kreiss–Lopatinskii theory. We introduce a new tool, the intrinsic Kreiss–Lopatinskii determinant, which possesses remarkable regularity properties. By applying standard results of complex analysis, we are able to relate the strong stability of numerical schemes to the computation of a winding number, which is robust and cheap. The study is illustrated with the Beam–Warming scheme together with the simplified inverse Lax–Wendroff procedure at the boundary.
ISSN:0272-4979
1464-3642
DOI:10.1093/imanum/drad040