The spectrum and the stability of the Chebyshev collocation operator for transonic flow
Author:
Dalia Fishelov
Journal:
Math. Comp. 51 (1988), 559579
MSC:
Primary 65M10; Secondary 76H05
MathSciNet review:
930225
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Abstract: The extension of spectral methods to the small disturbance equation of transonic flow is considered. It is shown that the real parts of the eigenvalues of its spatial operator are nonpositive. Two schemes are considered; the first is spectral in the x and y variables, while the second is spectral in x and of second order in y. Stability for the second scheme is proved. Similar results hold for the twodimensional heat equation.
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 W. F. Ballhaus & P. M. Goorjian, Implicit Finite Difference Computation of Unsteady Transonic Flow about Airfoils, Including the Treatment of Irregular Shock Wave Methods, AIAA paper no. 77205, (1977).
 [2]
 J. D. Cole, "Modern developments in Transonic flow," SIAM J. Appl. Math., v. 29, 1975, pp. 763787. MR 0386435 (52:7289)
 [3]
 J. D. Cole & A. F. Messiter, "Expansion procedures and similarity laws for transonic flow, Part I: Slender bodies at zero incidence," Z. Angew. Math. Phys., v. 8, 1959, pp. 125. MR 0085034 (18:966j)
 [4]
 B. Engquist & S. Osher, "Stable and entropy satisfying approximations for transonic flow calculations," Math. of Comp., v. 34, 1980, pp. 4575. MR 551290 (81b:65082)
 [5]
 D. Fishelov, Application of Spectral Methods to Time Dependent Problems with Application to Transonic Flows, Ph.D. Thesis, Tel Aviv University, 1985.
 [6]
 D. Fishelov, "Spectral methods for the small disturbance equation of transonic flows," SIAM J. Sci. Statist. Comput., v. 9, 1988, pp. 232251. MR 930043 (89h:76025)
 [7]
 D. Gottlieb, "The stability of pseudospectral Chebyshev methods," Math. Comp., v. 36, 1981, pp. 107118. MR 595045 (82b:65123)
 [8]
 D. Gottlieb, "Strangtype difference schemes for multidimensional problems," SIAM J. Numer. Anal., v. 9, 1972, pp. 650661. MR 0314274 (47:2826)
 [9]
 D. Gottlieb & L. Lustman, "The spectrum of Chebyshev collocation operator for the heat equation," SIAM J. Numer. Anal., v. 20, 1983, pp. 909921. MR 714688 (85g:65107)
 [10]
 D. Gottlieb & S. Orszag, Numerical Analysis of Spectral Methods: Theory and Applications, CBMS Regional Conference Series in Applied Mathematics, SIAM, Philadelphia, Pa., 1977. MR 0520152 (58:24983)
 [11]
 D. Gottlieb & E. Turkel, private communications, 1983.
 [12]
 A. Jameson, "Numerical solution of nonlinear partial differential equations of mixed type," in: Numerical Solution of Partial Differential Equations III, Academic Press, New York, 1976, pp. 275307. MR 0468255 (57:8093)
 [13]
 E. Murman & J. Cole, "Calculations of plane steady transonic flows," AIAA J., v. 9, 1971, pp. 114121.
 [14]
 T. N. Phillipps, T. A. Zang & M. Y. Haussaini, "Preconditioners for the spectral multigrid method," IMA J. Numer. Anal., v. 6, 1986, pp. 273292. MR 967669 (89h:65063)
 [15]
 A. Solomonoff & E. Turkel, Global Collocation Methods for Approximation and the Solution of Partial Differential Equations, ICASE Report No. 8660, 1986.
 [16]
 H. TalEzer, "A pseudospectral Legendre method for hyperbolic equations with an improved stability condition," J. Comput. Phys., v. 67, 1986, pp. 145172. MR 867974 (88f:65209)
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Additional Information
DOI:
http://dx.doi.org/10.1090/S00255718198809302250
PII:
S 00255718(1988)09302250
Keywords:
Spectral methods,
transonic flow,
stability,
Chebyshev polynomials,
eigenvalue problems
Article copyright:
© Copyright 1988
American Mathematical Society
