Numerical boundary layers for hyperbolic systems in 1-D
ESAIM: Modélisation mathématique et analyse numérique, Tome 35 (2001) no. 1, pp. 91-106.

The aim of this paper is to investigate the stability of boundary layers which appear in numerical solutions of hyperbolic systems of conservation laws in one space dimension on regular meshes. We prove stability under a size condition for Lax Friedrichs type schemes and inconditionnal stability in the scalar case. Examples of unstable boundary layers are also given.

Classification : 65M, 35L
Mots-clés : boundary layers stability 
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Chainais-Hillairet, Claire; Grenier, Emmanuel. Numerical boundary layers for hyperbolic systems in 1-D. ESAIM: Modélisation mathématique et analyse numérique, Tome 35 (2001) no. 1, pp. 91-106. http://archive.numdam.org/item/M2AN_2001__35_1_91_0/

[1] C. Bardos, A.-Y. Leroux and J.-C. Nédélec, First order quasilinear equations with boundary conditions. Partial Differential Equations 4 (1979) 1017-1034. | Zbl

[2] Y. Coudière, J.-P. Vila and P. Villedieu, Convergence of a finite-volume time-explicit scheme for symmetric linear hyperbolic systems on bounded domains. C. R. Acad. Sci. Paris, Sér. I Math. 331 (2000) 95-100. | Zbl

[3] F. Dubois and P. Lefloch, Boundary conditions for nonlinear hyperbolic systems of conservation laws. J. Differential Equations 71 (1988) 93-122. | Zbl

[4] M. Gisclon, Étude des conditions aux limites pour un système strictement hyperbolique, via l'approximation parabolique. J. Math. Pures Appl. 75 (1996) 485-508. | Zbl

[5] M. Gisclon and D. Serre, Étude des conditions aux limites pour un système strictement hyberbolique via l'approximation parabolique. C. R. Acad. Sci. Paris Sér. I Math. 319 (1994) 377-382. | Zbl

[6] M. Gisclon and D. Serre, Conditions aux limites pour un système strictement hyperbolique fournies par le schéma de Godunov. RAIRO-Modél. Math. Anal. Numér. 31 (1997) 359-380. | Numdam | Zbl

[7] J. Goodman, Nonlinear asymptotic stability of viscous shock profiles for conservation laws. Arch. Rational Mech. Anal. 95 (1986) 325-344. | Zbl

[8] E. Grenier and O. Guès, Boundary layers for viscous perturbations of noncharacteristic quasilinear hyperbolic problems. J. Differential Equations 143 (1998) 110-146. | Zbl

[9] K.T. Joseph and P.G. Lefloch, Boundary layers in weak solutions of hyperbolic conservation laws. Arch. Ration. Mech. Anal. 147 (1999) 47-88. | Zbl

[10] T.T. Li and W.C. Yu, Boundary value problems for quasilinear hyperbolic systems. Math. series V. Duke Univ., Durham (1985). | Zbl

[11] T.P. Liu, Nonlinear stability of shock waves for viscous conservation laws. Mem. Amer. Math. Soc. 56 (1985) 108 p. | MR | Zbl

[12] J.B. Rauch and F.J. Massey, Iii, Differentiability of solutions to hyperbolic initial boundary value problems. Trans. Amer. Math. Soc. 189 (1974) 303-318. | Zbl

[13] D. Serre, Sur la stabilité des couches limites de viscosité, preprint. | Numdam | MR

[14] M. Shub, A. Fathi and R. Langevin, Global stability of dynamical systems. Springer-Verlag, New-York, Berlin, 1987. | MR