The stability theory for hyperbolic initial boundary value problems relies most of the time on the Laplace transform with respect to the time variable. For technical reasons, this usually restricts the validity of stability estimates to the case of zero initial data. In this article, we consider the class of non-glancing finite difference approximations to the hyperbolic operator. We show that the maximal stability estimates that are known for zero initial data and nonzero boundary source term extend to the case of nonzero initial data in . The main novelty of our approach is to cover finite difference schemes with an arbitrary number of time levels. As an easy corollary of our main trace estimate, we recover former stability results in the semigroup sense by Kreiss [11] and Osher [17].
@article{CML_2015__7_2_17_0,
author = {Coulombel, Jean-Fran\c cois},
title = {Fully discrete hyperbolic initial boundary value problems with nonzero initial data},
journal = {Confluentes Mathematici},
volume = {7},
year = {2015},
pages = {17-47},
doi = {10.5802/cml.22},
language = {en},
url = {http://dml.mathdoc.fr/item/CML_2015__7_2_17_0}
}
Coulombel, Jean-François. Fully discrete hyperbolic initial boundary value problems with nonzero initial data. Confluentes Mathematici, Tome 7 (2015) pp. 17-47. doi : 10.5802/cml.22. http://gdmltest.u-ga.fr/item/CML_2015__7_2_17_0/
[1] On mixed initial-boundary value problems for systems that are not strictly hyperbolic, Appl. Math. Lett., Tome 24 (2011) no. 5, pp. 757-761 | MR 2765156 | Zbl 1213.35294
[2] Multidimensional hyperbolic partial differential equations, Oxford University Press (2007) (First-order systems and applications) | MR 2284507 | Zbl 1113.35001
[3] Quelques inégalités concernant les fonctions analytiques, Ark. Mat. Astr. Fys., Tome 29B (1943) no. 11, pp. 6 | MR 11717
[4] Stability of finite difference schemes for hyperbolic initial boundary value problems, SIAM J. Numer. Anal., Tome 47 (2009) no. 4, pp. 2844-2871 | MR 2551149 | Zbl 1205.65245
[5] Stability of finite difference schemes for hyperbolic initial boundary value problems II, Ann. Sc. Norm. Super. Pisa Cl. Sci. (5), Tome X (2011) no. 1, pp. 37-98 | MR 2829318 | Zbl 1225.65089
[6] Stability of finite difference schemes for hyperbolic initial boundary value problems, HCDTE Lecture Notes. Part I. Nonlinear Hyperbolic PDEs, Dispersive and Transport Equations, American Institute of Mathematical Sciences (2013), pp. 97-225 | MR 3340992 | Zbl 1284.65116
[7] Semigroup stability of finite difference schemes for multidimensional hyperbolic initial boundary value problems, Math. Comp., Tome 80 (2011) no. 273, pp. 165-203 | MR 2728976
[8] Time dependent problems and difference methods, John Wiley & Sons (1995) | MR 1377057 | Zbl 1275.65048
[9] Stability theory of difference approximations for mixed initial boundary value problems. II, Math. Comp., Tome 26 (1972) no. 119, pp. 649-686 | MR 341888 | Zbl 0293.65076
[10] Initial-boundary value problems for first order hyperbolic systems, Publ. Res. Inst. Math. Sci., Tome 7 (1971/72), pp. 181-204 | MR 348283 | Zbl 0246.35054
[11] Stability theory for difference approximations of mixed initial boundary value problems. I, Math. Comp., Tome 22 (1968), pp. 703-714 | MR 241010 | Zbl 0197.13704
[12] Initial boundary value problems for hyperbolic systems, Comm. Pure Appl. Math., Tome 23 (1970), pp. 277-298 | MR 437941 | Zbl 0193.06902
[13] On the stability definition of difference approximations for the initial-boundary value problem, Appl. Numer. Math., Tome 12 (1993) no. 1-3, pp. 213-227 | MR 1227187 | Zbl 0782.65119
[14] Boundary problems for wave equations with grazing and gliding rays, Unpublished notes
[15] On the well posedness of hyperbolic initial boundary value problems, Preprint (2014)
[16] Stability of difference approximations of dissipative type for mixed initial boundary value problems. I, Math. Comp., Tome 23 (1969), pp. 335-340 | MR 246530 | Zbl 0177.20403
[17] Systems of difference equations with general homogeneous boundary conditions, Trans. Amer. Math. Soc., Tome 137 (1969), pp. 177-201 | MR 237982 | Zbl 0174.41701
[18] is a continuable initial condition for Kreiss’ mixed problems, Comm. Pure Appl. Math., Tome 25 (1972), pp. 265-285 | MR 298232 | Zbl 0226.35056
[19] On hyperbolic mixed problems, Arch. Rational Mech. Anal., Tome 18 (1965), pp. 310-334 | MR 172002 | Zbl 0137.06506
[20] Hyperbolic and other symmetrizable systems in regions with corners and edges, Indiana Univ. Math. J., Tome 26 (1977) no. 1, pp. 1-39 | MR 442495 | Zbl 0366.35061
[21] On the construction and comparison of difference schemes, SIAM J. Numer. Anal., Tome 5 (1968), pp. 506-517 | MR 235754 | Zbl 0184.38503
[22] Complex symmetric matrices with strongly stable iterates, Linear Algebra Appl., Tome 78 (1986), pp. 65-77 | MR 840168 | Zbl 0591.15018
[23] Group velocity in finite difference schemes, SIAM Rev., Tome 24 (1982) no. 2, pp. 113-136 | MR 652463 | Zbl 0487.65055
[24] Instability of difference models for hyperbolic initial boundary value problems, Comm. Pure Appl. Math., Tome 37 (1984), pp. 329-367 | MR 739924 | Zbl 0575.65095
[25] Spectra and pseudospectra, Princeton University Press (2005) (The behavior of nonnormal matrices and operators) | MR 2155029 | Zbl 1085.15009
[26] The semigroup stability of the difference approximations for initial-boundary value problems, Math. Comp., Tome 64 (1995) no. 209, pp. 71-88 | MR 1257582 | Zbl 0820.65053