Dissipative two-four methods for time-dependent problems
HTML articles powered by AMS MathViewer
- by David Gottlieb and Eli Turkel PDF
- Math. Comp. 30 (1976), 703-723 Request permission
Abstract:
A generalization of the Lax-Wendroff method is presented. This generalization bears the same relationship to the two-step Richtmyer method as the Kreiss-Oliger scheme does to the leapfrog method. Variants based on the MacCormack method are considered as well as extensions to parabolic problems. Extensions to two dimensions are analyzed, and a proof is presented for the stability of a Thommen-type algorithm. Numerical results show that the phase error is considerably reduced from that of second-order methods and is similar to that of the Kreiss-Oliger method. Furthermore, the (2, 4) dissipative scheme can handle shocks without the necessity for an artificial viscosity.References
- 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 388794, DOI 10.1137/0129029 S. Z. BURSTEIN, "High order accurate difference methods in hydrodynamics," Nonlinear Partial Differential Equations, W. F. Ames, Editor, Academic Press, New York, 1967, pp. 279-290. MR 36 #510. W. P. CROWLEY, "Numerical advection experiments," Monthly Weather Review, v. 96, 1968, pp. 1-11.
- B. Fornberg, On the instability of leap-frog and Crank-Nicolson approximations of a nonlinear partial differential equation, Math. Comp. 27 (1973), 45–57. MR 395249, DOI 10.1090/S0025-5718-1973-0395249-2 J. GAZDAG, "Numerical convective schemes based on accurate computation of space derivatives," J. Computational Phys., v. 13, 1973, pp. 100-113. J. P. GERRITY, JR., R. P. McPHERSON & P. D. POLGER, "On the efficient reduction of truncation error in numerical prediction models," Monthly Weather Review, v. 100, 1972, pp. 637-643.
- M. K. Gol′dberg, Multigraphs with a chromatic index that is nearly maximal, Diskret. Analiz 23 (1973), 3–7, 72 (Russian). MR 354429
- David Gottlieb, Strang-type difference schemes for multidimensional problems, SIAM J. Numer. Anal. 9 (1972), 650–661. MR 314274, DOI 10.1137/0709054
- Heinz-Otto Kreiss and Joseph Oliger, Comparison of accurate methods for the integration of hyperbolic equations, Tellus 24 (1972), 199–215 (English, with Russian summary). MR 319382, DOI 10.3402/tellusa.v24i3.10634 H.-O. KREISS & J. OLIGER, Methods for the Approximate Solution of Time Dependent Problems, Global Atmospheric Research Programme Publications Series, no. 10, 1973.
- Peter D. Lax and Burton Wendroff, Difference schemes for hyperbolic equations with high order of accuracy, Comm. Pure Appl. Math. 17 (1964), 381–398. MR 170484, DOI 10.1002/cpa.3160170311
- Alain Lerat and Roger Peyret, Noncentered schemes and shock propagation problems, Internat. J. Comput. & Fluids 2 (1974), no. 1, 35–52. MR 363148, DOI 10.1016/0045-7930(74)90004-8 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), Springer-Verlag, Lecture Notes in Phys., vol. 8, 1970, pp. 151-163. MR 43 #4216.
- G. R. McGuire and J. Ll. Morris, A class of second-order accurate methods for the solution of systems of conservation laws, J. Comput. Phys. 11 (1973), 531–549. MR 331808, DOI 10.1016/0021-9991(73)90136-8
- G. I. Marchuk, On the theory of the splitting-up method, Numerical Solution of Partial Differential Equations, II (SYNSPADE 1970) (Proc. Sympos., Univ. of Maryland, College Park, Md., 1970) Academic Press, New York, 1971, pp. 469–500. MR 0284004
- Joseph Oliger, Fourth order difference methods for the initial boundary-value problem for hyperbolic equations, Math. Comp. 28 (1974), 15–25. MR 359344, DOI 10.1090/S0025-5718-1974-0359344-7
- Robert D. Richtmyer and K. W. Morton, Difference methods for initial-value problems, 2nd ed., Interscience Tracts in Pure and Applied Mathematics, No. 4, Interscience Publishers John Wiley & Sons, Inc., New York-London-Sydney, 1967. MR 0220455
- K. V. Roberts and N. O. Weiss, Convective difference schemes, Math. Comp. 20 (1966), 272–299. MR 198702, DOI 10.1090/S0025-5718-1966-0198702-6
- Gilbert Strang, On the construction and comparison of difference schemes, SIAM J. Numer. Anal. 5 (1968), 506–517. MR 235754, DOI 10.1137/0705041
- Hans U. Thommen, Numerical integration of the Navier-Stokes equations, Z. Angew. Math. Phys. 17 (1966), 369–384 (English, with German summary). MR 205560, DOI 10.1007/BF01594529
- E. Turkel, Symmetric hyperbolic difference schemes and matrix problems, Linear Algebra Appl. 16 (1977), no. 2, 109–129. MR 464603, DOI 10.1016/0024-3795(77)90025-8
- Eli Turkel, Composite methods for hyperbolic equations, SIAM J. Numer. Anal. 14 (1977), no. 4, 744–759. MR 443365, DOI 10.1137/0714051
- Lars B. Wahlbin, A dissipative Galerkin method for the numerical solution of first order hyperbolic equations, Mathematical aspects of finite elements in partial differential equations (Proc. Sympos., Math. Res. Center, Univ. Wisconsin, Madison, Wis., 1974) Publication No. 33, Math. Res. Center, Univ. of Wisconsin-Madison, Academic Press, New York, 1974, pp. 147–169. MR 0658322
Additional Information
- © Copyright 1976 American Mathematical Society
- Journal: Math. Comp. 30 (1976), 703-723
- MSC: Primary 65M05
- DOI: https://doi.org/10.1090/S0025-5718-1976-0443362-6
- MathSciNet review: 0443362