On the convergence of difference approximations to scalar conservation laws
Authors:
Stanley Osher and Eitan Tadmor
Journal:
Math. Comp. 50 (1988), 1951
MSC:
Primary 65M10; Secondary 35L65, 65M05
MathSciNet review:
917817
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Abstract: We present a unified treatment of explicit in time, twolevel, secondorder resolution (SOR), totalvariation diminishing (TVD), approximations to scalar conservation laws. The schemes are assumed only to have conservation form and incremental form. We introduce a modified flux and a viscosity coefficient and obtain results in terms of the latter. The existence of a cell entropy inequality is discussed and such an equality for all entropies is shown to imply that the scheme is an E scheme on monotone (actually more general) data, hence at most only firstorder accurate in general. Convergence for TVDSOR schemes approximating convex or concave conservation laws is shown by enforcing a single discrete entropy inequality.
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 [1]
 J. P. Boris & D. L. Book, "Flux corrected transport: I. SHASTA, a fluid transport algorithm that works," J. Comput. Phys., v. 11, 1973, pp. 3869.
 [2]
 S. R. Charravarthy & S. Osher, "Computing with high resolution upwind schemes for hyperbolic equations," to appear in Proceedings of AMS/SIAM, 1983 Summer Seminar, La Jolla, Calif. (B. Engquist, S. Osher, and R. C. J. Somerville, eds.).
 [3]
 M. Crandall & A. Majda, "Monotone difference approximations for scalar conservative laws," Math. Comp., v. 34, 1980, pp. 121. MR 551288 (81b:65079)
 [4]
 P. J. Davis & P. Rabinowitz, Methods of Numerical Integration, Academic Press, New York, 1975. MR 0448814 (56:7119)
 [5]
 R. J. DiPerna, "Convergence of approximate solutions to conservation laws," Arch. Rational Mech. Anal., v. 82, 1983, pp. 2770 MR 684413 (84k:35091)
 [6]
 B. Engquist & S. Osher, "Onesided difference approximations for nonlinear conservation laws," Math. Comp., v. 36, 1981, pp. 321351. MR 606500 (82c:65056)
 [7]
 S. K. Godunov, "A finite difference method for the numerical computation of discontinuous solutions of the equations of fluid dynamics," Mat. Sb., v. 47, 1959, pp. 271290. MR 0119433 (22:10194)
 [8]
 A. Harten, The Method of Artificial Compression: I. Shocks and Contact Discontinuities, AEC Research & Development Report C00307750, Courant Institute, New York University, June 1984.
 [9]
 A. Harten, "The artificial compression method for computation of shocks and contact discontinuities: III. Selfadjusting hybrid schemes," Math. Comp., v. 32, 1983, pp. 363389. MR 0489360 (58:8789)
 [10]
 A. Harten, "High resolution schemes for hyperbolic conservation laws," J. Comput. Phys., v. 49, 1983, pp. 357393. MR 701178 (84g:65115)
 [11]
 A. Harten, "On a class of high resolution totalvariationstable finitedifference schemes," SIAM J. Numer. Anal., v. 21, 1984, pp. 123. MR 731210 (85f:65085)
 [12]
 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)
 [13]
 S. N. Kružkov, "First order quasilinear equations in several independent variables," Math. USSR Sb., v. 10, 1970, pp. 217243.
 [14]
 P. D. Lax, Hyperbolic Systems of Conservation Laws and the Mathematical Theory of Shock Waves, SIAM Regional Conference Lectures in Applied Mathematics, No. 11, 1972. MR 0350216 (50:2709)
 [15]
 P. D. Lax & B. Wendroff, "Systems of conservation laws," Comm. Pure Appl. Math., v. 13, 1960, pp. 217237. MR 0120774 (22:11523)
 [16]
 E. M. Murman, "Analysis of embedded shock waves calculated by relaxation methods," AIAA J., v. 12, 1974, pp. 626633.
 [17]
 O. A. Oleĭnik, Discontinuous Solutions of Nonlinear Differential Equations, Amer. Math. Soc. Transl. (2), vol. 26, Amer. Math. Soc., Providence, R. I., 1963, pp. 95172.
 [18]
 S. Osher, "Riemann solvers, the entropy condition, and difference approximations," SIAM J. Numer. Anal., v. 21, 1984, pp. 217235. MR 736327 (86d:65119)
 [19]
 S. Osher, Convergence of Generalized MUSCL Schemes, NASA Langley Contractor Report 172306, 1984, SIAM J. Numer. Anal., v. 22, 1984, pp. 947961. MR 799122 (87b:65147)
 [20]
 S. Osher & S. R. Chakravarthy, "High resolution schemes and the entropy condition," SIAM J. Numer. Anal., v. 21, 1984, pp. 955984. MR 760626 (86a:65086)
 [21]
 S. Osher & S. R. Chakravarthy, Very High Order Accurate TVD Schemes, ICASE Report 8444, 1984, IMA Volumes in Mathematics and its Applications, 2, SpringerVerlag, 1986, pp. 229294. MR 869827
 [22]
 P. L. Roe, "Approximate Riemann solvers, parameter vectors, and difference schemes," J. Comput. Phys., v. 43, 1981, pp. 357372. MR 640362 (82k:65055)
 [23]
 R. Sanders, "On convergence of monotone finite difference schemes with variable spatial differencing," Math. Comp., v. 40, 1983, pp. 91106. MR 679435 (84a:65075)
 [24]
 P. K. Sweby, "High resolution schemes using flux limiters for hyperbolic conservation laws," SIAM J. Numer. Anal., v. 21, 1984, pp. 9951011. MR 760628 (85m:65085)
 [25]
 E. Tadmor, "The large time behavior of the scalar, genuinely nonlinear LaxFriedrichs scheme," Math. Comp., v. 43, 1984, pp. 353368. MR 758188 (86g:65162)
 [26]
 E. Tadmor, "Numerical viscosity and the entropy condition for conservative difference schemes," Math. Comp., v. 43, 1984, pp. 369382. MR 758189 (86g:65163)
 [27]
 B. Van Leer, "Towards the ultimate conservative difference scheme. V. A secondorder sequel to Godunov's method," J. Comput. Phys., v. 32, 1979, pp. 101136. MR 1703646 (2000h:65120)
 [28]
 B. Van Leer, "Towards the ultimate conservative difference scheme, II. Monotonicity and conservation combined in a secondorder scheme," J. Comput. Phys., v. 14, 1974, pp. 361376.
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Additional Information
DOI:
http://dx.doi.org/10.1090/S0025571819880917817X
PII:
S 00255718(1988)0917817X
Article copyright:
© Copyright 1988
American Mathematical Society
