On the convergence of difference approximations to scalar conservation laws

Authors:
Stanley Osher and Eitan Tadmor

Journal:
Math. Comp. **50** (1988), 19-51

MSC:
Primary 65M10; Secondary 35L65, 65M05

DOI:
https://doi.org/10.1090/S0025-5718-1988-0917817-X

MathSciNet review:
917817

Full-text PDF Free Access

Abstract | References | Similar Articles | Additional Information

Abstract: We present a unified treatment of explicit in time, two-level, second-order resolution (SOR), total-variation 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 first-order accurate in general. Convergence for TVD-SOR schemes approximating convex or concave conservation laws is shown by enforcing a single discrete entropy inequality.

**[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. 38-69.**[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. 1-21. 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. 27-70 MR**684413 (84k:35091)****[6]**B. Engquist & S. Osher, "One-sided difference approximations for nonlinear conservation laws,"*Math. Comp.*, v. 36, 1981, pp. 321-351. 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. 271-290. MR**0119433 (22:10194)****[8]**A. Harten,*The Method of Artificial Compression*: I.*Shocks and Contact Discontinuities*, AEC Research & Development Report C00-3077-50, Courant Institute, New York University, June 1984.**[9]**A. Harten, "The artificial compression method for computation of shocks and contact discontinuities: III. Self-adjusting hybrid schemes,"*Math. Comp.*, v. 32, 1983, pp. 363-389. MR**0489360 (58:8789)****[10]**A. Harten, "High resolution schemes for hyperbolic conservation laws,"*J. Comput. Phys.*, v. 49, 1983, pp. 357-393. MR**701178 (84g:65115)****[11]**A. Harten, "On a class of high resolution total-variation-stable finite-difference schemes,"*SIAM J. Numer. Anal.*, v. 21, 1984, pp. 1-23. 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. 297-322. MR**0413526 (54:1640)****[13]**S. N. Kružkov, "First order quasilinear equations in several independent variables,"*Math. USSR Sb.*, v. 10, 1970, pp. 217-243.**[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. 217-237. MR**0120774 (22:11523)****[16]**E. M. Murman, "Analysis of embedded shock waves calculated by relaxation methods,"*AIAA J.*, v. 12, 1974, pp. 626-633.**[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. 95-172.**[18]**S. Osher, "Riemann solvers, the entropy condition, and difference approximations,"*SIAM J. Numer. Anal.*, v. 21, 1984, pp. 217-235. 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. 947-961. 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. 955-984. MR**760626 (86a:65086)****[21]**S. Osher & S. R. Chakravarthy,*Very High Order Accurate TVD Schemes*, ICASE Report 84-44, 1984,*IMA Volumes in Mathematics and its Applications*, 2, Springer-Verlag, 1986, pp. 229-294. MR**869827****[22]**P. L. Roe, "Approximate Riemann solvers, parameter vectors, and difference schemes,"*J. Comput. Phys.*, v. 43, 1981, pp. 357-372. MR**640362 (82k:65055)****[23]**R. Sanders, "On convergence of monotone finite difference schemes with variable spatial differencing,"*Math. Comp.*, v. 40, 1983, pp. 91-106. 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. 995-1011. MR**760628 (85m:65085)****[25]**E. Tadmor, "The large time behavior of the scalar, genuinely nonlinear Lax-Friedrichs scheme,"*Math. Comp.*, v. 43, 1984, pp. 353-368. MR**758188 (86g:65162)****[26]**E. Tadmor, "Numerical viscosity and the entropy condition for conservative difference schemes,"*Math. Comp.*, v. 43, 1984, pp. 369-382. MR**758189 (86g:65163)****[27]**B. Van Leer, "Towards the ultimate conservative difference scheme. V. A second-order sequel to Godunov's method,"*J. Comput. Phys.*, v. 32, 1979, pp. 101-136. MR**1703646 (2000h:65120)****[28]**B. Van Leer, "Towards the ultimate conservative difference scheme, II. Monotonicity and conservation combined in a second-order scheme,"*J. Comput. Phys.*, v. 14, 1974, pp. 361-376.

Retrieve articles in *Mathematics of Computation*
with MSC:
65M10,
35L65,
65M05

Retrieve articles in all journals with MSC: 65M10, 35L65, 65M05

Additional Information

DOI:
https://doi.org/10.1090/S0025-5718-1988-0917817-X

Article copyright:
© Copyright 1988
American Mathematical Society