A fronttracking alternative to the random choice method
Author:
Nils Henrik Risebro
Journal:
Proc. Amer. Math. Soc. 117 (1993), 11251139
MSC:
Primary 35L65; Secondary 76L05, 76M99
MathSciNet review:
1120511
Fulltext PDF Free Access
Abstract 
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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.
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 J. Glimm, Solutions in the large for nonlinear hyperbolic systems of equations, Comm. Pure Appl. Math. 18 (1965), 697715. MR 0194770 (33:2976)
 [3]
 T. P. Liu, The deterministic version of the Glimm scheme, Comm. Math. Phys. 57 (1977), 135148. MR 0470508 (57:10259)
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 A. J. Chorin, Random choice solution of hyperbolic systems, J. Comp. Phys. 22 (1976), 517533. MR 0471342 (57:11077)
 [5]
 P. D. Lax, Shock waves and entropy, Contributions to Nonlinear Functional Analysis (E. H. Zarantonello, ed.), Academic Press, New York and London, 1971, pp. 603635. MR 0393870 (52:14677)
 [6]
 J. H. Bick and G. F. Newell, A continuum model for twodirectional traffic flow, Quart. J. Appl. Math. 18 (1961), 191204.
 [7]
 A. J. Chorin and J. E. Marsden, A mathematical introduction to fluid dynamics, Springer, New York, 1979. MR 551053 (81m:76001)
 [8]
 D. W. Peaceman, Fundamentals of numerical reservoir simulation, Elsevier, Amsterdam, 1977.
 [9]
 C. M. Dafermos, Polygonal approximation of solution to the initial value problem for a conservation law, J. Math. Anal. Appl. 38 (1972), 3341. MR 0303068 (46:2210)
 [10]
 R. J. LeVeque, A large time step shockcapturing technique for scalar conservation laws, SIAM J. Numer. Anal. 19 (1982), 10511073. MR 679654 (84e:65099)
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 H. Holden, L. Holden, and R. HøeghKrohn, A numerical method for first order nonlinear scalar conservation laws in one dimension, Comput. Math. Appl. 15 (1988), 595602. MR 953567 (90c:65112)
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 R. HøeghKrohn and N. H. Risebro, The Riemann problem for a single conservation law in two space dimensions, Oslo Univ. Preprint series, 1988.
 [13]
 B. K. Swartz and B. Wendroff, Aztec: A front tracking code based on Godunovs method, Appl. Numer. Math. 2 (1986), 385397. MR 863995
 [14]
 J. Smoller, Shock waves and reactiondiffusion equations, Springer, New York, 1983. MR 688146 (84d:35002)
 [15]
 T. P. Liu, Large time behavior of solutions of initial and initialboundary problems of a general system of hyperbolic conservation laws, Comm. Math. Phys. 55 (1977), 163177. MR 0447825 (56:6135)
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Additional Information
DOI:
http://dx.doi.org/10.1090/S0002993919931120511X
PII:
S 00029939(1993)1120511X
Article copyright:
© Copyright 1993 American Mathematical Society
