A third-order accurate variation nonexpansive difference scheme for single nonlinear conservation laws

Author:
Richard Sanders

Journal:
Math. Comp. **51** (1988), 535-558

MSC:
Primary 65M10; Secondary 35L65

MathSciNet review:
935073

Full-text PDF Free Access

Abstract | References | Similar Articles | Additional Information

Abstract: It was widely believed that all variation nonexpansive finite difference schemes for single conservation laws must reduce to first-order at extreme points of the approximation. It is shown here that this belief is in fact false. A third-order scheme, which at worst may reduce to second order at extreme points, is developed and analyzed. Moreover, extensive numerical experiments indicate that the third-order scheme introduced here yields superior approximations when compared with other variation nonexpansive difference schemes.

**[1]**Matania Ben-Artzi and Joseph Falcovitz,*A second-order Godunov-type scheme for compressible fluid dynamics*, J. Comput. Phys.**55**(1984), no. 1, 1–32. MR**757422**, 10.1016/0021-9991(84)90013-5**[2]**S. R. Chakravarthy, A. Harten & S. Osher,*Essentially Non-Oscillatory Shock-Capturing Schemes of Arbitrarily High Accuracy*, AIAA 24th Aerospace Sciences Meeting, January 6-9, 1986, Reno, Nevada.**[3]**P. Colella & P. R. Woodward, "The piecewise-parabolic method (PPM) for gas-dynamical simulations,"*J. Comput. Phys.*, v. 54, 1984, pp. 174-201.**[4]**S. D. Conte & C. de Boor,*Elementary Numerical Analysis*, 3rd ed., McGraw-Hill, New York, 1980.**[5]**Michael G. Crandall and Andrew Majda,*Monotone difference approximations for scalar conservation laws*, Math. Comp.**34**(1980), no. 149, 1–21. MR**551288**, 10.1090/S0025-5718-1980-0551288-3**[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]**Ami Harten,*High resolution schemes for hyperbolic conservation laws*, J. Comput. Phys.**49**(1983), no. 3, 357–393. MR**701178**, 10.1016/0021-9991(83)90136-5**[8]**Ami Harten, Björn Engquist, Stanley Osher, and Sukumar R. Chakravarthy,*Uniformly high-order accurate essentially nonoscillatory schemes. III*, J. Comput. Phys.**71**(1987), no. 2, 231–303. MR**897244**, 10.1016/0021-9991(87)90031-3**[9]**A. Harten, J. M. Hyman, and P. D. Lax,*On finite-difference approximations and entropy conditions for shocks*, Comm. Pure Appl. Math.**29**(1976), no. 3, 297–322. With an appendix by B. Keyfitz. MR**0413526****[10]**Ami Harten and Stanley Osher,*Uniformly high-order accurate nonoscillatory schemes. I*, SIAM J. Numer. Anal.**24**(1987), no. 2, 279–309. MR**881365**, 10.1137/0724022**[11]**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****[12]**Richard Sanders,*On convergence of monotone finite difference schemes with variable spatial differencing*, Math. Comp.**40**(1983), no. 161, 91–106. MR**679435**, 10.1090/S0025-5718-1983-0679435-6**[13]**Richard Sanders,*The moving grid method for nonlinear hyperbolic conservation laws*, SIAM J. Numer. Anal.**22**(1985), no. 4, 713–728. MR**795949**, 10.1137/0722043**[14]**P. K. Sweby,*High resolution schemes using flux limiters for hyperbolic conservation laws*, SIAM J. Numer. Anal.**21**(1984), no. 5, 995–1011. MR**760628**, 10.1137/0721062**[15]**B. van Leer, "Towards the ultimate conservative 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

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

Additional Information

DOI:
http://dx.doi.org/10.1090/S0025-5718-1988-0935073-3

Article copyright:
© Copyright 1988
American Mathematical Society