Remote Access Proceedings of the American Mathematical Society
Green Open Access

Proceedings of the American Mathematical Society

ISSN 1088-6826(online) ISSN 0002-9939(print)



A front-tracking alternative to the random choice method

Author: Nils Henrik Risebro
Journal: Proc. Amer. Math. Soc. 117 (1993), 1125-1139
MSC: Primary 35L65; Secondary 76L05, 76M99
MathSciNet review: 1120511
Full-text PDF Free Access

Abstract | References | Similar Articles | Additional Information

Abstract: An alternative to Glimm's proof of the existence of solutions to systems of hyperbolic conservation laws is presented. The proof is based on an idea by Dafermos for the single conservation law and in some respects simplifies Glimm's original argument. The proof is based on construction of approximate solutions of which a subsequence converges. It is shown that the constructed solution satisfies Lax's entropy inequalities. The construction also gives a numerical method for solving such systems.

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

  • [1] P. D. Lax, Hyperbolic systems of conservation laws. II, Comm. Pure Appl. Math. 10 (1957), 537–566. MR 0093653
  • [2] James Glimm, Solutions in the large for nonlinear hyperbolic systems of equations, Comm. Pure Appl. Math. 18 (1965), 697–715. MR 0194770
  • [3] Tai Ping Liu, The deterministic version of the Glimm scheme, Comm. Math. Phys. 57 (1977), no. 2, 135–148. MR 0470508
  • [4] Alexandre Joel Chorin, Random choice solution of hyperbolic systems, J. Computational Phys. 22 (1976), no. 4, 517–533. MR 0471342
  • [5] Peter Lax, Shock waves and entropy, Contributions to nonlinear functional analysis (Proc. Sympos., Math. Res. Center, Univ. Wisconsin, Madison, Wis., 1971) Academic Press, New York, 1971, pp. 603–634. MR 0393870
  • [6] J. H. Bick and G. F. Newell, A continuum model for two-directional traffic flow, Quart. J. Appl. Math. 18 (1961), 191-204.
  • [7] A. J. Chorin and J. E. Marsden, A mathematical introduction to fluid mechanics, Springer-Verlag, New York-Heidelberg, 1979. MR 551053
  • [8] D. W. Peaceman, Fundamentals of numerical reservoir simulation, Elsevier, Amsterdam, 1977.
  • [9] Constantine M. Dafermos, Polygonal approximations of solutions of the initial value problem for a conservation law, J. Math. Anal. Appl. 38 (1972), 33–41. MR 0303068
  • [10] Randall J. LeVeque, Large time step shock-capturing techniques for scalar conservation laws, SIAM J. Numer. Anal. 19 (1982), no. 6, 1091–1109. MR 679654, 10.1137/0719080
  • [11] H. Holden, L. Holden, and R. Høegh-Krohn, A numerical method for first order nonlinear scalar conservation laws in one dimension, Comput. Math. Appl. 15 (1988), no. 6-8, 595–602. Hyperbolic partial differential equations. V. MR 953567, 10.1016/0898-1221(88)90282-9
  • [12] R. Høegh-Krohn and N. H. Risebro, The Riemann problem for a single conservation law in two space dimensions, Oslo Univ. Preprint series, 1988.
  • [13] Blair K. Swartz and Burton Wendroff, AZTEC: a front tracking code based on Godunov’s method, Appl. Numer. Math. 2 (1986), no. 3-5, 385–397. MR 863995, 10.1016/0168-9274(86)90041-3
  • [14] Joel Smoller, Shock waves and reaction-diffusion equations, Grundlehren der Mathematischen Wissenschaften [Fundamental Principles of Mathematical Science], vol. 258, Springer-Verlag, New York-Berlin, 1983. MR 688146
  • [15] Tai Ping Liu, Large-time behavior of solutions of initial and initial-boundary value problems of a general system of hyperbolic conservation laws, Comm. Math. Phys. 55 (1977), no. 2, 163–177. MR 0447825

Similar Articles

Retrieve articles in Proceedings of the American Mathematical Society with MSC: 35L65, 76L05, 76M99

Retrieve articles in all journals with MSC: 35L65, 76L05, 76M99

Additional Information

Article copyright: © Copyright 1993 American Mathematical Society