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Journal Articles (Review Article) IMA Journal of Numerical Analysis Year : 2023

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 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 elate 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.
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Dates and versions

hal-03732720 , version 1 (21-07-2022)
hal-03732720 , version 2 (18-01-2023)
hal-03732720 , version 3 (16-06-2023)

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Boutin Benjamin, Pierre Le Barbenchon, Seguin Nicolas. On the stability of totally upwind schemes for the hyperbolic initial boundary value problem. IMA Journal of Numerical Analysis, inPress, ⟨10.1093/imanum/drad040⟩. ⟨hal-03732720v3⟩
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