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Stability of finite difference schemes for the hyperbolic initial boundary value problem by winding number computations
Benjamin Boutin1 ; Pierre Le Barbenchon1 ; Nicolas Seguin2
1 Univ Rennes, CNRS, IRMAR - UMR 6625, 35000 Rennes, France
2 Antenne Inria de l’Université de Montpellier, IMAG, CNRS, université de Montpellier, France
Annales de la Faculté des sciences de Toulouse : Mathématiques, Série 6, Tome 34 (2025) no. 1, pp. 193-224.

In this paper, we present a numerical strategy to check the strong stability (or GKS-stability) of one-step explicit finite difference schemes for the one-dimensional advection equation with an inflow boundary condition. The strong stability is studied using the Kreiss–Lopatinskii theory. We introduce a new tool, the intrinsic Kreiss–Lopatinskii determinant, which possesses the same regularity as the vector bundle of discrete stable solutions. By applying standard results of complex analysis to this determinant, 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 O3 scheme and the fifth-order Lax–Wendroff (LW5) scheme together with a reconstruction procedure at the boundary.

Dans cet article, nous présentons une stratégie numérique permettant de vérifier la stabilité forte (ou stabilité GKS) des schémas de différences finies explicites à un pas pour l’équation d’advection unidimensionnelle avec une condition de bord entrante. La stabilité forte est étudiée à l’aide de la théorie de Kreiss–Lopatinskii. Nous introduisons un nouvel outil, le déterminant intrinsèque de Kreiss–Lopatinskii, qui possède la même régularité que le fibré vectoriel des solutions stables. En lui appliquant les résultats usuels de l’analyse complexe, nous sommes en mesure de ramener l’étude de la stabilité forte au calcul d’indice d’un lacet, procédure fiable et peu coûteuse. L’étude est illustrée avec les schémas O3 et de Lax–Wendroff d’ordre cinq (LW5) avec une condition de bord obtenue par reconstruction.

Reçu le :
Accepté le :
Publié le :
DOI : 10.5802/afst.1809
Classification : 65M12, 65M06, 30E10
Keywords: IBVP, Kreiss–Lopatinskii determinant, GKS-stability, finite difference methods, winding number

Benjamin Boutin 1 ; Pierre Le Barbenchon 1 ; Nicolas Seguin 2

1 Univ Rennes, CNRS, IRMAR - UMR 6625, 35000 Rennes, France
2 Antenne Inria de l’Université de Montpellier, IMAG, CNRS, université de Montpellier, France
Licence : CC-BY 4.0
Droits d'auteur : Les auteurs conservent leurs droits 
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     journal = {Annales de la Facult\'e des sciences de Toulouse : Math\'ematiques},
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     publisher = {Universit\'e Paul Sabatier, Toulouse},
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Benjamin Boutin; Pierre Le Barbenchon; Nicolas Seguin. Stability of finite difference schemes for the hyperbolic initial boundary value problem by winding number computations. Annales de la Faculté des sciences de Toulouse : Mathématiques, Série 6, Tome 34 (2025) no. 1, pp. 193-224. doi : 10.5802/afst.1809. https://afst.centre-mersenne.org/articles/10.5802/afst.1809/

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