Numerical approximations to nonlinear conservation laws with locally varying time and space grids

Authors:
Stanley Osher and Richard Sanders

Journal:
Math. Comp. **41** (1983), 321-336

MSC:
Primary 65M10; Secondary 65M05, 76-08

DOI:
https://doi.org/10.1090/S0025-5718-1983-0717689-8

MathSciNet review:
717689

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Abstract: An explicit time differencing technique is introduced to approximate nonlinear conservation laws. This differencing technique links together an arbitrary number of space regimes containing fine and coarse time increments. Previous stability requirements, i.e. the CFL condition, placed a *global* bound on the size of the time increments. For scalar, monotone, approximations in one space dimension, using this variable step time differencing, convergence to the correct physical solution is proven given only a *local* CFL condition.

**[1]**Sukumar R. Chakravarthy and Stanley Osher,*Numerical experiments with the Osher upwind scheme for the Euler equations*, AIAA J.**21**(1983), no. 9, 1241–1248. MR**714762**, https://doi.org/10.2514/3.60143**[2]**Michael G. Crandall and Andrew Majda,*Monotone difference approximations for scalar conservation laws*, Math. Comp.**34**(1980), no. 149, 1–21. MR**551288**, https://doi.org/10.1090/S0025-5718-1980-0551288-3**[3]**Björn Engquist and Stanley Osher,*Stable and entropy satisfying approximations for transonic flow calculations*, Math. Comp.**34**(1980), no. 149, 45–75. MR**551290**, https://doi.org/10.1090/S0025-5718-1980-0551290-1**[4]**Björn Engquist and Stanley Osher,*One-sided difference approximations for nonlinear conservation laws*, Math. Comp.**36**(1981), no. 154, 321–351. MR**606500**, https://doi.org/10.1090/S0025-5718-1981-0606500-X**[5]**B. Engquist, S. Osher, P. Roe & B. Van Leer, "Discrete shocks and upwind schemes." (To appear.)**[6]**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****[7]**P. M. Goorjian & R. Van Buskirk,*Implicit Calculations of Transonic Flows Using Monotone Methods*, AIAA-81-0331, St. Louis, Mo., Jan. 1981.**[8]**Gray Jennings,*Discrete shocks*, Comm. Pure Appl. Math.**27**(1974), 25–37. MR**0338594**, https://doi.org/10.1002/cpa.3160270103**[9]**S. N. Kruzkov, "First order quasi-linear equations in several independent variables,"*Math. USSR Sb.*, v. 10, 1970, pp. 217-243.**[10]**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****[11]**Peter D. Lax,*Hyperbolic systems of conservation laws and the mathematical theory of shock waves*, Society for Industrial and Applied Mathematics, Philadelphia, Pa., 1973. Conference Board of the Mathematical Sciences Regional Conference Series in Applied Mathematics, No. 11. MR**0350216****[12]**Peter Lax and Burton Wendroff,*Systems of conservation laws*, Comm. Pure Appl. Math.**13**(1960), 217–237. MR**0120774**, https://doi.org/10.1002/cpa.3160130205**[13]**O. A. Oleĭnik,*Discontinuous solutions of non-linear differential equations*, Amer. Math. Soc. Transl. (2)**26**(1963), 95–172. MR**0151737****[14]**Stanley Osher,*Numerical solution of singular perturbation problems and hyperbolic systems of conservation laws*, Analytical and numerical approaches to asymptotic problems in analysis (Proc. Conf., Univ. Nijmegen, Nijmegen, 1980) North-Holland Math. Stud., vol. 47, North-Holland, Amsterdam-New York, 1981, pp. 179–204. MR**605507****[15]**Richard Sanders,*On convergence of monotone finite difference schemes with variable spatial differencing*, Math. Comp.**40**(1983), no. 161, 91–106. MR**679435**, https://doi.org/10.1090/S0025-5718-1983-0679435-6**[16]**B. Van Leer,*On the Relation Between the Upwind Differencing Schemes of Godunov, Engquist-Osher, and Roe*, ICASE Report #81-11, 1981.

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DOI:
https://doi.org/10.1090/S0025-5718-1983-0717689-8

Article copyright:
© Copyright 1983
American Mathematical Society