Stability of finite difference schemes for hyperbolic initial boundary value problems II
Annali della Scuola Normale Superiore di Pisa - Classe di Scienze, Serie 5, Volume 10 (2011) no. 1, p. 37-98

We study the stability of finite difference schemes for hyperbolic initial boundary value problems in one space dimension. Assuming ${\ell }^{2}$-stability for the discretization of the hyperbolic operator as well as a geometric regularity condition, we show that the uniform Kreiss-Lopatinskii condition yields strong stability for the discretized initial boundary value problem. The present work extends the results of [4,7] to the widest possible class of finite difference schemes by dropping the technical assumptions of our former work [4]. We give some new examples of numerical schemes for which our results apply.

Published online : 2018-06-21
Classification:  65M12,  65M06,  35L50
@article{ASNSP_2011_5_10_1_37_0,
author = {Coulombel, Jean-Fran\c cois},
title = {Stability of finite difference schemes for hyperbolic initial boundary value problems II},
journal = {Annali della Scuola Normale Superiore di Pisa - Classe di Scienze},
publisher = {Scuola Normale Superiore, Pisa},
volume = {Ser. 5, 10},
number = {1},
year = {2011},
pages = {37-98},
zbl = {1225.65089},
mrnumber = {2829318},
language = {en},
url = {http://www.numdam.org/item/ASNSP_2011_5_10_1_37_0}
}

Coulombel, Jean-François. Stability of finite difference schemes for hyperbolic initial boundary value problems II. Annali della Scuola Normale Superiore di Pisa - Classe di Scienze, Serie 5, Volume 10 (2011) no. 1, pp. 37-98. http://www.numdam.org/item/ASNSP_2011_5_10_1_37_0/

[1] H. Baumgärtel, “Analytic Perturbation Theory for Matrices and Operators”, Birkhäuser Verlag, 1985. | MR 878974 | Zbl 0591.47013

[2] S. Benzoni-Gavage and D. Serre, “Multidimensional Hyperbolic Partial Differential Equations”, Oxford Mathematical Monographs, Oxford University Press, 2007. | MR 2284507 | Zbl 1113.35001

[3] J. Chazarain and A. Piriou, “Introduction to the Theory of Linear Partial Differential Equations”, North-Holland, 1982. | MR 678605 | Zbl 0487.35002

[4] J.-F. Coulombel, Stability of finite difference schemes for hyperbolic initial boundary value problems, SIAM J. Numer. Anal. 47 (2009), 2844–2871. | MR 2551149 | Zbl 1205.65245

[5] E. Godlewski and P.-A. Raviart, “Numerical Approximation of Hyperbolic Systems of Conservation Laws”, Springer-Verlag, 1996. | MR 1410987 | Zbl 0860.65075

[6] B. Gustafsson, H.-O. Kreiss and J. Oliger, “Time Dependent Problems and Difference Methods”, John Wiley & Sons, 1995. | MR 1377057 | Zbl 1275.65048

[7] B. Gustafsson, H.-O. Kreiss and A. Sundström, Stability theory of difference approximations for mixed initial boundary value problems II, Math. Comp. 26 (1972), 649–686. | MR 341888 | Zbl 0293.65076

[8] H.-O. Kreiss, Stability theory for difference approximations of mixed initial boundary value problems, I, Math. Comp. 22 (1968), 703–714. | MR 241010 | Zbl 0197.13704

[9] H.-O. Kreiss, Initial boundary value problems for hyperbolic systems, Comm. Pure Appl. Math. 23 (1970), 277–298. | MR 437941 | Zbl 0193.06902

[10] G. Métivier and K. Zumbrun, Symmetrizers and continuity of stable subspaces for parabolic-hyperbolic boundary value problems, Discrete Contin. Dyn. Syst. 11 (2004), 205–220. | MR 2073953 | Zbl 1102.35332

[11] G. Métivier and K. Zumbrun, Hyperbolic boundary value problems for symmetric systems with variable multiplicities, J. Differential Equations 211 (2005), 61–134. | MR 2121110 | Zbl 1073.35155

[12] D. Michelson, Stability theory of difference approximations for multidimensional initial-boundary value problems, Math. Comp. 40 (1983), 1–45. | MR 679433 | Zbl 0563.65064

[13] R. Mneimné and F. Testard, “Introduction à la théorie des groupes de Lie classiques”, Hermann, 1986. | MR 903847 | Zbl 0598.22001

[14] J. Ralston, Note on a paper of Kreiss, Comm. Pure Appl. Math. 24 (1971), 759–762. | MR 606239 | Zbl 0215.16802

[15] R. Sakamoto, Mixed problems for hyperbolic equations, I, Energy inequalities, J. Math. Kyoto Univ. 10 (1970), 349–373. | MR 283400 | Zbl 0203.10001