Numerical viscosity and the entropy condition for conservative difference schemes
Author:
Eitan Tadmor
Journal:
Math. Comp. 43 (1984), 369381
MSC:
Primary 65M05; Secondary 35L65
MathSciNet review:
758189
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Abstract: Consider a scalar, nonlinear conservative difference scheme satisfying the entropy condition. It is shown that difference schemes containing more numerical viscosity will necessarily converge to the unique, physically relevant weak solution of the approximated conservative equation. In particular, entropy satisfying convergence follows for E schemesthose containing more numerical viscosity than Godunov's scheme.
 [1]
Michael
G. Crandall and Andrew
Majda, Monotone difference approximations for
scalar conservation laws, Math. Comp.
34 (1980), no. 149, 1–21. MR 551288
(81b:65079), http://dx.doi.org/10.1090/S00255718198005512883
 [2]
K.
O. Friedrichs, Symmetric hyperbolic linear differential
equations, Comm. Pure Appl. Math. 7 (1954),
345–392. MR 0062932
(16,44c)
 [3]
James
Glimm, Solutions in the large for nonlinear hyperbolic systems of
equations, Comm. Pure Appl. Math. 18 (1965),
697–715. MR 0194770
(33 #2976)
 [4]
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
(22 #10194)
 [5]
Ami
Harten, High resolution schemes for hyperbolic conservation
laws, J. Comput. Phys. 49 (1983), no. 3,
357–393. MR
701178 (84g:65115), http://dx.doi.org/10.1016/00219991(83)901365
 [6]
Peter
D. Lax, Weak solutions of nonlinear hyperbolic equations and their
numerical computation, Comm. Pure Appl. Math. 7
(1954), 159–193. MR 0066040
(16,524g)
 [7]
Peter
Lax, Shock waves and entropy, Contributions to nonlinear
functional analysis (Proc. Sympos., Math. Res. Center, Univ. Wisconsin,
Madison, Wis., 1971) Academic Press, New York, 1971,
pp. 603–634. MR 0393870
(52 #14677)
 [8]
Andrew
Majda and Stanley
Osher, Numerical viscosity and the entropy condition, Comm.
Pure Appl. Math. 32 (1979), no. 6, 797–838. MR 539160
(80j:65031), http://dx.doi.org/10.1002/cpa.3160320605
 [9]
Richard
Sanders, On convergence of monotone finite
difference schemes with variable spatial differencing, Math. Comp. 40 (1983), no. 161, 91–106. MR 679435
(84a:65075), http://dx.doi.org/10.1090/S00255718198306794356
 [10]
Eitan
Tadmor, The largetime behavior of the scalar,
genuinely nonlinear LaxFriedrichs scheme, Math. Comp. 43 (1984), no. 168, 353–368. MR 758188
(86g:65162), http://dx.doi.org/10.1090/S00255718198407581888
 [11]
Vidar
Thomée, Stability theory for partial difference
operators, SIAM Rev. 11 (1969), 152–195. MR 0250505
(40 #3739)
 [12]
Stanley
Osher, Riemann solvers, the entropy condition, and difference
approximations, SIAM J. Numer. Anal. 21 (1984),
no. 2, 217–235. MR 736327
(86d:65119), http://dx.doi.org/10.1137/0721016
 [1]
 M. Crandall & A. Majda, "Monotone difference approximations for scalar conservation laws," Math. Comp., v. 34, 1980, pp. 121. MR 551288 (81b:65079)
 [2]
 K. O. Friedrichs, "Symmetric hyperbolic linear differential equations," Comm. Pure Appl. Math., v. 7, 1954, pp. 345392. MR 0062932 (16:44c)
 [3]
 J. Glimm, "Solution in the large for nonlinear hyperbolic systems of equations," Comm. Pure Appl. Math., v. 18, 1965, pp. 697715. MR 0194770 (33:2976)
 [4]
 S. K. Godunov, "A finite difference method for the numerical computation of discontinuous solutions of the equations of flow dynamics," Mat. Sb., v. 47, 1959, pp. 271290. MR 0119433 (22:10194)
 [5]
 A. Harten, "High resolution schemes for hyperbolic conservation laws," J. Comput. Phys., v. 49, 1983, pp. 357393. MR 701178 (84g:65115)
 [6]
 P. D. Lax, "Weak solutions of nonlinear hyperbolic equations and their numerical computation," Comm. Pure Appl. Math., v. 7, 1954, pp. 159193. MR 0066040 (16:524g)
 [7]
 P. D. Lax, "Shock waves and entropy," in Contributions to Nonlinear Functional Analysis (E. A. Zarantonello, ed.), Academic Press, New York, 1971, pp. 603634. MR 0393870 (52:14677)
 [8]
 A. Majda & S. Osher, "Numerical viscosity and the entropy condition," Comm. Pure Appl. Math., v. 32, 1979, pp. 797838. MR 539160 (80j:65031)
 [9]
 R. Sanders, "On convergence of monotone finite difference schemes with variable spatial efficiency," Math. Comp., v. 40, 1983, pp. 91106. MR 679435 (84a:65075)
 [10]
 E. Tadmor, "The largetime behavior of the scalar, genuinely nonlinear LaxFriedrichs scheme," Math. Comp., this issue. MR 758188 (86g:65162)
 [11]
 V. Thomée, "Stability theory for partial difference operators," SIAM Rev., v. 11, 1969, pp. 152195. MR 0250505 (40:3739)
 [12]
 S. Osher, "Riemann solvers, the entropy condition and difference approximations," SIAM J. Numer. Anal., v. 21, 1984, pp. 217235. MR 736327 (86d:65119)
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Additional Information
DOI:
http://dx.doi.org/10.1090/S0025571819840758189X
PII:
S 00255718(1984)0758189X
Article copyright:
© Copyright 1984
American Mathematical Society
