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Green's function pointwise estimates for the modified Lax–Friedrichs scheme

Published online by Cambridge University Press:  15 March 2003

Pauline Godillon
Pauline Godillon*
Affiliation:
Unité de Mathématiques Pures et Appliquées, CNRS UMR # 5669, ENS Lyon, 46 allée d'Italie, 69364 Lyon Cedex 07, France. pgodillo@umpa.ens-lyon.fr.
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Abstract

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 [CITE] in the continuous viscous setting.

Keywords

Linear stabilitydiscrete shock profilesLaplace transform.
Type
Research Article
Information
ESAIM: Mathematical Modelling and Numerical Analysis , Volume 37 , Issue 1 , January 2003 , pp. 1 - 39
Copyright
© EDP Sciences, SMAI, 2003

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References

Benzoni-Gavage, S., Stability of semi-discrete shock profiles by means of an Evans function in infinite dimensions. J. Dynam. Differential Equations 14 (2002) 613-674. CrossRef
Benzoni-Gavage, S., Serre, D. and Zumbrun, K., Alternate Evans functions and viscous shock waves. SIAM J. Math. Anal. 32 (2001) 929-962. CrossRef
Bultelle, M., Grassin, M. and Serre, D., Unstable Godunov discrete profiles for steady shock waves. SIAM J. Numer. Anal. 35 (1998) 2272-2297. CrossRef
Chainais-Hillairet, C. and Grenier, E., Numerical boundary layers for hyperbolic systems in 1-D. ESAIM: M2AN 35 (2001) 91-106. CrossRef
C. Dafermos, Hyperbolic conservation laws in continuum physics. Springer (2000).
Gardner, R. A. and Zumbrun, K., The gap lemma and geometric criteria for instability of viscous shock profiles. Comm. Pure Appl. Math. 51 (1998) 797-855. 3.0.CO;2-1>CrossRef
Gisclon, M. and Serre, D., 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.
Gisclon, M. and Serre, D., Conditions aux limites pour un système strictement hyperbolique fournies par le schéma de Godunov. RAIRO Modél. Math. Anal. Numér. 31 (1997) 359-380. CrossRef
P. Godillon, Necessary condition of spectral stability for a stationary Lax-Wendroff shock profile. Preprint UMPA, ENS Lyon, 295 (2001).
Godillon, P., Linear stability of shock profiles for systems of conservation laws with semi-linear relaxation. Phys. D 148 (2001) 289-316. CrossRef
E. Grenier and O. Guès, Boundary layers for viscous perturbations of non-characteristic quasilinear hyperbolic problems. J. Differential Equations (1998).
Grenier, E. and Rousset, F., Stability of one-dimensional boundary layers by using Green's functions. Comm. Pure Appl. Math. 54 (2001) 1343-1385. CrossRef
Jennings, G., Discrete shocks. Comm. Pure Appl. Math. 27 (1974) 25-37. CrossRef
Jones, C.K.R.T., Stability of the travelling wave solution of the FitzHugh-Nagumo system. Trans. Amer. Math. Soc. 286 (1984) 431-469. CrossRef
T. Kato, Perturbation theory for linear operators. Springer-Verlag (1985).
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).
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.
Liu, T.-P. and Continuum, S.-H. Yu shock profiles for discrete conservation laws. I. Construction. Comm. Pure Appl. Math. 52 (1999) 85-127. 3.0.CO;2-U>CrossRef
Liu, T.-P. and Continuum, S.-H. Yu shock profiles for discrete conservation laws. II. Stability. Comm. Pure Appl. Math. 52 (1999) 1047-1073. 3.0.CO;2-4>CrossRef
Majda, A. and Ralston, J., Discrete shock profiles for systems of conservation laws. Comm. Pure Appl. Math. 32 (1979) 445-482. CrossRef
Mascia, C. and Zumbrun, K., Pointwise green's function bounds and stability of relaxation shocks. Indiana Univ. Math. J. 51 (2002) 773-904. CrossRef
Michelson, D., Discrete shocks for difference approximations to systems of conservation laws. Adv. in Appl. Math. 5 (1984) 433-469. CrossRef
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).
Serre, D., 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).
D. Serre, Discrete shock profiles and their stability, in Hyperbolic problems: theory, numerics, applications, Vol. II (Zürich, 1998), pp. 843-853. Birkhäuser, Basel (1999).
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.
Zumbrun, K. and Howard, P., Pointwise semigroup methods and stability of viscous shock waves. Indiana Univ. Math. J. 47 (1998) 741-871. CrossRef
Zumbrun, K. and Serre, D., Viscous and inviscid stability of multidimensional planar shock fronts. Indiana Univ. Math. J. 48 (1999) 937-992. CrossRef