Upwind difference schemes for hyperbolic systems of conservation laws
Authors:
Stanley Osher and Fred Solomon
Journal:
Math. Comp. 38 (1982), 339374
MSC:
Primary 65M05
MathSciNet review:
645656
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Abstract: We derive a new upwind finite difference approximation to systems of nonlinear hyperbolic conservation laws. The scheme has desirable properties for shock calculations. Under fairly general hypotheses we prove that limit solutions satisfy the entropy condition and that discrete steady shocks exist which are unique and sharp. Numerical examples involving the Euler and Lagrange equations of compressible gas dynamics in one and two space dimensions are given.
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(83g:65098)
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Stanley
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difference schemes, SIAM J. Numer. Anal. 18 (1981),
no. 1, 129–144. MR 603435
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S. Osher, B. Engquist & M. W. Mann, "Upwind difference schemes for the potential equation of transonic flow." (In preparation.)
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P.
L. Roe, Approximate Riemann solvers, parameter vectors, and
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no. 2, 357–372. MR 640362
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 [1]
 B. Engquist & S. Osher, "Stable and entropy satisfying approximations for transonic flow calculations," Math. Comp., v. 34, 1980, pp. 4575. MR 551290 (81b:65082)
 [2]
 B. Engquist & S. Osher, "One sided difference schemes and transonic flow," Proc. Nat. Acad. Sci. U.S.A., v. 77, 1980, pp. 30713074. MR 574380 (83b:76052)
 [3]
 B. Engquist & S. Osher, "One sided difference approximations for nonlinear conservation laws," Math. Comp., v. 36, 1981, pp. 321352. MR 606500 (82c:65056)
 [4]
 B. Engquist & S. Osher, Upwind Difference Equations for Systems of Conservation LawsPotential Flow Equations, MRC Technical Report #2186, Univ. of Wisconsin, 1981.
 [5]
 K. O. Friedrichs & P. D. Lax, "Systems of conservation laws with a convex extension," Proc. Nat. Acad. Sci. U.S.A., v. 68, 1971, pp. 16861688. MR 0285799 (44:3016)
 [6]
 S. K. Godunov, "A finitedifference method for the numerical computation of discontinuous solutions of the equations of fluid dynamics," Mat. Sb., v. 47, 1959, pp. 271290. (Russian) MR 0119433 (22:10194)
 [7]
 P. M. Goorjian & R. van Buskirk, Implicit Calculations of Transonic Flow Using Monotone Methods, AIAA810331, 1981.
 [8]
 A. Harten, J. M. Hyman & P. D. Lax, "On finite difference approximations and entropy conditions for shocks," Comm. Pure Appl. Math., v. 29, 1976, pp. 297322. MR 0413526 (54:1640)
 [9]
 A. Harten, P. D. Lax & B. van Leer, "On upstream differencing and Godunov type schemes for hyperbolic conservation laws." (Preprint.) MR 693713 (85h:65188)
 [10]
 G. Jennings, "Discrete shocks," Comm. Pure Appl. Math., v. 27, 1974, pp. 2537. MR 0338594 (49:3358)
 [11]
 N. N. Kuznetsov, "On stable methods for solving nonlinear first order partial differential equations in the class of discontinuous functions," Topics in Numerical Analysis II (Proc. Roy. Irish Acad. Conf. on Numerical Analysis, 1976), Ed. J. J. H. Miller, Academic Press, New York, 1977, pp. 183197. MR 0657786 (58:31874)
 [12]
 P. D. Lax, Hyperbolic Systems of Conservation Laws and the Mathematical Theory of Shock Waves, SIAM Regional Conf. Series Lectures in Appl. Math. No. 11, SIAM, Philadelphia, Pa., 1973. MR 0350216 (50:2709)
 [13]
 P. D. Lax & B. Wendroff, "Systems of conservation laws," Comm. Pure Appl. Math., v. 13, 1960, pp. 217237. MR 0120774 (22:11523)
 [14]
 A. Majda & S. Osher, "Numerical viscosity and the entropy condition," Comm. Pure Appl. Math., v. 32, 1979, pp. 797838. MR 539160 (80j:65031)
 [15]
 A. Majda & J. Ralston, "Discrete shock profiles for systems of conservation laws," Comm. Pure Appl. Math., v. 22, 1979, pp. 445482. MR 528630 (81i:35108)
 [16]
 M. S. Mock, "Some higherorder difference schemes enforcing an entropy inequality," Michigan Math. J., v. 25, 1978, pp. 325344. MR 512903 (80a:65198)
 [17]
 E. M. Murman, "Analysis of embedded Shockwaves calculated by relaxation methods," AIAA J., v. 12, 1974, pp. 626633.
 [18]
 S. Osher, "Approximation par éléments finis avec décentrage pour des lois de conservation hyperboliques nonlinéares multidimensionelles," C.R. Acad. Sci. Paris, Sér. A, v. 290, 1980, pp. 819821. MR 580574 (81e:65059)
 [19]
 S. Osher, Numerical Solution of Singular Perturbation Problems and Hyperbolic Systems of Conservation Laws, Math. Studies No. 47 (0. Axelsson, L. S. Frank, A. van der Sluis, Eds.), NorthHolland, Amsterdam, 1981, pp. 179205. MR 605507 (83g:65098)
 [20]
 S. Osher, "Nonlinear singular perturbation problems and onesided difference schemes," SIAM J. Numer. Anal., v. 18, 1981, pp. 129144. MR 603435 (83c:65188)
 [21]
 S. Osher, B. Engquist & M. W. Mann, "Upwind difference schemes for the potential equation of transonic flow." (In preparation.)
 [22]
 P. L. Roe, "Approximate Riemann solvers, parameter vectors, and difference schemes," J. Comput. Phys. (To appear.) MR 640362 (82k:65055)
 [23]
 B. van Leer, Upwind Differencing for Hyperbolic Systems of Conservation Laws, ICASE Internal Report Document #12, 1980. MR 660670 (83g:65088)
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Additional Information
DOI:
http://dx.doi.org/10.1090/S00255718198206456560
PII:
S 00255718(1982)06456560
Keywords:
Finite difference approximation,
upwind schemes,
hyperbolic conservation laws
Article copyright:
© Copyright 1982
American Mathematical Society
