Remote Access Mathematics of Computation
Green Open Access

Mathematics of Computation

ISSN 1088-6842(online) ISSN 0025-5718(print)



Existence and asymptotic stability of relaxation discrete shock profiles

Author: Mao Ye
Journal: Math. Comp. 73 (2004), 1261-1296
MSC (2000): Primary 65M12; Secondary 35L65
Published electronically: March 3, 2004
MathSciNet review: 2047087
Full-text PDF

Abstract | References | Similar Articles | Additional Information

Abstract: In this paper we study the asymptotic nonlinear stability of discrete shocks of the relaxing scheme for approximating the general system of nonlinear hyperbolic conservation laws. The existence of discrete shocks is established by suitable manifold construction, and it is shown that weak single discrete shocks for such a scheme are nonlinearly stable in $L^2$, provided that the sums of the initial perturbations equal zero. These results should shed light on the convergence of the numerical solution constructed by the relaxing scheme for the single shock solution of the system of hyperbolic conservation laws. These results are proved by using both a weighted norm estimate and a characteristic energy method based on the internal structures of the discrete shocks.

References [Enhancements On Off] (What's this?)

  • 1. M. Bultelle, M. Grassin, D. Serre, Unstable Godunov discrete profiles for steady shock waves. SIAM J. Numer. Anal. 35 (1998), 2272-2292. MR 2000k:35189
  • 2. R. Diperna, Convergence of approximate solutions to conservation laws. Arch. Rational Mech. Anal. 82 (1983), 27-70. MR 84k:35091
  • 3. B. Enquist, S. Osher, One-sided difference approximations for nonlinear conservation laws. Math. Comp. 35 (1981), 321-351. MR 82c:65056
  • 4. H. T. Fan, Existence of discrete shock profiles of a class of monotonicity preserving schemes for conservation laws. Math. Comp. 70 (2001), 1043-1069. MR 2001m:65105
  • 5. J. Goodman, Nonlinear asymptotic stability of viscous shock profiles to conservation laws. Arch. Rational Mech. Anal. 95 (1986), 325-344. MR 88b:35127
  • 6. J. Goodman, Z. P. Xin, Viscous limits for piecewise smooth solutions to systems of conservation laws. Arch. Rational Mech. Anal. 121 (1992), 235-265. MR 93k:35167
  • 7. G. H. Hardy, J. E. Littlewood, G. Polya, Inequalities. Cambridge University Press, London, New York, 1943; 2nd edition, 1952. MR 13:727e
  • 8. G. S. Jiang, S. H. Yu, Discrete shocks for finite difference approximations to scalar conservation laws. SIAM J. Numer. Anal. 35 (1998), 749-772. MR 99a:65099
  • 9. S. Jin, Z. P. Xin, The relaxation schemes for system of conservation laws in arbititary space dimensions. Comm. Pure Appl. Math. 48 (1995), 235-276. MR 96c:65134
  • 10. G. Jennings, Discrete shocks. Comm. Pure Appl. Math. 27 (1974), 25-37. MR 49:3358
  • 11. P. D. Lax, Weak solutions of nonlinear hyperbolic equations and their numerical computation. Comm. Pure Appl. Math. 7 (1954), 159-193. MR 16:524g
  • 12. P. D. Lax, Hyperbolic Systems of Conservation Laws and the Mathematical Theory of Shock Waves. SIAM, Philadelphia, 1973. MR 50:2709
  • 13. H. L. Liu, J. H. Wang, T. Yang, Existence of the discrete travelling waves for a relaxing scheme. Appl. Math. Lett. 10 (1997), 117-122.
  • 14. H. L. Liu, Convergence rates to the discrete travelling wave for relaxation schemes. Math. Comp. 69 (2000), 583-608. MR 2000i:65132
  • 15. H. L. Liu, J. H. Wang, T. Yang, Nonlinear stability and existence of stationary discrete travelling waves for the relaxing schemes. Japan J. Indust. Appl. Math. 16 (1999), 195-224. MR 2000d:65157
  • 16. J. G. Liu, Z. P. Xin, Nonlinear stability of discrete shocks for system of conservation laws. Arch. Rational Mech. Anal. 123 (1993), 217-256. MR 95c:35166
  • 17. J. G. Liu, Z. P. Xin, $L^1$-stability of stationary discrete shocks. Math. Comp. 60 (1993), 233-244. MR 93d:35097
  • 18. T. P. Liu, Hyperbolic conservation laws with relaxation. Commun. Math. Phys. 108 (1987), 153-175. MR 88f:35092
  • 19. T. P. Liu, S. H. Yu, Continuum shock profiles for discrete conservation laws. I: Construction. Comm. Pure Appl. Math. 52 (1999), 85-127. MR 2000b:65154
  • 20. T. P. Liu, S. H. Yu, Continuum shock profiles for discrete conservation laws. II. Stability. Comm. Pure Appl. Math. 52 (1999), 1047-1073. MR 2000f:35095
  • 21. A. Majda, J. Ralston, Discrete shock profiles for system of conservation laws. Comm. Pure Appl. Math. 32 (1979), 445-482. MR 81i:35108
  • 22. D. Michelson, Discrete shocks for difference approximations to system of conservation laws. Adv. in Appl. Math. 4 (1984), 433-469. MR 86f:65159
  • 23. R. D. Richtmyer, K. W. Morton, Difference Methods for Initial Value Problems. 2nd ed., Wiley-Interscience, New York, 1967. MR 36:3515
  • 24. Y. S. Smyrlis, Existence and stability of stationary profiles of the LW scheme. Comm. Pure Appl. Math. 43 (1990), 509-545. MR 91d:65143
  • 25. A. Szepessy, Z. P. Xin, Nonlinear stability of viscous shock waves. Arch. Rational Mech. Anal. 122 (1993), 53-103. MR 93m:35125
  • 26. E. Tadmor, The large time behavior of the scalar genuinely nonlinear Lax-Friedrichs scheme. Math. Comp. 43 (1984), 353-368. MR 86g:65162

Similar Articles

Retrieve articles in Mathematics of Computation with MSC (2000): 65M12, 35L65

Retrieve articles in all journals with MSC (2000): 65M12, 35L65

Additional Information

Mao Ye
Affiliation: Institute of Mathematics Science and Department of Mathematics, Chinese University of Hong Kong, Hong Kong
Address at time of publication: School of Computer Science and Engineering, University of Electronic Science and Technology of China, Sichuan, China 610054

Keywords: Relaxing scheme, hyperbolic systems of conservation laws, discrete shock profiles, nonlinear stability
Received by editor(s): September 3, 2002
Published electronically: March 3, 2004
Additional Notes: This work was supported by the Youth Science and Technology Foundation, UESTC YF020801.
Article copyright: © Copyright 2004 American Mathematical Society

American Mathematical Society