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The artificial compression method for computation of shocks and contact discontinuities. III. Self-adjusting hybrid schemes

Author: Amiram Harten
Journal: Math. Comp. 32 (1978), 363-389
MSC: Primary 76.65; Secondary 65H10
MathSciNet review: 0489360
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Abstract: This paper presents a new computational method for the calculation of discontinuous solutions of hyperbolic systems of conservation laws, which deal effectively with both shock and contact discontinuities. The method consists of two stages: in the first stage a standard finite-difference scheme is hybridized with a nonoscillatory first order accurate method to provide for the monotonic variation of the solution near discontinuities, and in the second stage artificial compression is applied to sharpen transitions at discontinuities. This modification of a standard finite-difference method results in a scheme which preserves the order of truncation error of the original method and yet yields a sharp and oscillation free transition for both shocks and contact discontinuities. The modification can be easily implemented in existing computer codes.

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  • [1] J. P. BORIS & D. L. BOOK, "Flux corrected transport. I. SHASTA, a fluid transport algorithm that works," J. Computational Phys., v. 11, 1973, pp. 38-69.
  • [2] S. Z. BURSTEIN, "Finite difference calculations for hydrodynamic flows containing discontinuities," J. Computational Phys., v. 1, 1966, pp. 198-222.
  • [3] A. J. CHORIN, "Random choice solution of hyperbolic systems," J. Computational Phys., v. 22, 1976, pp. 517-533. MR 0471342 (57:11077)
  • [4] R. COURANT & K. O. FRIEDRICHS, Supersonic Flow and Shock Waves, Interscience-Wiley, New York, 1948. MR 0029615 (10:637c)
  • [5] S. K. GODUNOV, "Finite difference methods for numerical computations of discontinuous solutions of equations of fluid dynamics," Mat. Sb., v. 47, 1959, pp. 271-295. (Russian)
  • [6] M. GOLDBERG & S. ABARBANEL, "Stable approximations for hyperbolic systems with moving internal boundary conditions," Math. Comp., v. 28, 1974, pp. 413-447. MR 0381343 (52:2240)
  • [7] D. GOTTLIEB, "Strang-type difference schemes for multi-dimensional problems," SIAM J. Numer. Anal., v. 9, 1972, pp. 650-661. MR 0314274 (47:2826)
  • [8] A. HARTEN, The Method of Artificial Compression: I. Shocks and Contract Discontinuities, AEC Research & Develop. Rep. COO-3077-50, Courant Inst., New York Univ., June 1974.
  • [9] A. HARTEN, "The artificial compression method for computation of shocks and contact discontinuities: I. Single conservation laws," Comm. Pure Appl. Math. (To appear; also to appear as an ICASE Report.) MR 0438730 (55:11637)
  • [10] A. HARTEN, "The artificial compression method for computation of shocks and contact discontinuities: II. Systems of conservation laws." (In preparation.)
  • [11] A. HARTEN, J. M. HYMAN & P. D. LAX, "On finite-difference approximations and entropy conditions for shocks," (with Appendix by B. Keyfitz), Comm. Pure Appl. Math., v. 29, 1976, pp. 297-322. MR 0413526 (54:1640)
  • [12] A. HARTEN & G. ZWAS, "Switched numerical Shuman filters for shock calculations," J. Engrg. Math., v. 6, 1972, pp. 207-216.
  • [13] A. HARTEN & G. ZWAS, "Self-adjusting hybrid schemes for shock computations," J. Computational Phys., v. 6, 1972, pp. 568-583. MR 0309339 (46:8449)
  • [14] G. JENNINGS, "Discrete shocks," Comm. Pure Appl. Math., v. 27, 1974, pp. 25-37. MR 0338594 (49:3358)
  • [15] A. LAPIDUS, "A detached shock calculation by second-order finite differences," J. Computational Phys., v. 2, 1967, pp. 154-177.
  • [16] P. D. LAX, "Hyperbolic systems of conservation laws. II," Comm. Pure Appl. Math., v.10, 1957, pp. 537-566. MR 0093653 (20:176)
  • [17] P. D. LAX & B. WENDROFF, "Systems of conservation laws," Comm. Pure Appl. Math., v. 13, 1960, pp. 217-237. MR 0120774 (22:11523)
  • [18] T. P. LIU, "The entropy condition and the admissibility of shocks," J. Math. Anal. Appl., v. 53, 1976, pp. 78-88. MR 0387830 (52:8669)
  • [19] R. W. MacCORMACK, Numerical Solution of the Interaction of a Shock Wave with a Laminar Boundary Layer (Proc. 2nd Internat. Conf. on Numerical Methods in Fluid Dynamics, M. Holt, Editor), Lecture Notes in Phys., v. 8, Springer-Verlag, New York, 1970, pp. 151-163.
  • [20] A. MUJDA & S. OSHER, "Propagation of errors into regions of smoothness for accurate difference approximations to hyperbolic equations," Comm. Pure Appl. Math. (To appear.) MR 0471345 (57:11080)
  • [21] O. A. OLEINIK, "Discontinuous solutions of nonlinear differential equations," Uspehi Mat. Nauk, v. 12, 1957, pp. 3-73; English transl., Amer. Math. Soc. Transl. (2), v. 26, 1963, pp. 95-172. MR 0151737 (27:1721)
  • [22] S. A. ORSZAG & L. W. JAYNE, "Local errors of difference approximations to hyperbolic equations," J. Computational Phys., v. 14, 1974, pp. 93-103. MR 0494992 (58:13762a)
  • [23] R. D. RICHTMYER & K. W. MORTON, Difference Methods for Initial Value Problems, 2nd ed., Interscience-Wiley, New York, 1967. MR 0220455 (36:3515)
  • [24] G. STRANG, "On the construction and comparison of difference schemes," SIAM J. Numer. Anal., v. 5, 1968, pp. 506-517. MR 0235754 (38:4057)
  • [25] H. U. THOMMEN, "Numerical integration of the Navier-Stokes equations," Z. Angew. Math. Phys., v. 17, 1966, pp. 369-384. MR 0205560 (34:5387)
  • [26] B. VAN LEER, "Towards the ultimate conservative difference scheme. II. Monotonicity and conservation combined in a second order scheme," J. Computational Phys., v. 14, 1974, pp. 361-370.
  • [27] J. von NEUMANN & R. D. RICHTMYER, "A method for the numerical calculation of hydrodynamic shocks," J. Appl. Phys., v. 21, 1950, pp. 232-237. MR 0037613 (12:289b)
  • [28] N. N. YANENKO & Yu. I. SHOKIN, "First differential approximation method and approximate viscosity of difference schemes," Phys. Fluids, suppl. II, v. 12, 1969, pp. II-28-11-33.

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Article copyright: © Copyright 1978 American Mathematical Society

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