The spectrum and the stability of the Chebyshev collocation operator for transonic flow

Fishelov, Dalia

doi:10.1090/S0025-5718-1988-0930225-0

The spectrum and the stability of the Chebyshev collocation operator for transonic flow
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Math. Comp. 51 (1988), 559-579 Request permission

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 two-dimensional heat equation.

References

Implicit Finite Difference Computation of Unsteady Transonic Flow about Airfoils, Including the Treatment of Irregular Shock Wave Methods

Julian D. Cole, Modern developments in transonic flow, SIAM J. Appl. Math. 29 (1975), no. 4, 763–787. MR 386435, DOI 10.1137/0129065
Julian D. Cole and Arthur F. Messiter, Expansion procedures and similarity laws for transonic flow. I. Slender bodies at zero incidence, Z. Angew. Math. Phys. 8 (1957), 1–25. MR 85034, DOI 10.1007/BF01601152
Björn Engquist and Stanley Osher, Stable and entropy satisfying approximations for transonic flow calculations, Math. Comp. 34 (1980), no. 149, 45–75. MR 551290, DOI 10.1090/S0025-5718-1980-0551290-1

Application of Spectral Methods to Time Dependent Problems with Application to Transonic Flows

Dalia Fishelov, Spectral methods for the small disturbance equation of transonic flows, SIAM J. Sci. Statist. Comput. 9 (1988), no. 2, 232–251. MR 930043, DOI 10.1137/0909015
David Gottlieb, The stability of pseudospectral-Chebyshev methods, Math. Comp. 36 (1981), no. 153, 107–118. MR 595045, DOI 10.1090/S0025-5718-1981-0595045-1
David Gottlieb, Strang-type difference schemes for multidimensional problems, SIAM J. Numer. Anal. 9 (1972), 650–661. MR 314274, DOI 10.1137/0709054
David Gottlieb and Liviu Lustman, The spectrum of the Chebyshev collocation operator for the heat equation, SIAM J. Numer. Anal. 20 (1983), no. 5, 909–921. MR 714688, DOI 10.1137/0720063
David Gottlieb and Steven A. Orszag, Numerical analysis of spectral methods: theory and applications, CBMS-NSF Regional Conference Series in Applied Mathematics, No. 26, Society for Industrial and Applied Mathematics, Philadelphia, Pa., 1977. MR 0520152
Antony Jameson, Numerical solution of nonlinear partial differential equations of mixed type, Numerical solution of partial differential equations, III (Proc. Third Sympos. (SYNSPADE), Univ. Maryland, College Park, Md., 1975) Academic Press, New York, 1976, pp. 275–320. MR 0468255

AIAA J.

Timothy N. Phillips, Thomas A. Zang, and M. Yousuff Hussaini, Preconditioners for the spectral multigrid method, IMA J. Numer. Anal. 6 (1986), no. 3, 273–292. MR 967669, DOI 10.1093/imanum/6.3.273

Global Collocation Methods for Approximation and the Solution of Partial Differential Equations

Hillel Tal-Ezer, A pseudospectral Legendre method for hyperbolic equations with an improved stability condition, J. Comput. Phys. 67 (1986), no. 1, 145–172. MR 867974, DOI 10.1016/0021-9991(86)90119-1