CONFLUENTES MATHEMATICI

Fully discrete hyperbolic initial boundary value problems with nonzero initial data
Confluentes Mathematici, Tome 7 (2015) no. 2, pp. 17-52.

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 ${\ell }^{2}$. 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].

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DOI : https://doi.org/10.5802/cml.22
Classification : 65M12,  65M06,  35L50
@article{CML_2015__7_2_17_0,
author = {Jean-Fran\c{c}ois Coulombel},
title = {Fully discrete hyperbolic initial boundary value problems with nonzero initial data},
journal = {Confluentes Mathematici},
pages = {17--52},
publisher = {Institut Camille Jordan},
volume = {7},
number = {2},
year = {2015},
doi = {10.5802/cml.22},
language = {en},
url = {https://cml.centre-mersenne.org/articles/10.5802/cml.22/}
}
Jean-François Coulombel. Fully discrete hyperbolic initial boundary value problems with nonzero initial data. Confluentes Mathematici, Tome 7 (2015) no. 2, pp. 17-52. doi : 10.5802/cml.22. https://cml.centre-mersenne.org/articles/10.5802/cml.22/

[1] C. Audiard On mixed initial-boundary value problems for systems that are not strictly hyperbolic, Appl. Math. Lett., Volume 24 (2011) no. 5, pp. 757-761

[2] S. Benzoni-Gavage; D. Serre Multidimensional hyperbolic partial differential equations, Oxford University Press, 2007 (First-order systems and applications)

[3] F. Carlson Quelques inégalités concernant les fonctions analytiques, Ark. Mat. Astr. Fys., Volume 29B (1943) no. 11, 6 pages

[4] J.-F. Coulombel Stability of finite difference schemes for hyperbolic initial boundary value problems, SIAM J. Numer. Anal., Volume 47 (2009) no. 4, pp. 2844-2871

[5] J.-F. Coulombel Stability of finite difference schemes for hyperbolic initial boundary value problems II, Ann. Sc. Norm. Super. Pisa Cl. Sci. (5), Volume X (2011) no. 1, pp. 37-98

[6] J.-F. Coulombel 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

[7] J.-F. Coulombel; A. Gloria Semigroup stability of finite difference schemes for multidimensional hyperbolic initial boundary value problems, Math. Comp., Volume 80 (2011) no. 273, pp. 165-203

[8] B. Gustafsson; H.-O. Kreiss; J. Oliger Time dependent problems and difference methods, John Wiley & Sons, 1995

[9] B. Gustafsson; H.-O. Kreiss; A. Sundström Stability theory of difference approximations for mixed initial boundary value problems. II, Math. Comp., Volume 26 (1972) no. 119, pp. 649-686

[10] K. Kajitani Initial-boundary value problems for first order hyperbolic systems, Publ. Res. Inst. Math. Sci., Volume 7 (1971/72), pp. 181-204

[11] H.-O. Kreiss Stability theory for difference approximations of mixed initial boundary value problems. I, Math. Comp., Volume 22 (1968), pp. 703-714

[12] H.-O. Kreiss Initial boundary value problems for hyperbolic systems, Comm. Pure Appl. Math., Volume 23 (1970), pp. 277-298

[13] H.-O. Kreiss; L. Wu On the stability definition of difference approximations for the initial-boundary value problem, Appl. Numer. Math., Volume 12 (1993) no. 1-3, pp. 213-227

[14] R. Melrose; M. Taylor Boundary problems for wave equations with grazing and gliding rays, Unpublished notes

[15] G. Métivier On the ${L}^{2}$ well posedness of hyperbolic initial boundary value problems, Preprint (2014)

[16] S. Osher Stability of difference approximations of dissipative type for mixed initial boundary value problems. I, Math. Comp., Volume 23 (1969), pp. 335-340

[17] S. Osher Systems of difference equations with general homogeneous boundary conditions, Trans. Amer. Math. Soc., Volume 137 (1969), pp. 177-201

[18] J. Rauch ${ℒ}^{2}$ is a continuable initial condition for Kreiss’ mixed problems, Comm. Pure Appl. Math., Volume 25 (1972), pp. 265-285

[19] L. Sarason On hyperbolic mixed problems, Arch. Rational Mech. Anal., Volume 18 (1965), pp. 310-334

[20] L. Sarason Hyperbolic and other symmetrizable systems in regions with corners and edges, Indiana Univ. Math. J., Volume 26 (1977) no. 1, pp. 1-39

[21] G. Strang On the construction and comparison of difference schemes, SIAM J. Numer. Anal., Volume 5 (1968), pp. 506-517

[22] E. Tadmor Complex symmetric matrices with strongly stable iterates, Linear Algebra Appl., Volume 78 (1986), pp. 65-77

[23] L. N. Trefethen Group velocity in finite difference schemes, SIAM Rev., Volume 24 (1982) no. 2, pp. 113-136

[24] L. N. Trefethen Instability of difference models for hyperbolic initial boundary value problems, Comm. Pure Appl. Math., Volume 37 (1984), pp. 329-367

[25] L. N. Trefethen; M. Embree Spectra and pseudospectra, Princeton University Press, 2005 (The behavior of nonnormal matrices and operators)

[26] L. Wu The semigroup stability of the difference approximations for initial-boundary value problems, Math. Comp., Volume 64 (1995) no. 209, pp. 71-88