Dissipative twofour methods for timedependent problems
Authors:
David Gottlieb and Eli Turkel
Journal:
Math. Comp. 30 (1976), 703723
MSC:
Primary 65M05
MathSciNet review:
0443362
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Abstract: A generalization of the LaxWendroff method is presented. This generalization bears the same relationship to the twostep Richtmyer method as the KreissOliger 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 Thommentype algorithm. Numerical results show that the phase error is considerably reduced from that of secondorder methods and is similar to that of the KreissOliger method. Furthermore, the (2, 4) dissipative scheme can handle shocks without the necessity for an artificial viscosity.
 [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
(52 #9628)
 [2]
S. Z. BURSTEIN, "High order accurate difference methods in hydrodynamics," Nonlinear Partial Differential Equations, W. F. Ames, Editor, Academic Press, New York, 1967, pp. 279290. MR 36 #510.
 [3]
W. P. CROWLEY, "Numerical advection experiments," Monthly Weather Review, v. 96, 1968, pp. 111.
 [4]
B.
Fornberg, On the instability of leapfrog and
CrankNicolson approximations of a nonlinear partial differential
equation, Math. Comp. 27 (1973), 45–57. MR 0395249
(52 #16046), http://dx.doi.org/10.1090/S00255718197303952492
 [5]
J. GAZDAG, "Numerical convective schemes based on accurate computation of space derivatives," J. Computational Phys., v. 13, 1973, pp. 100113.
 [6]
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. 637643.
 [7]
M.
K. Gol′dberg, Multigraphs with a chromatic index that is
nearly maximal, Diskret. Analiz 23 (1973), 3–7,
72 (Russian). A collection of articles dedicated to the memory of
Vitaliĭ\ Konstantinovič Korobkov. MR 0354429
(50 #6907)
 [8]
David
Gottlieb, Strangtype difference schemes for multidimensional
problems, SIAM J. Numer. Anal. 9 (1972),
650–661. MR 0314274
(47 #2826)
 [9]
HeinzOtto
Kreiss and Joseph
Oliger, Comparison of accurate methods for the integration of
hyperbolic equations, Tellus 24 (1972), 199–215
(English, with Russian summary). MR 0319382
(47 #7926)
 [10]
H.O. KREISS & J. OLIGER, Methods for the Approximate Solution of Time Dependent Problems, Global Atmospheric Research Programme Publications Series, no. 10, 1973.
 [11]
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 0170484
(30 #722)
 [12]
Alain
Lerat and Roger
Peyret, Noncentered schemes and shock propagation problems,
Internat. J. Comput. & Fluids 2 (1974), no. 1,
35–52. MR
0363148 (50 #15586)
 [13]
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), SpringerVerlag, Lecture Notes in Phys., vol. 8, 1970, pp. 151163. MR 43 #4216.
 [14]
G.
R. McGuire and J.
Ll. Morris, A class of secondorder accurate methods for the
solution of systems of conservation laws, J. Computational Phys.
11 (1973), 531–549. MR 0331808
(48 #10140)
 [15]
G.
I. Marchuk, On the theory of the splittingup 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
(44 #1234)
 [16]
Joseph
Oliger, Fourth order difference methods for
the initial boundaryvalue problem for hyperbolic equations, Math. Comp. 28 (1974), 15–25. MR 0359344
(50 #11798), http://dx.doi.org/10.1090/S00255718197403593447
 [17]
Robert
D. Richtmyer and K.
W. Morton, Difference methods for initialvalue problems,
Second edition. Interscience Tracts in Pure and Applied Mathematics, No. 4,
Interscience Publishers John Wiley & Sons, Inc., New
YorkLondonSydney, 1967. MR 0220455
(36 #3515)
 [18]
K.
V. Roberts and N.
O. Weiss, Convective difference
schemes, Math. Comp. 20 (1966), 272–299. MR 0198702
(33 #6857), http://dx.doi.org/10.1090/S00255718196601987026
 [19]
Gilbert
Strang, On the construction and comparison of difference
schemes, SIAM J. Numer. Anal. 5 (1968),
506–517. MR 0235754
(38 #4057)
 [20]
Hans
U. Thommen, Numerical integration of the NavierStokes
equations, Z. Angew. Math. Phys. 17 (1966),
369–384 (English, with German summary). MR 0205560
(34 #5387)
 [21]
E.
Turkel, Symmetric hyperbolic difference schemes and matrix
problems, Linear Algebra and Appl. 16 (1977),
no. 2, 109–129. MR 0464603
(57 #4530)
 [22]
Eli
Turkel, Composite methods for hyperbolic equations, SIAM J.
Numer. Anal. 14 (1977), no. 4, 744–759. MR 0443365
(56 #1735)
 [23]
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), Math. Res. Center,
Univ. of WisconsinMadison, Academic Press, New York, 1974,
pp. 147–169. Publication No. 33. MR 0658322
(58 #31929)
 [1]
 S. ABARBANEL, D. GOTTLIEB & E. TURKEL, "Difference schemes with fourth order accuracy for hyperbolic equations," SIAM J. Appl. Math., v. 29, 1975, pp. 329351. MR 0388794 (52:9628)
 [2]
 S. Z. BURSTEIN, "High order accurate difference methods in hydrodynamics," Nonlinear Partial Differential Equations, W. F. Ames, Editor, Academic Press, New York, 1967, pp. 279290. MR 36 #510.
 [3]
 W. P. CROWLEY, "Numerical advection experiments," Monthly Weather Review, v. 96, 1968, pp. 111.
 [4]
 B. FORNBERG, "On the instability of leapfrog and CrankNicholson approximations of a nonlinear partial differential equation," Math. Comp., v. 27, 1973, pp 4557. MR 0395249 (52:16046)
 [5]
 J. GAZDAG, "Numerical convective schemes based on accurate computation of space derivatives," J. Computational Phys., v. 13, 1973, pp. 100113.
 [6]
 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. 637643.
 [7]
 M. GOLDBERG. (To appear.) MR 0354429 (50:6907)
 [8]
 D. GOTTLIEB, "Strangtype difference schemes for multidimensional problems," SIAM J. Numer. Anal., v. 9, 1972, pp. 650661. MR 47 #2826. MR 0314274 (47:2826)
 [9]
 H.O. KREISS & J. OLIGER, "Comparison of accurate methods for the integration of hyperbolic equations," Tellus, v. 24, 1972, pp. 199215. MR 47 #7926. MR 0319382 (47:7926)
 [10]
 H.O. KREISS & J. OLIGER, Methods for the Approximate Solution of Time Dependent Problems, Global Atmospheric Research Programme Publications Series, no. 10, 1973.
 [11]
 P. D. LAX & B. WENDROFF, "Difference schemes for hyperbolic equations with high order of accuracy," Comm. Pure Appl. Math., v. 17, 1964, pp. 381398. MR 30 #722. MR 0170484 (30:722)
 [12]
 A. LERAT & R. PEYRET, "Noncentered schemes and shock propagation problems," Internat. J. Comput. & Fluids, v. 2, 1974, pp. 3552. MR 50 #15586. MR 0363148 (50:15586)
 [13]
 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), SpringerVerlag, Lecture Notes in Phys., vol. 8, 1970, pp. 151163. MR 43 #4216.
 [14]
 G. R. McGUIRE & J. Ll. MORRIS, "A class of secondorder accurate methods for the solution of systems of conservation laws," J. Computational Phys., v. 11, 1973, pp. 531549. MR 48 #10140. MR 0331808 (48:10140)
 [15]
 G. I. MARČUK, "On the theory of the splittingup method," Numerical Solution of Partial Differential Equations, II (SYNSPADE 1970), B. Hubbard, Editor, (Proc. Sympos., Univ. of Maryland, College Park, Md., 1970), Academic Press, New York, 1971, pp. 469500. MR 44 #1234. MR 0284004 (44:1234)
 [16]
 J. OLIGER, "Fourth order difference methods for the initial boundaryvalue problem for hyperbolic equations," Math. Comp., v. 28, 1974, pp. 1525. MR 50 #11798. MR 0359344 (50:11798)
 [17]
 R. D. RICHTMYER & K. W. MORTON, Difference Methods for InitialValue Problems, 2nd ed., Interscience Tracts in Pure and Appl. Math., no. 4, Interscience, New York, 1967. MR 36 #3515. MR 0220455 (36:3515)
 [18]
 K. V. ROBERTS & N. O. WEISS, "Convective difference schemes," Math. Comp., v. 20, 1966, pp. 272299. MR 33 #6857. MR 0198702 (33:6857)
 [19]
 G. W. STRANG, "On the construction and comparison of difference schemes," SIAM J. Numer. Anal., v. 5, 1968, pp. 506517. MR 38 #4057. MR 0235754 (38:4057)
 [20]
 H. U. THOMMEN, "Numerical integration of the NavierStokes equations," Z. Angew Math. Phys., v. 17, 1966, pp. 369384. MR 34 #5387. MR 0205560 (34:5387)
 [21]
 E. TURKEL, "Symmetric hyperbolic difference schemes," Linear Algebra and Appl. (To appear.) MR 0464603 (57:4530)
 [22]
 E. TURKEL, "Composite methods for hyperbolic equations," SIAM J. Numer. Anal. (To appear.) MR 0443365 (56:1735)
 [23]
 L. 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., Univ. of Wisconsin, C. de Boor, Editor), Publ. No. 33 of the Mathematics Research Center, Univ. of Wisconsin, Academic Press, New York, 1974, pp. 147169. MR 50 #1525. MR 0658322 (58:31929)
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DOI:
http://dx.doi.org/10.1090/S00255718197604433626
PII:
S 00255718(1976)04433626
Article copyright:
© Copyright 1976
American Mathematical Society
