On a fourth order accurate implicit finite difference scheme for hyperbolic conservation laws. I. Nonstiff strongly dynamic problems

Authors:
Amiram Harten and Hillel Tal-Ezer

Journal:
Math. Comp. **36** (1981), 353-373

MSC:
Primary 65M05; Secondary 35L65, 76L05

DOI:
https://doi.org/10.1090/S0025-5718-1981-0606501-1

MathSciNet review:
606501

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Abstract: An implicit finite difference method of fourth order accuracy (in space and time) is introduced for the numerical solution of one-dimensional systems of hyperbolic conservation laws. The basic form of this method is a straightforward generalization of the Crank-Nicholson scheme: it is a two-level scheme which is unconditionally stable and nondissipative. The scheme is compact, i.e., it uses only 3 mesh points at level *t* and 3 mesh points at level .

In this paper, the first in a series, we present a dissipative version of the basic method which is conditionally stable under the CFL (Courant-Friedrichs-Lewy) condition. This version is particularly useful for numerical solution of problems with strong but nonstiff dynamic features, where the CFL restriction is reasonable on accuracy grounds. Numerical results are presented to illustrate properties of the proposed scheme.

**[1]**S. Abarbanel, D. Gottlieb, and E. Turkel,*Difference schemes with fourth order accuracy for hyperbolic equations*, SIAM J. Appl. Math.**29**(1975), no. 2, 329–351. MR**0388794**, https://doi.org/10.1137/0129029**[2]**Richard M. Beam and R. F. Warming,*An implicit finite-difference algorithm for hyperbolic systems in conservation-law form*, J. Computational Phys.**22**(1976), no. 1, 87–110. MR**0455435****[3]**Philip J. Davis and Philip Rabinowitz,*Methods of numerical integration*, 2nd ed., Computer Science and Applied Mathematics, Academic Press, Inc., Orlando, FL, 1984. MR**760629****[4]**Bertil Gustafsson,*An alternating direction implicit method for solving the shallow water equations*, J. Computational Phys.**7**(1971), 239–254. MR**0282548****[5]**Amiram Harten,*The artificial compression method for computation of shocks and contact discontinuities. III. Self-adjusting hybrid schemes*, Math. Comp.**32**(1978), no. 142, 363–389. MR**0489360**, https://doi.org/10.1090/S0025-5718-1978-0489360-X**[6]**A. Harten and G. Zwas,*Self-adjusting hybrid schemes for shock computations*, J. Computational Phys.**9**(1972), 568–583. MR**0309339****[7]**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**, https://doi.org/10.1002/cpa.3160290305**[8]**Richard S. Hirsh and David H. Rudy,*The role of diagonal dominance and cell Reynolds number in implicit difference methods for fluid mechanics problems*, J. Computational Phys.**16**(1974), 304–310. MR**0381512****[9]**Eugene Isaacson and Herbert Bishop Keller,*Analysis of numerical methods*, John Wiley & Sons, Inc., New York-London-Sydney, 1966. MR**0201039****[10]**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****[11]**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**[12]**Andrew Majda and Stanley Osher,*A systematic approach for correcting nonlinear instabilities. The Lax-Wendroff scheme for scalar conservation laws*, Numer. Math.**30**(1978), no. 4, 429–452. MR**502526**, https://doi.org/10.1007/BF01398510**[13]**K. W. Morton,*Initial-value problems by finite difference and other methods*, The state of the art in numerical analysis (Proc. Conf., Univ. York, Heslington, 1976) Academic Press, London, 1977, pp. 699–756. MR**0451754****[14]**Gary A. Sod,*A survey of several finite difference methods for systems of nonlinear hyperbolic conservation laws*, J. Computational Phys.**27**(1978), no. 1, 1–31. MR**0495002****[15]**E. Turkel,*On the Practical Use of High Order Methods for Hyperbolic Problems*, ICASE Report No. 78-19, NASA Langley Research Center, November 1978.**[16]**H. J. Wirz, F. De Schutter, and A. Turi,*An implicit, compact, finite difference method to solve hyperbolic equations*, Math. Comput. Simulation**19**(1977), no. 4, 241–261. MR**0478650**, https://doi.org/10.1016/0378-4754(77)90042-8**[17]**A. Harten & H. Tal-Ezer,*On a Fourth Order Accurate Implicit Finite Difference Scheme for Hyperbolic Conservation Laws*: I.*Nonstiff Strongly Dynamic Problems*, ICASE Report No. 79-1, NASA Langley Research Center, January 1979.

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Additional Information

DOI:
https://doi.org/10.1090/S0025-5718-1981-0606501-1

Keywords:
Implicit finite difference scheme,
fourth order accuracy,
hyperbolic conservation laws

Article copyright:
© Copyright 1981
American Mathematical Society