Remote Access Mathematics of Computation
Green Open Access

Mathematics of Computation

ISSN 1088-6842(online) ISSN 0025-5718(print)



Numerical viscosity and the entropy condition for conservative difference schemes

Author: Eitan Tadmor
Journal: Math. Comp. 43 (1984), 369-381
MSC: Primary 65M05; Secondary 35L65
MathSciNet review: 758189
Full-text PDF

Abstract | References | Similar Articles | Additional Information

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 schemes--those containing more numerical viscosity than Godunov's scheme.

References [Enhancements On Off] (What's this?)

  • [1] M. Crandall & A. Majda, "Monotone difference approximations for scalar conservation laws," Math. Comp., v. 34, 1980, pp. 1-21. MR 551288 (81b:65079)
  • [2] K. O. Friedrichs, "Symmetric hyperbolic linear differential equations," Comm. Pure Appl. Math., v. 7, 1954, pp. 345-392. 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. 697-715. 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. 271-290. MR 0119433 (22:10194)
  • [5] A. Harten, "High resolution schemes for hyperbolic conservation laws," J. Comput. Phys., v. 49, 1983, pp. 357-393. 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. 159-193. 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. 603-634. MR 0393870 (52:14677)
  • [8] A. Majda & S. Osher, "Numerical viscosity and the entropy condition," Comm. Pure Appl. Math., v. 32, 1979, pp. 797-838. MR 539160 (80j:65031)
  • [9] R. Sanders, "On convergence of monotone finite difference schemes with variable spatial efficiency," Math. Comp., v. 40, 1983, pp. 91-106. MR 679435 (84a:65075)
  • [10] E. Tadmor, "The large-time behavior of the scalar, genuinely nonlinear Lax-Friedrichs scheme," Math. Comp., this issue. MR 758188 (86g:65162)
  • [11] V. Thomée, "Stability theory for partial difference operators," SIAM Rev., v. 11, 1969, pp. 152-195. MR 0250505 (40:3739)
  • [12] S. Osher, "Riemann solvers, the entropy condition and difference approximations," SIAM J. Numer. Anal., v. 21, 1984, pp. 217-235. MR 736327 (86d:65119)

Similar Articles

Retrieve articles in Mathematics of Computation with MSC: 65M05, 35L65

Retrieve articles in all journals with MSC: 65M05, 35L65

Additional Information

Article copyright: © Copyright 1984 American Mathematical Society

American Mathematical Society