Remote Access Mathematics of Computation
Green Open Access

Mathematics of Computation

ISSN 1088-6842(online) ISSN 0025-5718(print)



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
MathSciNet review: 606501
Full-text PDF Free Access

Abstract | References | Similar Articles | Additional Information

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 $ t + \Delta t$.

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.

References [Enhancements On Off] (What's this?)

  • [1] S. Abarbanel, D. Gottlieb & E. Turkel, "Difference schemes with fourth order accuracy for hyperbolic equations," SIAM J. Appl. Math., v. 29, 1975, pp. 329-351. MR 0388794 (52:9628)
  • [2] R. M. Beam & R. F. Warming, "An implicit finite difference algorithm for hyperbolic systems in conservation-law form," J. Comput. Phys., v. 22, 1976, pp. 87-110. MR 0455435 (56:13673)
  • [3] P. Davis & P. Rabinowitz, Methods of Numerical Integration, Academic Press, New York, 1974. MR 760629 (86d:65004)
  • [4] B. Gustafsson, "An alternating direction implicit method for solving the shallow water equations," J. Comput. Phys., v. 7, 1971, pp. 239-254. MR 0282548 (43:8258)
  • [5] A. Harten, "The artificial compression method for computation of shocks and contact discontinuities: III. Self-adjusting hybrid scheme," Math. Comp., v. 32, 1978, pp. 363-389. MR 0489360 (58:8789)
  • [6] A. Harten & G. Zwas, "Self-adjusting hybrid schemes for shock computations," J. Comput. Phys., v. 9, 1972, pp. 568-583. MR 0309339 (46:8449)
  • [7] 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)
  • [8] R. S. Hirsh & D. H. Rudy, "The role of diagonal dominance and cell Reynolds number in implicit difference methods for fluid mechaics problems," J. Comput. Phys., v. 16, 1976, pp. 304-310. MR 0381512 (52:2405)
  • [9] E. Isaacson & H. Keller, Analysis of Numerical Methods, Wiley, New York, 1966. MR 0201039 (34:924)
  • [10] P. D. Lax, Hyperbolic Systems of Conservation Laws and the Mathematical Theory of Shock Waves, SIAM Regional Conference in Applied Math., 1973. MR 0350216 (50:2709)
  • [11] P. D. Lax & B. Wendroff, "Systems of conservation laws," Comm. Pure Appl. Math., v. 23, 1960, pp. 217-237. MR 0120774 (22:11523)
  • [12] A. Majda & S. Osher, "A systematic approach for correcting nonlinear instabilities: The Lax-Wendroff scheme for scalar conservation laws," Numer. Math. (To appear.) MR 502526 (80g:65101)
  • [13] K. W. Morton, "Initial value problems by finite difference and other methods," The State of the Art in Numerical Analysis, Academic Press, New York, 1977, pp. 699-756. MR 0451754 (56:10036)
  • [14] G. A. Sod, "A survey of several finite difference methods for systems of nonlinear hyperbolic conservation laws," J. Comput. Phys., v. 27, 1978, pp. 1-31. MR 0495002 (58:13770)
  • [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. DeSchutter & A. Turi, "An implicit, compact, finite difference method to solve hyperbolic equations," Math. Comput. Simulation, v. 19, 1977, pp. 241-261. MR 0478650 (57:18127)
  • [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.

Similar Articles

Retrieve articles in Mathematics of Computation with MSC: 65M05, 35L65, 76L05

Retrieve articles in all journals with MSC: 65M05, 35L65, 76L05

Additional Information

Keywords: Implicit finite difference scheme, fourth order accuracy, hyperbolic conservation laws
Article copyright: © Copyright 1981 American Mathematical Society

American Mathematical Society