Remote Access Mathematics of Computation
Green Open Access

Mathematics of Computation

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



One-sided difference approximations for nonlinear conservation laws

Authors: Björn Engquist and Stanley Osher
Journal: Math. Comp. 36 (1981), 321-351
MSC: Primary 65M10; Secondary 35L67, 65M05
MathSciNet review: 606500
Full-text PDF

Abstract | References | Similar Articles | Additional Information

Abstract: We analyze one-sided or upwind finite difference approximations to hyperbolic partial differential equations and, in particular, nonlinear conservation laws. Second order schemes are designed for which we prove both nonlinear stability and that the entropy condition is satisfied for limit solutions. We show that no such stable approximation of order higher than two is possible. These one-sided schemes have desirable properties for shock calculations. We show that the proper switch used to change the direction in the upwind differencing across a shock is of great importance. New and simple schemes are developed for which we prove qualitative properties such as sharp monotone shock profiles, existence, uniqueness, and stability of discrete shocks. Numerical examples are given.

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

  • [1] M. G. Crandall & A. Majda, "Monotone difference approximations for scalar conservation laws," Math. Comp., v. 34, 1980, pp. 1-21. MR 551288 (81b:65079)
  • [2] G. Dahlquist, "A special stability problem for linear multistep methods," BIT, v. 3, 1963, pp. 27-43. MR 0170477 (30:715)
  • [3] G. Dahlquist, "Positive functions and some applications to stability questions for numerical methods," Recent Advances in Numerical Analysis (C. de Boor and G. Golub, Eds.), Academic Press, New York, 1978, pp. 1-29. MR 519054 (80c:65118)
  • [4] B. Engquist & S. Osher, "Stable and entropy satisfying approximations for transonic flow calculations," Math. Comp., v. 34, 1980, pp. 45-75. MR 551290 (81b:65082)
  • [5] A. Harten, "The artificial compression method for computation of shocks and contact discontinuities," Comm. Pure Appl. Math., v. 30, 1977, pp. 611-638. MR 0438730 (55:11637)
  • [6] 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. 297-332. MR 0413526 (54:1640)
  • [7] A. Jameson, "A numerical solution of nonlinear partial differential equations of mixed type," Numerical Solutions of Partial Differential Equations III, Academic Press, New York, 1976, pp. 275-320. MR 0468255 (57:8093)
  • [8] G. Jennings, "Discrete shocks," Comm. Pure Appl. Math., v. 27, 1974, pp. 25-37. MR 0338594 (49:3358)
  • [9] J. A. Krupp & J. D. Cole, Studies in Transonic Flow IV, Unsteady Transonic Flow, UCLA Eng. Dept. Rep., 76/04, 1976.
  • [10] P. D. Lax, "Shock waves and entropy," Contributions to Nonlinear Functional Analysis (E. H. Zarantonello, Ed.), Academic Press, New York, 1971, pp. 603-634. MR 0393870 (52:14677)
  • [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] E. M. Murman & J. D. Cole, "Calculations of steady transonic flows," AIAA J., v. 9, 1971, pp. 114-121.
  • [13] P. Roache, Computational Fluid Dynamics, Hermosa, Albuquerque, N.M., 1972. MR 0411358 (53:15094)
  • [14] J. Steger, "Coefficient matrices for implicit finite difference solutions of the inviscid fluid conservation law equations," Comput. Methods Appl. Mech. Engrg., v. 13, 1978, pp. 175-188. MR 497272 (80a:76015)
  • [15] G. Strang, "Accurate partial difference methods. II: Non-linear problems," Numer. Math., v. 6, 1964, pp. 37-46. MR 0166942 (29:4215)
  • [16] B. van Leer, "Towards the ultimate conservative difference scheme III-Upstream-centered finite-difference schemes for ideal compressible flow," J. Comput. Phys., v. 3, 1977, pp. 263-275.
  • [17] B. van Leer, "Towards the ultimate conservative difference scheme IV; A new approach to numerical convection," J. Comput. Phys., v. 23, 1977, pp. 276-299.
  • [18] R. F. Warming & R. M. Beam, "Upwind second-order difference schemes and applications in aerodynamical flows," AIAA J., v. 14, 1976, pp. 1241-1249. MR 0459301 (56:17495)
  • [19] R. F. Warming & R. M. Beam, "On the construction and application of implicit factored schemes for conservation laws," Computational Fluid Dynamics, SIAM-AMS Proceedings, vol. 11, 1978, pp. 85 - 129. MR 0520170 (58:24998)

Similar Articles

Retrieve articles in Mathematics of Computation with MSC: 65M10, 35L67, 65M05

Retrieve articles in all journals with MSC: 65M10, 35L67, 65M05

Additional Information

Article copyright: © Copyright 1981 American Mathematical Society

American Mathematical Society