The Leray-Gårding method for finite difference schemes
[La méthode de Leray et Gårding pour les schémas aux différences finies]
Journal de l’École polytechnique — Mathématiques, Tome 2 (2015) , pp. 297-331.

Dans les années 1950, Leray et Gårding ont développé une technique de multiplicateur pour obtenir des estimations a priori de solutions d’équations hyperboliques scalaires. L’existence d’un multiplicateur est le point de départ du travail de Rauch  pour montrer des estimations de semi-groupe pour les problèmes aux limites hyperboliques. Dans cet article, nous expliquons comment cette technique de multiplicateur peut être adaptée au cadre des schémas aux différences finies pour les équations de transport. Ce travail s’applique à des schémas numériques multi-pas en temps. L’existence et les propriétés du multiplicateur nous permettent d’obtenir des estimations de semi-groupe optimales pour des versions totalement discrètes des problèmes aux limites hyperboliques.

In the fifties, Leray and Gårding have developed a multiplier technique for deriving a priori estimates for solutions to scalar hyperbolic equations. The existence of such a multiplier is the starting point of the argument by Rauch  for the derivation of semigroup estimates for hyperbolic initial boundary value problems. In this article, we explain how this multiplier technique can be adapted to the framework of finite difference approximations of transport equations. The technique applies to numerical schemes with arbitrarily many time levels. The existence and properties of the multiplier enable us to derive optimal semigroup estimates for fully discrete hyperbolic initial boundary value problems.

Reçu le : 2015-06-12
Accepté le : 2015-11-06
DOI : https://doi.org/10.5802/jep.25
Classification : 65M06,  65M12,  35L03,  35L04
Mots clés: Équations hyperboliques, différences finies, stabilité, conditions aux limites, semi-groupe
@article{JEP_2015__2__297_0,
author = {Jean-Fran\c cois Coulombel},
title = {The Leray-G\aa rding method for finite~difference schemes},
journal = {Journal de l'\'Ecole polytechnique --- Math\'ematiques},
pages = {297--331},
publisher = {\'Ecole polytechnique},
volume = {2},
year = {2015},
doi = {10.5802/jep.25},
zbl = {1328.65175},
language = {en},
url = {jep.centre-mersenne.org/item/JEP_2015__2__297_0/}
}
Coulombel, Jean-François. The Leray-Gårding method for finite difference schemes. Journal de l’École polytechnique — Mathématiques, Tome 2 (2015) , pp. 297-331. doi : 10.5802/jep.25. https://jep.centre-mersenne.org/item/JEP_2015__2__297_0/

 S. Abarbanel & D. Gottlieb - “A note on the leap-frog scheme in two and three space dimensions”, J. Comput. Phys. 21 (1976) no. 3, p. 351-355 | Article | Zbl 0331.65057

 S. Abarbanel & D. Gottlieb - “Stability of two-dimensional initial boundary value problems using leap-frog type schemes”, Math. Comp. 33 (1979) no. 148, p. 1145-1155 | MR 537962 | Zbl 0447.65055

 S. Benzoni-Gavage & D. Serre - Multidimensional hyperbolic partial differential equations. First-order systems and applications, Oxford Mathematical Monographs, The Clarendon Press, Oxford University Press, Oxford, 2007 | Zbl 1113.35001

 J.-F. Coulombel - “Fully discrete hyperbolic initial boundary value problems with nonzero initial data”, to appear in Confluentes Math. | Article | Zbl 1355.65116

 J.-F. Coulombel - “Stability of finite difference schemes for hyperbolic initial boundary value problems”, SIAM J. Numer. Anal. 47 (2009) no. 4, p. 2844-2871 | Article | MR 2551149

 J.-F. Coulombel - “Stability of finite difference schemes for hyperbolic initial boundary value problems”, in HCDTE lecture notes. Part I. Nonlinear hyperbolic PDEs, dispersive and transport equations, AIMS Ser. Appl. Math., vol. 6, Am. Inst. Math. Sci. (AIMS), Springfield, MO, 2013, 146 pages | MR 3340992

 J.-F. Coulombel & A. Gloria - “Semigroup stability of finite difference schemes for multidimensional hyperbolic initial boundary value problems”, Math. Comp. 80 (2011) no. 273, p. 165-203 | Article | MR 2728976 | Zbl 1308.65142

 E. Emmrich - “Convergence of the variable two-step BDF time discretisation of nonlinear evolution problems governed by a monotone potential operator”, BIT 49 (2009) no. 2, p. 297-323 | Article | MR 2507603 | Zbl 1172.65026

 E. Emmrich - “Two-step BDF time discretisation of nonlinear evolution problems governed by monotone operators with strongly continuous perturbations”, Comput. Methods Math. 9 (2009) no. 1, p. 37-62 | MR 2641310 | Zbl 1169.65046

 L. Gårding - “Solution directe du problème de Cauchy pour les équations hyperboliques”, in La théorie des équations aux dérivées partielles, Colloques Internationaux du C.N.R.S., C.N.R.S., Paris, 1956, p. 71-90

 M. Goldberg & E. Tadmor - “Scheme-independent stability criteria for difference approximations of hyperbolic initial-boundary value problems. II”, Math. Comp. 36 (1981) no. 154, p. 603-626 | Article | MR 606519 | Zbl 0466.65054

 B. Gustafsson, H.-O. Kreiss & J. Oliger - Time dependent problems and difference methods, Pure and Applied Mathematics (New York), John Wiley & Sons, Inc., New York, 1995

 B. Gustafsson, H.-O. Kreiss & A. Sundström - “Stability theory of difference approximations for mixed initial boundary value problems. II”, Math. Comp. 26 (1972) no. 119, p. 649-686 | Article | MR 341888 | Zbl 0293.65076

 E. Hairer, S. P. Nørsett & G. Wanner - Solving ordinary differential equations I. Nonstiff problems, Springer Series in Computational Mathematics, vol. 8, Springer-Verlag, Berlin, 1993 | Zbl 0789.65048

 E. Hairer & G. Wanner - Solving ordinary differential equations II. Stiff and differential-algebraic problems, Springer Series in Computational Mathematics, vol. 14, Springer-Verlag, Berlin, 1996 | Article | Zbl 0859.65067

 H.-O. Kreiss - “Stability theory for difference approximations of mixed initial boundary value problems. I”, Math. Comp. 22 (1968), p. 703-714 | Article | MR 241010 | Zbl 0197.13704

 H.-O. Kreiss & L. Wu - “On the stability definition of difference approximations for the initial-boundary value problem”, Appl. Numer. Math. 12 (1993) no. 1-3, p. 213-227 | Article | MR 1227187 | Zbl 0782.65119

 J. Leray - Hyperbolic differential equations, The Institute for Advanced Study, Princeton, N.J., 1953 | MR 80849

 D. Michelson - “Stability theory of difference approximations for multidimensional initial-boundary value problems”, Math. Comp. 40 (1983) no. 161, p. 1-45 | Article | MR 679433 | Zbl 0563.65064

 J. Oliger - “Fourth order difference methods for the initial boundary-value problem for hyperbolic equations”, Math. Comp. 28 (1974), p. 15-25 | Article | MR 359344 | Zbl 0284.65074

 S. Osher - “Stability of difference approximations of dissipative type for mixed initial boundary value problems. I”, Math. Comp. 23 (1969), p. 335-340 | MR 246530 | Zbl 0177.20403

 S. Osher - “Systems of difference equations with general homogeneous boundary conditions”, Trans. Amer. Math. Soc. 137 (1969), p. 177-201 | Article | MR 237982 | Zbl 0174.41701

 J. Rauch - “${ℒ}^{2}$ is a continuable initial condition for Kreiss’ mixed problems”, Comm. Pure Appl. Math. 25 (1972), p. 265-285 | Article | MR 298232 | Zbl 0226.35056

 R. D. Richtmyer & K. W. Morton - Difference methods for initial-value problems. Theory and applications, Interscience Tracts in Pure and Applied Mathematics, vol. 4, Interscience Publishers John Wiley & Sons, Inc., New York-London-Sydney, 1967 | Zbl 0155.47502

 D. M. Sloan - “Boundary conditions for a fourth order hyperbolic difference scheme”, Math. Comp. 41 (1983), p. 1-11 | Article | MR 701620 | Zbl 0536.65077

 J. C. Strikwerda & B. A. Wade - “A survey of the Kreiss matrix theorem for power bounded families of matrices and its extensions”, in Linear operators (Warsaw, 1994), Banach Center Publ., vol. 38, Polish Acad. Sci., Warsaw, 1997, p. 339-360 | MR 1457017 | Zbl 0877.15029

 J. M. Thomas - “Discrétisation des conditions aux limites dans les schémas saute-mouton”, ESAIM Math. Model. Numer. Anal. 6 (1972) no. R-2, p. 31-44 | Numdam

 L. N. Trefethen - “Instability of difference models for hyperbolic initial boundary value problems”, Comm. Pure Appl. Math. 37 (1984), p. 329-367 | Article | MR 739924 | Zbl 0575.65095

 L. N. Trefethen & M. Embree - Spectra and pseudospectra. The behavior of nonnormal matrices and operators, Princeton University Press, Princeton, N.J., 2005 | Zbl 1085.15009

 B. A. Wade - “Symmetrizable finite difference operators”, Math. Comp. 54 (1990) no. 190, p. 525-543 | Article | MR 1011447 | Zbl 0697.65069

 L. Wu - “The semigroup stability of the difference approximations for initial-boundary value problems”, Math. Comp. 64 (1995) no. 209, p. 71-88 | MR 1257582