The numerical viscosity of entropy stable schemes for systems of conservation laws. I
Author:
Eitan Tadmor
Journal:
Math. Comp. 49 (1987), 91103
MSC:
Primary 65M10; Secondary 35L65
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 secondorder 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 andfor threepoint schemesonly if they contain more viscosity than that present in the abovementioned entropy conservative ones.
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355–359. MR
877818 (88d:35039), http://dx.doi.org/10.1016/0022247X(87)902654
 [1]
 R. J. DiPerna, "Convergence of approximate solutions to conservation laws," Arch. Rational Mech. Anal., v. 82, 1983, pp. 2770. MR 684413 (84k:35091)
 [2]
 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)
 [3]
 A. Harten, "High resolution schemes for hyperbolic conservation laws," J. Comput. Phys., v. 49, 1983, pp. 357393. MR 701178 (84g:65115)
 [4]
 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)
 [5]
 A. Harten & P. D. Lax, "A random choice finite difference scheme for hyperbolic conservation laws," SIAM J. Numer. Anal., v. 18, 1981, pp. 289315. MR 612144 (83b:65090)
 [6]
 A. Harten & S. Osher, Uniformly HighOrder Accurate NonOscillatory Schemes, I, MRC Technical Summary Report No. 2823, May 1985.
 [7]
 T. J. R. Hughes, M. Mallet & L. P. Franca, "Entropystable finite element methods for compressible fluids; application to high mach number flows with shocks," Finite Element Methods for Nonlinear Problems, SpringerVerlag. (To appear.)
 [8]
 S. N. Kružkov, "Firstorder quasilinear equations in several independent variables," Math. USSRSb., v. 10, 1970, pp. 217243.
 [9]
 P. D. Lax, "Hyperbolic systems of conservation laws, II," Comm. Pure Appl. Math., v. 10, 1957, pp. 537566. MR 0093653 (20:176)
 [10]
 P. D. Lax, Hyperbolic Systems of Conservation Laws and the Mathematical Theory of Shock Waves, SIAM Regional Conference Lecturers in Applied Mathematics, No. 11, 1972. MR 0350216 (50:2709)
 [11]
 M. S. Mock, A Difference Scheme Employing FourthOrder Viscosity to Enforce an Entropy Inequality, Proc. BatSheva Conference, TelAviv University, 1977.
 [12]
 M. S. Mock, "Systems of conservation laws of mixed type," J. Differential Equations, v. 37, 1980, pp. 7088. MR 583340 (81m:35088)
 [13]
 J. von Neumann & R. D. Richtmyer, "A method for the numerical calculation of hydrodynamic shocks," J. Appl. Phys., v. 21, 1950, pp. 232237. MR 0037613 (12:289b)
 [14]
 S. Osher, "Riemann solvers, the entropy condition, and difference approximations," SIAM J. Numer. Anal., v. 21, 1984, pp. 217235. MR 736327 (86d:65119)
 [15]
 S. Osher, "Convergence of generalized MUSCL schemes," SIAM J. Numer. Anal., v. 22, 1984, pp. 947961. MR 799122 (87b:65147)
 [16]
 S. Osher & S. Chakravarthy, "High resolution schemes and the entropy condition," SIAM J. Numer. Anal., v. 21, 1984, pp. 955984. MR 760626 (86a:65086)
 [17]
 S. Osher & E. Tadmor, The Convergence of Difference Approximations to Scalar Conservation Laws, ICASE Report No. 8528, NASA CR172614.
 [18]
 E. Tadmor, "Skewselfadjoint form for systems of cońservation laws," J. Math. Anal. Appl., v. 103, 1984, pp. 428442. MR 762567 (86c:35100)
 [19]
 E. Tadmor, "The largetime behavior of the scalar, genuinely nonlinear LaxFriedrichs scheme," Math. Comp., v. 43, 1984, pp. 353368. MR 758188 (86g:65162)
 [20]
 E. Tadmor, "Numerical viscosity and the entropy condition for conservative difference schemes," Math. Comp., v. 43, 1984, pp. 369381. MR 758189 (86g:65163)
 [21]
 E. Tadmor, "Entropy functions for symmetric systems of conservation laws," J. Math. Anal. Appl., v. 121, 1987. MR 877818 (88d:35039)
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Additional Information
DOI:
http://dx.doi.org/10.1090/S00255718198708902553
PII:
S 00255718(1987)08902553
Article copyright:
© Copyright 1987
American Mathematical Society
