The spectrum and the stability of the Chebyshev collocation operator for transonic flow. (English) Zbl 0699.76070
Summary: 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 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.
MSC:
| 76H05 | Transonic flows |
| 33C45 | Orthogonal polynomials and functions of hypergeometric type (Jacobi, Laguerre, Hermite, Askey scheme, etc.) |
| 35P15 | Estimates of eigenvalues in context of PDEs |
Keywords:
extension of spectral methods; small disturbance equation of transonic flow; eigenvalues; spatial operatorReferences:
| [1] | 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. 77-205, (1977). · Zbl 0394.76008 |
| [2] | Julian D. Cole, Modern developments in transonic flow, SIAM J. Appl. Math. 29 (1975), no. 4, 763 – 787. · Zbl 0324.76048 · doi:10.1137/0129065 |
| [3] | 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. · Zbl 0077.19003 · doi:10.1007/BF01601152 |
| [4] | Björn Engquist and Stanley Osher, Stable and entropy satisfying approximations for transonic flow calculations, Math. Comp. 34 (1980), no. 149, 45 – 75. · Zbl 0438.76051 |
| [5] | D. Fishelov, Application of Spectral Methods to Time Dependent Problems with Application to Transonic Flows, Ph.D. Thesis, Tel Aviv University, 1985. |
| [6] | Dalia Fishelov, Spectral methods for the small disturbance equation of transonic flows, SIAM J. Sci. Statist. Comput. 9 (1988), no. 2, 232 – 251. · Zbl 0657.76056 · doi:10.1137/0909015 |
| [7] | David Gottlieb, The stability of pseudospectral-Chebyshev methods, Math. Comp. 36 (1981), no. 153, 107 – 118. · Zbl 0469.65076 |
| [8] | David Gottlieb, Strang-type difference schemes for multidimensional problems, SIAM J. Numer. Anal. 9 (1972), 650 – 661. · Zbl 0272.65074 · doi:10.1137/0709054 |
| [9] | 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. · Zbl 0537.65085 · doi:10.1137/0720063 |
| [10] | David Gottlieb and Steven A. Orszag, Numerical analysis of spectral methods: theory and applications, Society for Industrial and Applied Mathematics, Philadelphia, Pa., 1977. CBMS-NSF Regional Conference Series in Applied Mathematics, No. 26. · Zbl 0412.65058 |
| [11] | D. Gottlieb & E. Turkel, private communications, 1983. |
| [12] | 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. |
| [13] | E. Murman & J. Cole, ”Calculations of plane steady transonic flows,” AIAA J., v. 9, 1971, pp. 114-121. · Zbl 0249.76033 |
| [14] | 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. · Zbl 0624.65119 · doi:10.1093/imanum/6.3.273 |
| [15] | A. Solomonoff & E. Turkel, Global Collocation Methods for Approximation and the Solution of Partial Differential Equations, ICASE Report No. 86-60, 1986. |
| [16] | Hillel Tal-Ezer, A pseudospectral Legendre method for hyperbolic equations with an improved stability condition, J. Comput. Phys. 67 (1986), no. 1, 145 – 172. · Zbl 0613.65092 · doi:10.1016/0021-9991(86)90119-1 |
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