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The origin and nature of spurious eigenvalues in the spectral tau method. (English) Zbl 0924.65077
The tau method, first proposed by Lanczos, is a means of solving boundary value problems for ordinary differential equations using truncated series expansions in a complete set of orthogonal functions. From the author’s summary: ‘The Chebyshev-tau spectral method for approximating eigenvalues of boundary value problems may produce spurious eigenvalues with large positive real parts, even when all true eigenvalues of the problem are known to have negative real parts. We explain the origin and nature of the ‘spurious eigenvalues’ in an example problem. The explanation will demonstrate that the large positive eigenvalues are an approximation of infinite eigenvalues in a nearby generalized eigenvalue problem’.

MSC:
65L15 Numerical solution of eigenvalue problems involving ordinary differential equations
65L10 Numerical solution of boundary value problems involving ordinary differential equations
34L15 Eigenvalues, estimation of eigenvalues, upper and lower bounds of ordinary differential operators
65L60 Finite element, Rayleigh-Ritz, Galerkin and collocation methods for ordinary differential equations
34B05 Linear boundary value problems for ordinary differential equations
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[1] Abramowitz, M.; Stegun, I., Handbook of mathematical functions, (1965)
[2] Andrews, L.C., Special functions for engineers and applied mathematicians, (1985)
[3] Boyd, J.P., Complex coordinate methods for hydrodynamic instabilities and sturm – liouville eigenproblems with an infinite singularity, J. comput. phys., 57, 454, (1985) · Zbl 0631.76038
[4] Boyd, J.P., Lecture notes in engineering, Chebyshev & Fourier spectral methods, 49, (1989)
[5] Brenier, B.; Roux, B.; Bontoux, P., Comparaison des méthodes tau – chebyshev et Galerkin dans l’étude de stabilité des mouvements de convection naturelle. problème des valeurs propres parasites, J. Méch. théor. appl., 5, 95, (1986) · Zbl 0595.76045
[6] Dawkins, P., Spurious eigenvalues in spectral – tau methods, (1997)
[7] Fox, L., Chebyshev methods for ordinary differential equations, Comput. J., 4, 318, (1962) · Zbl 0103.34203
[8] Fox, L.; Parker, I., Chebyshev polynomials in numerical analysis, (1968) · Zbl 0153.17502
[9] Gardner, D.B.; Trogdon, S.A.; Douglass, R.W., A modified tau spectral method that eliminates spurious eigenvalues, J. comput. phys., 80, 137, (1989) · Zbl 0661.65084
[10] Gardner, D.R., Linear stability and bifurcation of natural convection flows in narrow-gap concentric spherical annulus enclosures, (1988)
[11] Gottlieb, D.; Orszag, S.A., Numerical anlaysis of spectral methods, (1977)
[12] Gradshteyn, I.S.; Ryzhik, I.M., Tables of integrals, series, and products, (1969) · Zbl 0918.65002
[13] Huaung, W.; Sloan, D.M., The pseudospectral method for solving differential eigenvalue problems, J. comput. phys., 111, 399, (1994)
[14] Lanczos, C., Trigonometric interpolation of empirical and analytical functions, J. math. phys., 7, 123, (1938) · Zbl 0020.01301
[15] Lanczos, C., Applied analysis, (1964) · Zbl 0111.12403
[16] McFadden, G.B.; Murray, B.T.; Boisvert, R.F., Elimination of spurious eigenvalues in the Chebyshev tau spectral method, J. comput. phys., 91, 228, (1990) · Zbl 0717.65063
[17] Orszag, S.A., Accurate solution of the orr—sommerfeld stability equation, J. fluid mech., 50, 689, (1971) · Zbl 0237.76027
[18] Ortiz, E.L., The tau method, SIAM J. numer. anal., 6, 480, (1969) · Zbl 0195.45701
[19] Rivlin, T.J., Chebyshev polynomials, (1990) · Zbl 0871.41022
[20] Stewart, G.W.; Sun, J.-G., Matrix pertubation theory, (1990)
[21] Straughan, B.; Walker, D.W., Two very accurate and efficient methods for computing eigenvalues and eigenfunctions in porous convection problems, J. comput. phys., 127, 128, (1996) · Zbl 0858.76064
[22] Su, Y.Y.; Khomami, B., Numerical solution of eigenvalue problems using spectral techniques, J. comput. phys., 100, 297, (1992) · Zbl 0757.76052
[23] Tebeest, K., Linear stability and bifurcation of natural convection flows within arbitrary-gap spherical annuli, (1992)
[24] Zebib, A., A Chebyshev method for the solution of boundary value problems, J. comput. phys., 53, 443, (1984) · Zbl 0541.76036
[25] Zebib, A., Removal of spurious modes encountered in solving stability problems by spectral methods, J. comput. phys., 70, 521, (1987) · Zbl 0612.65049
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