The numerical viscosity of entropy stable schemes for systems of conservation laws. I

Author:
Eitan Tadmor

Journal:
Math. Comp. **49** (1987), 91-103

MSC:
Primary 65M10; Secondary 35L65

DOI:
https://doi.org/10.1090/S0025-5718-1987-0890255-3

MathSciNet review:
890255

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Abstract: Discrete approximations to hyperbolic systems of conservation laws are studied. We quantify the amount of numerical viscosity present in such schemes, and relate it to their entropy stability by means of *comparison*. To this end, conservative schemes which are also entropy conservative are constructed. These *entropy conservative* schemes enjoy second-order accuracy; moreover, they can be interpreted as piecewise linear finite element methods, and hence can be formulated on various mesh configurations. We then show that conservative schemes are entropy stable, if and--for three-point schemes--only if they contain *more* viscosity than that present in the above-mentioned entropy conservative ones.

**[1]**R. J. DiPerna,*Convergence of approximate solutions to conservation laws*, Arch. Rational Mech. Anal.**82**(1983), no. 1, 27–70. MR**684413**, https://doi.org/10.1007/BF00251724**[2]**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****[3]**Ami Harten,*High resolution schemes for hyperbolic conservation laws*, J. Comput. Phys.**49**(1983), no. 3, 357–393. MR**701178**, https://doi.org/10.1016/0021-9991(83)90136-5**[4]**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**, https://doi.org/10.1002/cpa.3160290305**[5]**Amiram Harten and Peter D. Lax,*A random choice finite difference scheme for hyperbolic conservation laws*, SIAM J. Numer. Anal.**18**(1981), no. 2, 289–315. MR**612144**, https://doi.org/10.1137/0718021**[6]**A. Harten & S. Osher,*Uniformly High-Order Accurate Non-Oscillatory Schemes*, I, MRC Technical Summary Report No. 2823, May 1985.**[7]**T. J. R. Hughes, M. Mallet & L. P. Franca, "Entropy-stable finite element methods for compressible fluids; application to high mach number flows with shocks,"*Finite Element Methods for Nonlinear Problems*, Springer-Verlag. (To appear.)**[8]**S. N. Kružkov, "First-order quasilinear equations in several independent variables,"*Math. USSR-Sb.*, v. 10, 1970, pp. 217-243.**[9]**P. D. Lax,*Hyperbolic systems of conservation laws. II*, Comm. Pure Appl. Math.**10**(1957), 537–566. MR**0093653**, https://doi.org/10.1002/cpa.3160100406**[10]**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****[11]**M. S. Mock,*A Difference Scheme Employing Fourth-Order Viscosity to Enforce an Entropy Inequality*, Proc. Bat-Sheva Conference, Tel-Aviv University, 1977.**[12]**M. S. Mock,*Systems of conservation laws of mixed type*, J. Differential Equations**37**(1980), no. 1, 70–88. MR**583340**, https://doi.org/10.1016/0022-0396(80)90089-3**[13]**J. Von Neumann and R. D. Richtmyer,*A method for the numerical calculation of hydrodynamic shocks*, J. Appl. Phys.**21**(1950), 232–237. MR**0037613****[14]**Stanley Osher,*Riemann solvers, the entropy condition, and difference approximations*, SIAM J. Numer. Anal.**21**(1984), no. 2, 217–235. MR**736327**, https://doi.org/10.1137/0721016**[15]**Stanley Osher,*Convergence of generalized MUSCL schemes*, SIAM J. Numer. Anal.**22**(1985), no. 5, 947–961. MR**799122**, https://doi.org/10.1137/0722057**[16]**Stanley Osher and Sukumar Chakravarthy,*High resolution schemes and the entropy condition*, SIAM J. Numer. Anal.**21**(1984), no. 5, 955–984. MR**760626**, https://doi.org/10.1137/0721060**[17]**S. Osher & E. Tadmor,*The Convergence of Difference Approximations to Scalar Conservation Laws*, ICASE Report No. 85-28, NASA CR-172614.**[18]**Eitan Tadmor,*Skew-selfadjoint form for systems of conservation laws*, J. Math. Anal. Appl.**103**(1984), no. 2, 428–442. MR**762567**, https://doi.org/10.1016/0022-247X(84)90139-2**[19]**Eitan Tadmor,*The large-time behavior of the scalar, genuinely nonlinear Lax-Friedrichs scheme*, Math. Comp.**43**(1984), no. 168, 353–368. MR**758188**, https://doi.org/10.1090/S0025-5718-1984-0758188-8**[20]**Eitan Tadmor,*Numerical viscosity and the entropy condition for conservative difference schemes*, Math. Comp.**43**(1984), no. 168, 369–381. MR**758189**, https://doi.org/10.1090/S0025-5718-1984-0758189-X**[21]**Eitan Tadmor,*Entropy functions for symmetry systems of conservation laws*, J. Math. Anal. Appl.**122**(1987), no. 2, 355–359. MR**877818**, https://doi.org/10.1016/0022-247X(87)90265-4

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DOI:
https://doi.org/10.1090/S0025-5718-1987-0890255-3

Article copyright:
© Copyright 1987
American Mathematical Society