On a large time-step high resolution scheme

Author:
Ami Harten

Journal:
Math. Comp. **46** (1986), 379-399

MSC:
Primary 65M05; Secondary 65M10, 76L05

MathSciNet review:
829615

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Abstract: This paper presents a class of new second-order accurate -point explicit schemes for the computation of weak solutions of hyperbolic conservation laws, that are total-variation-diminishing under a CFL restriction of *K*. These highly nonlinear schemes are obtained by applying a nonoscillatory first-order accurate -point scheme to a modified flux. The so-derived second-order accurate schemes achieve high resolution, while retaining the robustness of the original first-order accurate scheme.

**[1]**Y. Brennier, private communication.**[2]**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**[3]**Ami Harten,*On a class of high resolution total-variation-stable finite-difference schemes*, SIAM J. Numer. Anal.**21**(1984), no. 1, 1–23. With an appendix by Peter D. Lax. MR**731210**, 10.1137/0721001**[4]**Amiram Harten, Peter D. Lax, and Bram van Leer,*On upstream differencing and Godunov-type schemes for hyperbolic conservation laws*, SIAM Rev.**25**(1983), no. 1, 35–61. MR**693713**, 10.1137/1025002**[5]**Randall J. LeVeque,*Large time step shock-capturing techniques for scalar conservation laws*, SIAM J. Numer. Anal.**19**(1982), no. 6, 1091–1109. MR**679654**, 10.1137/0719080**[6]**R. J. LeVeque,*Towards a Large Time-Step Algorithm for Systems of Conservation Laws*:*Preliminary Results Ignoring Interactions*, Numerical Analysis Report 7/82, University of Reading, 1982.**[7]**Stanley Osher and Richard Sanders,*Numerical approximations to nonlinear conservation laws with locally varying time and space grids*, Math. Comp.**41**(1983), no. 164, 321–336. MR**717689**, 10.1090/S0025-5718-1983-0717689-8**[8]**P. L. Roe, Proc. Seventh Internat. Conf. on Numerical Methods in Fluid Dynamics, Stanford/NASA Ames, June 1980, Springer-Verlag.**[9]**P. L. Roe,*Approximate Riemann solvers, parameter vectors, and difference schemes*, J. Comput. Phys.**43**(1981), no. 2, 357–372. MR**640362**, 10.1016/0021-9991(81)90128-5**[10]**H. C. Yee, R. F. Warming, and Ami Harten,*Application of TVD schemes for the Euler equations of gas dynamics*, Large-scale computations in fluid mechanics, Part 2 (La Jolla, Calif., 1983), Lectures in Appl. Math., vol. 22, Amer. Math. Soc., Providence, RI, 1985, pp. 357–377. MR**818797**

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Additional Information

DOI:
https://doi.org/10.1090/S0025-5718-1986-0829615-4

Article copyright:
© Copyright 1986
American Mathematical Society