Upwind difference schemes for hyperbolic systems of conservation laws

Authors:
Stanley Osher and Fred Solomon

Journal:
Math. Comp. **38** (1982), 339-374

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.

**[1]**Björn Engquist and Stanley Osher,*Stable and entropy satisfying approximations for transonic flow calculations*, Math. Comp.**34**(1980), no. 149, 45–75. MR**551290**, 10.1090/S0025-5718-1980-0551290-1**[2]**Björn Engquist and Stanley Osher,*One-sided difference schemes and transonic flow*, Proc. Nat. Acad. Sci. U.S.A.**77**(1980), no. 6, 3071–3074. MR**574380****[3]**Björn Engquist and Stanley Osher,*One-sided difference approximations for nonlinear conservation laws*, Math. Comp.**36**(1981), no. 154, 321–351. MR**606500**, 10.1090/S0025-5718-1981-0606500-X**[4]**B. Engquist & S. Osher,*Upwind Difference Equations for Systems of Conservation Laws-Potential Flow Equations*, MRC Technical Report #2186, Univ. of Wisconsin, 1981.**[5]**K. O. Friedrichs and P. D. Lax,*Systems of conservation equations with a convex extension*, Proc. Nat. Acad. Sci. U.S.A.**68**(1971), 1686–1688. MR**0285799****[6]**S. K. Godunov,*A difference method for numerical calculation of discontinuous solutions of the equations of hydrodynamics*, Mat. Sb. (N.S.)**47 (89)**(1959), 271–306 (Russian). MR**0119433****[7]**P. M. Goorjian & R. van Buskirk,*Implicit Calculations of Transonic Flow Using Monotone Methods*, AIAA-81-0331, 1981.**[8]**A. Harten, J. M. Hyman, and P. D. Lax,*On finite-difference approximations and entropy conditions for shocks*, Comm. Pure Appl. Math.**29**(1976), no. 3, 297–322. With an appendix by B. Keyfitz. MR**0413526****[9]**Amiram Harten, Peter D. Lax, and Bram van Leer,*On upstream differencing and Godunov-type schemes for hyperbolic conservation laws*, SIAM Rev.**25**(1983), no. 1, 35–61. MR**693713**, 10.1137/1025002**[10]**Gray Jennings,*Discrete shocks*, Comm. Pure Appl. Math.**27**(1974), 25–37. MR**0338594****[11]**N. N. Kuznetsov,*On stable methods for solving non-linear first order partial differential equations in the class of discontinuous functions*, Topics in numerical analysis, III (Proc. Roy. Irish Acad. Conf., Trinity Coll., Dublin, 1976) Academic Press, London, 1977, pp. 183–197. MR**0657786****[12]**Peter D. Lax,*Hyperbolic systems of conservation laws and the mathematical theory of shock waves*, Society for Industrial and Applied Mathematics, Philadelphia, Pa., 1973. Conference Board of the Mathematical Sciences Regional Conference Series in Applied Mathematics, No. 11. MR**0350216****[13]**Peter Lax and Burton Wendroff,*Systems of conservation laws*, Comm. Pure Appl. Math.**13**(1960), 217–237. MR**0120774****[14]**Andrew Majda and Stanley Osher,*Numerical viscosity and the entropy condition*, Comm. Pure Appl. Math.**32**(1979), no. 6, 797–838. MR**539160**, 10.1002/cpa.3160320605**[15]**Andrew Majda and James Ralston,*Discrete shock profiles for systems of conservation laws*, Comm. Pure Appl. Math.**32**(1979), no. 4, 445–482. MR**528630**, 10.1002/cpa.3160320402**[16]**M. S. Mock,*Some higher order difference schemes enforcing an entropy inequality*, Michigan Math. J.**25**(1978), no. 3, 325–344. MR**512903****[17]**E. M. Murman, "Analysis of embedded Shockwaves calculated by relaxation methods,"*AIAA J.*, v. 12, 1974, pp. 626-633.**[18]**Stanley Osher,*Approximation par éléments finis avec décentrage pour des lois de conservation hyperboliques non linéaires multi-dimensionnelles*, C. R. Acad. Sci. Paris Sér. A-B**290**(1980), no. 17, A819–A821 (French, with English summary). MR**580574****[19]**Stanley Osher,*Numerical solution of singular perturbation problems and hyperbolic systems of conservation laws*, Analytical and numerical approaches to asymptotic problems in analysis (Proc. Conf., Univ. Nijmegen, Nijmegen, 1980) North-Holland Math. Stud., vol. 47, North-Holland, Amsterdam-New York, 1981, pp. 179–204. MR**605507****[20]**Stanley Osher,*Nonlinear singular perturbation problems and one-sided difference schemes*, SIAM J. Numer. Anal.**18**(1981), no. 1, 129–144. MR**603435**, 10.1137/0718010**[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.**43**(1981), no. 2, 357–372. MR**640362**, 10.1016/0021-9991(81)90128-5**[23]**B. van Leer,*Upwind differencing for hyperbolic systems of conservation laws*, Numerical methods for engineering, 1 (Paris, 1980) Dunod, Paris, 1980, pp. 137–149. MR**660670**

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DOI:
https://doi.org/10.1090/S0025-5718-1982-0645656-0

Keywords:
Finite difference approximation,
upwind schemes,
hyperbolic conservation laws

Article copyright:
© Copyright 1982
American Mathematical Society