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

Pauline Godillon

ESAIM: Mathematical Modelling and Numerical Analysis - Modélisation Mathématique et Analyse Numérique (2003)

  • Volume: 37, Issue: 1, page 1-39
  • ISSN: 0764-583X

Abstract

top
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.

How to cite

top

Godillon, Pauline. "Green’s function pointwise estimates for the modified Lax–Friedrichs scheme." ESAIM: Mathematical Modelling and Numerical Analysis - Modélisation Mathématique et Analyse Numérique 37.1 (2003): 1-39. <http://eudml.org/doc/245429>.

@article{Godillon2003,
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 [27] in the continuous viscous setting.},
author = {Godillon, Pauline},
journal = {ESAIM: Mathematical Modelling and Numerical Analysis - Modélisation Mathématique et Analyse Numérique},
keywords = {linear stability; discrete shock profiles; Laplace transform; shock profiles; boundary layer; Euler equations; isentropic fluids; concentration; cavitation},
language = {eng},
number = {1},
pages = {1-39},
publisher = {EDP-Sciences},
title = {Green’s function pointwise estimates for the modified Lax–Friedrichs scheme},
url = {http://eudml.org/doc/245429},
volume = {37},
year = {2003},
}

TY - JOUR
AU - Godillon, Pauline
TI - Green’s function pointwise estimates for the modified Lax–Friedrichs scheme
JO - ESAIM: Mathematical Modelling and Numerical Analysis - Modélisation Mathématique et Analyse Numérique
PY - 2003
PB - EDP-Sciences
VL - 37
IS - 1
SP - 1
EP - 39
AB - 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.
LA - eng
KW - linear stability; discrete shock profiles; Laplace transform; shock profiles; boundary layer; Euler equations; isentropic fluids; concentration; cavitation
UR - http://eudml.org/doc/245429
ER -

References

top
  1. [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. Zbl1001.35081
  2. [2] S. Benzoni-Gavage, D. Serre and K. Zumbrun, Alternate Evans functions and viscous shock waves. SIAM J. Math. Anal. 32 (2001) 929–962. Zbl0985.34075
  3. [3] M. Bultelle, M. Grassin and D. Serre, Unstable Godunov discrete profiles for steady shock waves. SIAM J. Numer. Anal. 35 (1998) 2272–2297. Zbl0929.76083
  4. [4] C. Chainais-Hillairet and E. Grenier, Numerical boundary layers for hyperbolic systems in 1-D. ESAIM: M2AN 35 (2001) 91–106. Zbl0980.65093
  5. [5] C. Dafermos, Hyperbolic conservation laws in continuum physics. Springer (2000). Zbl0940.35002MR1763936
  6. [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. Zbl0933.35136
  7. [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. Zbl0808.35075
  8. [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. Zbl0873.65087
  9. [9] P. Godillon, Necessary condition of spectral stability for a stationary Lax-Wendroff shock profile. Preprint UMPA, ENS Lyon, 295 (2001). 
  10. [10] P. Godillon, Linear stability of shock profiles for systems of conservation laws with semi-linear relaxation. Phys. D 148 (2001) 289–316. Zbl1076.76033
  11. [11] E. Grenier and O. Guès, Boundary layers for viscous perturbations of non-characteristic quasilinear hyperbolic problems. J. Differential Equations (1998). Zbl0896.35078MR1604888
  12. [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. Zbl1026.35015
  13. [13] G. Jennings, Discrete shocks. Comm. Pure Appl. Math. 27 (1974) 25–37. Zbl0304.65063
  14. [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. Zbl0567.35044
  15. [15] T. Kato, Perturbation theory for linear operators. Springer-Verlag (1985). Zbl0531.47014
  16. [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). Zbl0729.76637
  17. [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. Zbl0731.35063
  18. [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. Zbl0933.35137
  19. [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. Zbl0965.35095
  20. [20] A. Majda and J. Ralston, Discrete shock profiles for systems of conservation laws. Comm. Pure Appl. Math. 32 (1979) 445–482. Zbl0388.35047
  21. [21] C. Mascia and K. Zumbrun, Pointwise green’s function bounds and stability of relaxation shocks. Indiana Univ. Math. J. 51 (2002) 773–904. Zbl1036.35135
  22. [22] D. Michelson, Discrete shocks for difference approximations to systems of conservation laws. Adv. in Appl. Math. 5 (1984) 433–469. Zbl0575.65087
  23. [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). Zbl0752.35035
  24. [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). Zbl0864.35074
  25. [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). Zbl0928.35102
  26. [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. Zbl0930.35001MR1707279
  27. [27] K. Zumbrun and P. Howard, Pointwise semigroup methods and stability of viscous shock waves. Indiana Univ. Math. J. 47 (1998) 741–871. Zbl0928.35018
  28. [28] K. Zumbrun and D. Serre, Viscous and inviscid stability of multidimensional planar shock fronts. Indiana Univ. Math. J. 48 (1999) 937–992. Zbl0944.76027

NotesEmbed ?

top

You must be logged in to post comments.

To embed these notes on your page include the following JavaScript code on your page where you want the notes to appear.

Only the controls for the widget will be shown in your chosen language. Notes will be shown in their authored language.

Tells the widget how many notes to show per page. You can cycle through additional notes using the next and previous controls.

    
                

Note: Best practice suggests putting the JavaScript code just before the closing </body> tag.