Essentially nonoscillatory spectral Fourier methods for shock wave calculations
Authors:
Wei Cai, David Gottlieb and ChiWang Shu
Journal:
Math. Comp. 52 (1989), 389410
MSC:
Primary 65M99
MathSciNet review:
955749
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Abstract: In this paper, we present an essentially nonoscillatory spectral Fourier method for the solution of hyperbolic partial differential equations. The method is based on adding a nonsmooth function to the trigonometric polynomials which are the usual basis functions for the Fourier method. The high accuracy away from the shock is enhanced by using filters. Numerical results confirm that essentially no oscillations develop in the solution. Also, the accuracy of the spectral solution of the inviscid Burgers equation is shown to be higher than a fixed order.
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 [3]
 M. Crandall & A. Majda, "Monotone difference approximations for scalar conservation laws," Math. Comp., v. 34, 1980, pp. 121. MR 551288 (81b:65079)
 [4]
 A. Harten, B. Engquist, S. Osher & S. Chakravarthy, "Uniformly high order accurate nonoscillatory schemes, III," J. Comput. Phys., v. 71, 1987, pp. 231303. MR 897244 (90a:65199)
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 A. Harten, Preliminary Results on the Extension of ENO Schemes to TwoDimensional Problems, Proc. Internat. Conf. on Hyperbolic Problems, SaintEtienne, January 1986. MR 910102 (88k:65085)
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 M. Hussaini, D. Kopriva, M. Salas & T. Zang, "Spectral method for Euler equation, Part 1: Fourier method and shock capturing," AIAA J., v. 23, 1985, pp. 234240.
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 D. Kopriva, "A practical assessment of spectral accuracy for hyperbolic problems with discontinuity," J. Sci. Comput., v. 2, 1987, pp. 249262.
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 C. Lanczos, Discourse on Fourier Series, Oliver & Boyd, Edinburgh, 1966. MR 0199629 (33:7772)
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 A. Majda, J. McDonough & S. Osher, "The Fourier method for nonsmooth initial data," Math. Comp., v. 32, 1978, pp. 10411081. MR 501995 (80a:65197)
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 B. McDonald, "Flux corrected pseudospectral methods for scalar hyperbolic conservation laws," J. Comput. Phys. (To appear.) MR 1003490 (90g:65114)
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 S. Osher, "Convergence of generalized MUSCL schemes," SIAM J. Numer. Anal., v. 22, 1985, pp. 947961. MR 799122 (87b:65147)
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 C.W. Shu, "TVB uniformly highorder schemes for conservation laws," Math. Comp., v. 49, 1987, pp. 105121. MR 890256 (89b:65208)
 [13]
 C.W. Shu & S. Osher, "Efficient implementation of essentially nonoscillatory shockcapturing schemes," J. Comput. Phys., v. 77, 1988, pp. 439471. MR 954915 (89g:65113)
 [14]
 A. Zygmund, Trigonometric Series, v. 1, Cambridge Univ. Press, New York, 1959.
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Additional Information
DOI:
http://dx.doi.org/10.1090/S00255718198909557492
PII:
S 00255718(1989)09557492
Article copyright:
© Copyright 1989
American Mathematical Society
