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.
Mots clés : linear stability, discrete shock profiles, Laplace transform
@article{M2AN_2003__37_1_1_0,
author = {Godillon, Pauline},
title = {Green's function pointwise estimates for the modified {Lax-Friedrichs} scheme},
journal = {ESAIM: Mod\'elisation math\'ematique et analyse num\'erique},
pages = {1--39},
publisher = {EDP-Sciences},
volume = {37},
number = {1},
year = {2003},
doi = {10.1051/m2an:2003022},
zbl = {1038.35036},
language = {en},
url = {http://www.numdam.org/articles/10.1051/m2an:2003022/}
}
TY - JOUR AU - Godillon, Pauline TI - Green's function pointwise estimates for the modified Lax-Friedrichs scheme JO - ESAIM: Modélisation mathématique et analyse numérique PY - 2003 SP - 1 EP - 39 VL - 37 IS - 1 PB - EDP-Sciences UR - http://www.numdam.org/articles/10.1051/m2an:2003022/ DO - 10.1051/m2an:2003022 LA - en ID - M2AN_2003__37_1_1_0 ER -
%0 Journal Article %A Godillon, Pauline %T Green's function pointwise estimates for the modified Lax-Friedrichs scheme %J ESAIM: Modélisation mathématique et analyse numérique %D 2003 %P 1-39 %V 37 %N 1 %I EDP-Sciences %U http://www.numdam.org/articles/10.1051/m2an:2003022/ %R 10.1051/m2an:2003022 %G en %F M2AN_2003__37_1_1_0
Godillon, Pauline. Green's function pointwise estimates for the modified Lax-Friedrichs scheme. ESAIM: Modélisation mathématique et analyse numérique, Tome 37 (2003) no. 1, pp. 1-39. doi : 10.1051/m2an:2003022. http://www.numdam.org/articles/10.1051/m2an:2003022/
[1] , Stability of semi-discrete shock profiles by means of an Evans function in infinite dimensions. J. Dynam. Differential Equations 14 (2002) 613-674. | Zbl
[2] , and, Alternate Evans functions and viscous shock waves. SIAM J. Math. Anal. 32 (2001) 929-962. | Zbl
[3] , and, Unstable Godunov discrete profiles for steady shock waves. SIAM J. Numer. Anal. 35 (1998) 2272-2297. | Zbl
[4] and, Numerical boundary layers for hyperbolic systems in 1-D. ESAIM: M2AN 35 (2001) 91-106. | Numdam | Zbl
[5] , Hyperbolic conservation laws in continuum physics. Springer (2000). | MR | Zbl
[6] and, The gap lemma and geometric criteria for instability of viscous shock profiles. Comm. Pure Appl. Math. 51 (1998) 797-855. | Zbl
[7] and, 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
[8] and, 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
[9] , Necessary condition of spectral stability for a stationary Lax-Wendroff shock profile. Preprint UMPA, ENS Lyon, 295 (2001).
[10] , Linear stability of shock profiles for systems of conservation laws with semi-linear relaxation. Phys. D 148 (2001) 289-316. | Zbl
[11] and, Boundary layers for viscous perturbations of non-characteristic quasilinear hyperbolic problems. J. Differential Equations (1998). | MR | Zbl
[12] and, Stability of one-dimensional boundary layers by using Green's functions. Comm. Pure Appl. Math. 54 (2001) 1343-1385. | Zbl
[13] , Discrete shocks. Comm. Pure Appl. Math. 27 (1974) 25-37. | Zbl
[14] , Stability of the travelling wave solution of the FitzHugh-Nagumo system. Trans. Amer. Math. Soc. 286 (1984) 431-469. | Zbl
[15] , Perturbation theory for linear operators. Springer-Verlag (1985). | Zbl
[16] , 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
[17] and, Overcompressive shock waves, in Nonlinear evolution equations that change type. Springer-Verlag, New York, IMA Vol. Math. Appl. 27 (1990) 139-145. | Zbl
[18] and, Continuum shock profiles for discrete conservation laws. I. Construction. Comm. Pure Appl. Math. 52 (1999) 85-127. | Zbl
[19] and, Continuum shock profiles for discrete conservation laws. II. Stability. Comm. Pure Appl. Math. 52 (1999) 1047-1073. | Zbl
[20] and, Discrete shock profiles for systems of conservation laws. Comm. Pure Appl. Math. 32 (1979) 445-482. | Zbl
[21] and, Pointwise green's function bounds and stability of relaxation shocks. Indiana Univ. Math. J. 51 (2002) 773-904. | Zbl
[22] , Discrete shocks for difference approximations to systems of conservation laws. Adv. in Appl. Math. 5 (1984) 433-469. | Zbl
[23] and, 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
[24] , 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
[25] , Discrete shock profiles and their stability, in Hyperbolic problems: theory, numerics, applications, Vol. II (Zürich, 1998), pp. 843-853. Birkhäuser, Basel (1999). | Zbl
[26] , 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 | Zbl
[27] and, Pointwise semigroup methods and stability of viscous shock waves. Indiana Univ. Math. J. 47 (1998) 741-871. | Zbl
[28] and, Viscous and inviscid stability of multidimensional planar shock fronts. Indiana Univ. Math. J. 48 (1999) 937-992. | Zbl
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