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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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References

[1] Courant, R., Friedrichs, K. and Lewy, H., Über die partiellen differenzengle-ichungen der mathematischen physik, Math. Ann., 100(1) (1928), pp. 32–74.Google Scholar
[2] Coulombel, J. F. and Gloria, A., Semigroup stability of finite difference schemes for multidimensional hyperbolic initial boundary value problems, Math. Comp., 80(273) (2011), pp. 165–203.CrossRefGoogle Scholar
[3] Chainais-Hillairet, C. and Grenier, E., Numerical boundary layers for hyperbolic systems in 1-D, M2AN Math. Model. Numer. Anal., 35(1) (2001), pp. 91–106.Google Scholar
[4] Coulombel, J. F., Stability of finite difference schemes for hyperbolic initial boundary value problems, In HCDTE Lecture Notes. Part I. Nonlinear Hyperbolic PDEs, Dispersive and Transport Equations, American Institute of Mathematical Sciences, (2013), pp. 97–225.Google Scholar
[5] Coulombel, J. F., The Leray-Gårding method for finite difference schemes, J. Éc. Polytech. Math., 2 (2015), pp. 297–331.Google Scholar
[6] Dubois, F. and LeFloch, P., Boundary conditions for nonlinear hyperbolic systems of conservation laws, J. Differ. Equations, 71(1)(1988), pp. 93–122.Google Scholar
[7] Gustafsson, B., Kreiss, H. O., and Oliger, J., Time dependent problems and difference methods, John Wiley & Sons, 1995.Google Scholar
[8] Gustafsson, B., Kreiss, H. O. and Sundström, A., Stability theory of difference approximations for mixed initial boundary value problems. II, Math. Comp., 26(119) (1972), pp. 649–686.CrossRefGoogle Scholar
[9] Gisclon, M. and Serre, D., Conditions aux limites pour un système strictement hyperbolique fournies par le schéma de Godunov, RAIRO Modél. Math. Anal. Numér., 31(3) (1997), pp. 359–380.Google Scholar
[10] Goldberg, M. and Tadmor, E., Scheme-independent stability criteria for difference approximations of hyperbolic initial-boundary value problems. II, Math. Comp., 36(154) (1981), pp. 603–626.CrossRefGoogle Scholar
[11] Gustafsson, B., The convergence rate for difference approximations to mixed initial boundary value problems, Math. Comp., 29(130) (1975), pp. 396–406.Google Scholar
[12] Gerard-Varet, D., Formal derivation of boundary layers in fluid mechanics, J. Math. Fluid Mech., 7(2) (2005), pp. 179–200.Google Scholar
[13] Hairer, E., Nørsett, S. P. and Wanner, G., Solving Ordinary Differential Equations. I, Spr. S. Comp., second ed., 1993.Google Scholar
[14] Hairer, E. and Wanner, G., Solving Ordinary Differential Equations. II, Spr. S. Comp., (14) 1996.Google Scholar
[15] Kreiss, H. O., Initial boundary value problems for hyperbolic systems, Comm. Pure Appl. Math., 23 (1970), pp. 277–298.Google Scholar
[16] Lubich, C. and Nevanlinna, O., On resolvent conditions and stability estimates, BIT, 31(2) (1991), pp. 293–313.Google Scholar
[17] Métivier, G., On the L2 well-posedness of hyperbolic initial boundary value problems, Hyperbolic, 2014.Google Scholar
[18] Strikwerda, J. C. and Wade, B. A., A survey of the Kreiss matrix theorem for power bounded families of matrices and its extensions, In Linear operators (Warsaw, 1994), Polish Acad. Sci., (1997), pp.339–360.Google Scholar
[19] Trefethen, L. N. and Embree, M., Spectra and Pseudospectra, Springer, 175(98) (2005), xviii.Google Scholar
[20] Trefethen, L. N., Group velocity in finite difference schemes, SIAM Rev., 24(2) (1982), pp. 113–136.Google Scholar
[21] Wu, L., The semigroup stability of the difference approximations for initial-boundary value problems, Math. Comp., 64(209) (1995), pp. 71–88.Google Scholar