Green's function pointwise estimates for the modified Lax-Friedrichs scheme
Godillon, Pauline
ESAIM: Mathematical Modelling and Numerical Analysis - Modélisation Mathématique et Analyse Numérique, Tome 37 (2003), p. 1-39 / Harvested from Numdam

The aim of this paper is to find estimates of the Green's function of stationary discrete shock profiles and discrete boundary layers of the modified Lax-Friedrichs numerical scheme, by using techniques developed by Zumbrun and Howard [27] in the continuous viscous setting.

Publié le : 2003-01-01
DOI : https://doi.org/10.1051/m2an:2003022
Classification:  35L65
@article{M2AN_2003__37_1_1_0,
     author = {Godillon, Pauline},
     title = {Green's function pointwise estimates for the modified Lax-Friedrichs scheme},
     journal = {ESAIM: Mathematical Modelling and Numerical Analysis - Mod\'elisation Math\'ematique et Analyse Num\'erique},
     volume = {37},
     year = {2003},
     pages = {1-39},
     doi = {10.1051/m2an:2003022},
     zbl = {1038.35036},
     language = {en},
     url = {http://dml.mathdoc.fr/item/M2AN_2003__37_1_1_0}
}
Godillon, Pauline. Green's function pointwise estimates for the modified Lax-Friedrichs scheme. ESAIM: Mathematical Modelling and Numerical Analysis - Modélisation Mathématique et Analyse Numérique, Tome 37 (2003) pp. 1-39. doi : 10.1051/m2an:2003022. http://gdmltest.u-ga.fr/item/M2AN_2003__37_1_1_0/

[1] S. Benzoni-Gavage, Stability of semi-discrete shock profiles by means of an Evans function in infinite dimensions. J. Dynam. Differential Equations 14 (2002) 613-674. | Zbl 1001.35081

[2] S. Benzoni-Gavage, D. Serre and K. Zumbrun, Alternate Evans functions and viscous shock waves. SIAM J. Math. Anal. 32 (2001) 929-962. | Zbl 0985.34075

[3] M. Bultelle, M. Grassin and D. Serre, Unstable Godunov discrete profiles for steady shock waves. SIAM J. Numer. Anal. 35 (1998) 2272-2297. | Zbl 0929.76083

[4] C. Chainais-Hillairet and E. Grenier, Numerical boundary layers for hyperbolic systems in 1-D. ESAIM: M2AN 35 (2001) 91-106. | Numdam | Zbl 0980.65093

[5] C. Dafermos, Hyperbolic conservation laws in continuum physics. Springer (2000). | MR 1763936 | Zbl 0940.35002

[6] R. A. Gardner and K. Zumbrun, The gap lemma and geometric criteria for instability of viscous shock profiles. Comm. Pure Appl. Math. 51 (1998) 797-855. | Zbl 0933.35136

[7] 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 0808.35075

[8] 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 0873.65087

[9] P. Godillon, Necessary condition of spectral stability for a stationary Lax-Wendroff shock profile. Preprint UMPA, ENS Lyon, 295 (2001).

[10] P. Godillon, Linear stability of shock profiles for systems of conservation laws with semi-linear relaxation. Phys. D 148 (2001) 289-316. | Zbl 1076.76033

[11] E. Grenier and O. Guès, Boundary layers for viscous perturbations of non-characteristic quasilinear hyperbolic problems. J. Differential Equations (1998). | MR 1604888 | Zbl 0896.35078

[12] E. Grenier and F. Rousset, Stability of one-dimensional boundary layers by using Green's functions. Comm. Pure Appl. Math. 54 (2001) 1343-1385. | Zbl 1026.35015

[13] G. Jennings, Discrete shocks. Comm. Pure Appl. Math. 27 (1974) 25-37. | Zbl 0304.65063

[14] C.K.R.T. Jones, Stability of the travelling wave solution of the FitzHugh-Nagumo system. Trans. Amer. Math. Soc. 286 (1984) 431-469. | Zbl 0567.35044

[15] T. Kato, Perturbation theory for linear operators. Springer-Verlag (1985). | Zbl 0531.47014

[16] T.-P. Liu, On the viscosity criterion for hyperbolic conservation laws, in Viscous profiles and numerical methods for shock waves (Raleigh, NC, 1990), pp. 105-114. SIAM, Philadelphia, PA (1991). | Zbl 0729.76637

[17] T.-P. Liu and Z. Xin, Overcompressive shock waves, in Nonlinear evolution equations that change type. Springer-Verlag, New York, IMA Vol. Math. Appl. 27 (1990) 139-145. | Zbl 0731.35063

[18] T.-P. Liu and S.-H. Yu, Continuum shock profiles for discrete conservation laws. I. Construction. Comm. Pure Appl. Math. 52 (1999) 85-127. | Zbl 0933.35137

[19] T.-P. Liu and S.-H. Yu, Continuum shock profiles for discrete conservation laws. II. Stability. Comm. Pure Appl. Math. 52 (1999) 1047-1073. | Zbl 0965.35095

[20] A. Majda and J. Ralston, Discrete shock profiles for systems of conservation laws. Comm. Pure Appl. Math. 32 (1979) 445-482. | Zbl 0388.35047

[21] C. Mascia and K. Zumbrun, Pointwise green's function bounds and stability of relaxation shocks. Indiana Univ. Math. J. 51 (2002) 773-904. | Zbl 1036.35135

[22] D. Michelson, Discrete shocks for difference approximations to systems of conservation laws. Adv. in Appl. Math. 5 (1984) 433-469. | Zbl 0575.65087

[23] S. Schecter and M. Shearer, Transversality for undercompressive shocks in Riemann problems, in Viscous profiles and numerical methods for shock waves (Raleigh, NC, 1990), pp. 142-154. SIAM, Philadelphia, PA (1991). | Zbl 0752.35035

[24] D. Serre, Remarks about the discrete profiles of shock waves. Mat. Contemp. 11 (1996) 153-170. Fourth Workshop on Partial Differential Equations, Part II (Rio de Janeiro, 1995). | Zbl 0864.35074

[25] D. Serre, Discrete shock profiles and their stability, in Hyperbolic problems: theory, numerics, applications, Vol. II (Zürich, 1998), pp. 843-853. Birkhäuser, Basel (1999). | Zbl 0928.35102

[26] D. Serre, Systems of conservation laws. 1. Cambridge University Press, Cambridge (1999). Hyperbolicity, entropies, shock waves. Translated from the 1996 French original by I.N. Sneddon. | MR 1707279 | Zbl 0930.35001

[27] K. Zumbrun and P. Howard, Pointwise semigroup methods and stability of viscous shock waves. Indiana Univ. Math. J. 47 (1998) 741-871. | Zbl 0928.35018

[28] K. Zumbrun and D. Serre, Viscous and inviscid stability of multidimensional planar shock fronts. Indiana Univ. Math. J. 48 (1999) 937-992. | Zbl 0944.76027