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Stability of Finite Difference Schemes for Hyperbolic Initial Boundary Value Problems: Numerical Boundary Layers

Part of: Partial differential equations, initial value and time-dependent initial-boundary value problems

Published online by Cambridge University Press:  20 June 2017

Benjamin Boutin  and
Jean-François Coulombel
Benjamin Boutin*
Affiliation:
IRMAR (UMR CNRS 6625), Université de Rennes, Campus de Beaulieu, 35042 Rennes Cedex, France
Jean-François Coulombel*
Affiliation:
CNRS, Université de Nantes, Laboratoire de Mathématiques Jean Leray (CNRS UMR6629), 2 rue de la Houssinière, BP 92208, 44322 Nantes Cedex 3, France
*
*Corresponding author. Email addresses: Benjamin.Boutin@univ-rennes1.fr (B. Boutin), Jean-Francois.Coulombel@univ-nantes.fr (J. F. Coulombel)
*Corresponding author. Email addresses: Benjamin.Boutin@univ-rennes1.fr (B. Boutin), Jean-Francois.Coulombel@univ-nantes.fr (J. F. Coulombel)
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Abstract

In this article, we give a unified theory for constructing boundary layer expansions for discretized transport equations with homogeneous Dirichlet boundary conditions. We exhibit a natural assumption on the discretization under which the numerical solution can be written approximately as a two-scale boundary layer expansion. In particular, this expansion yields discrete semigroup estimates that are compatible with the continuous semigroup estimates in the limit where the space and time steps tend to zero. The novelty of our approach is to cover numerical schemes with arbitrarily many time levels.

Keywords

Transport equationsnumerical schemesDirichlet boundary conditionboundary layersstability

MSC classification

Secondary: 65M12: Stability and convergence of numerical methods 65M06: Finite difference methods 65M20: Method of lines
Type
Research Article
Information
Numerical Mathematics: Theory, Methods and Applications , Volume 10 , Issue 3 , August 2017 , pp. 489 - 519
Copyright
Copyright © Global-Science Press 2017 

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