A thirdorder accurate variation nonexpansive difference scheme for single nonlinear conservation laws
Author:
Richard Sanders
Journal:
Math. Comp. 51 (1988), 535558
MSC:
Primary 65M10; Secondary 35L65
MathSciNet review:
935073
Fulltext PDF Free Access
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Abstract: It was widely believed that all variation nonexpansive finite difference schemes for single conservation laws must reduce to firstorder at extreme points of the approximation. It is shown here that this belief is in fact false. A thirdorder scheme, which at worst may reduce to second order at extreme points, is developed and analyzed. Moreover, extensive numerical experiments indicate that the thirdorder scheme introduced here yields superior approximations when compared with other variation nonexpansive difference schemes.
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 [1]
 M. BenArtzi & J. Falcovitz, "A secondorder Godunovtype scheme for compressible fluid dynamics," J. Comput. Phys., v. 55, 1984, pp. 132. MR 757422 (86f:65146)
 [2]
 S. R. Chakravarthy, A. Harten & S. Osher, Essentially NonOscillatory ShockCapturing Schemes of Arbitrarily High Accuracy, AIAA 24th Aerospace Sciences Meeting, January 69, 1986, Reno, Nevada.
 [3]
 P. Colella & P. R. Woodward, "The piecewiseparabolic method (PPM) for gasdynamical simulations," J. Comput. Phys., v. 54, 1984, pp. 174201.
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 S. D. Conte & C. de Boor, Elementary Numerical Analysis, 3rd ed., McGrawHill, New York, 1980.
 [5]
 M. Crandall & A. Majda, "Monotone difference approximations for scalar conservation laws," Math. Comp., v. 34, 1980, pp. 122. MR 551288 (81b:65079)
 [6]
 S. K. Godunov, "A finite difference method for the numerical computation of discontinuous solutions of the equations of fluid dynamics," Mat. Sb., v. 47, 1959, pp. 271295. MR 0119433 (22:10194)
 [7]
 A. Harten, "High resolution schemes for hyperbolic conservation laws," J. Comput. Phys., v. 49, 1983, pp. 357393. MR 701178 (84g:65115)
 [8]
 A. Harten, B. Engquist, S. Osher & S. R. Chakravarthy, "Uniformly high order accurate essentially nonoscillatory schemes III," J. Comput. Phys., v. 71, 1987, pp. 231303. MR 897244 (90a:65199)
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 P. D. Lax, "Shock waves and entropy," Contributions to Nonlinear Functional Analysis (E. H. Zarantonello, ed.), Academic Press, New York, 1971, pp. 603634. MR 0393870 (52:14677)
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 [13]
 R. Sanders, "The moving grid method for nonlinear hyperbolic conservation laws," SIAM J. Numer. Anal., v. 22, 1985, pp. 713728. MR 795949 (87f:65110)
 [14]
 P. K. Sweby, "High resolution schemes using flux limiters for hyperbolic conservation laws," SIAM J. Numer. Anal., v. 21, 1984, pp. 9951011. MR 760628 (85m:65085)
 [15]
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Additional Information
DOI:
http://dx.doi.org/10.1090/S00255718198809350733
PII:
S 00255718(1988)09350733
Article copyright:
© Copyright 1988
American Mathematical Society
