JAMP  Vol.2 No.5 , April 2014
The Asymptotic Eigenvalues of First-Order Spectral Differentiation Matrices
Abstract: We complete and extend the asymptotic analysis of the spectrum of Jacobi Tau approximations that were first considered by Dubiner. The asymptotic formulas for Jacobi polynomials PN(α ,β ) ,α ,β > -1 are derived and confirmed by numerical approximations. More accurate results for the slowest decaying mode are obtained. We explain where the large negative eigenvalues come from. Furthermore, we show that a large negative eigenvalue of order N2 appears for -1 <α < 0 ; there are no large negative eigenvalues for collocations at Gauss-Lobatto points. The asymptotic results indicate unstable eigenvalues for α > 1 . The eigenvalues for Legendre polynomials are directly related to the roots of the spherical Bessel and Hankel functions that are involved in solving Helmholtz equation inspherical coordinates.
Cite this paper: Wang, J. and Waleffe, F. (2014) The Asymptotic Eigenvalues of First-Order Spectral Differentiation Matrices. Journal of Applied Mathematics and Physics, 2, 176-188. doi: 10.4236/jamp.2014.25022.

[1]   Canuto, C., Hussaini, M.Y., Quarteroni, A. and Zang, T.A. (1988) Spectral Methods in Fluid Dynamics. Springer, New York.

[2]   Gottlieb, D. (1981) The Stability of Pseudospectral Chebyshev Methods. Mathematics of Computation, 36, 107-118.

[3]   Trefethen, L.N. and Embree, M. (2005) Spectra and Pseudospectra: The Behavior of Nonnormal Matrices and Operators. Princeton University Press.

[4]   Trefethen, L.N. and Trummer, M.R. (1987) An Instability Phenomenon in Spectral Methods. SIAM Journal on Numerical Analysis, 24, 1008-1023.

[5]   Weideman, J.A.C. and Trefethen, L.N. (1988) The Eigenvalues of Second-Order Spectral Differentiation Matrices. SIAM Journal on Numerical Analysis, 25, 1279-1298.

[6]   Dubiner, M. (1987) Asymptotic Analysis of Spectral Methods. Journal of Scientific Computing, 2, 3-31.

[7]   Tal-Ezer, H. (1986) Spectral Methods in Time for Hyperbolic Equations. SIAM Journal on Numerical Analysis, 23, 11-26.

[8]   Jackiewicz, Z. and Welfert, B.D. (2003) Stability of Gauss-RadauPseudospectral Approximations of the One-Dimensional Wave Equation. Journal of Scientific Computing, 18, 287-313.

[9]   Csordas, G., Charalambides, M. and Waleffe, F. (2005) A New Property of a Class of Jacobi Polynomials. Proceedings of the AMS, 133, 3351-3560.

[10]   Weideman, J.A.C. and Reddy, S.C. (2000) A Matlab Differentiation MatrixSuite. ACM Transactions on Mathematical Software, 26, 465-519.

[11]   Arfken, G.B. and Weber, H.J. (1995) Mathematical Methods for Physicists. Academic Press.

[12]   Szego, G. (1939) Orthogonal Polynomials. AMS Colloquium Publication, 23.

[13]   Waston, G.N. (1995) A Treatise on the Theory of Bessel Functions. 2nd Edition, Cambridge University Press.

[14]   Olver, F.W.J. (1970) Why Steepest Descents? SIAM Review, 12, 228-247.

[15]   Chester, C., Friedman, B. and Ursell, F. (1957) An Extension of The Method of Steepest Descents. Proc. Cambridge Philos. Soc., 53, 599-611.

[16]   Driver, K.A. and Temme, N.M. (1999) Zero and Pole Distribution of Diagonal Padé Approximants to the Exponential Function. Questiones Mathematicae, 22, 7-17.

[17]   Abramowitz, M. and Stegun, I.A. (Eds.) (1965) Handbook of Mathematical Functions: With Formulas, Graphs, and Mathematical Tables. Dover Publications.

[18]   Colton, D. and Kress, R. (1998) Inverse Acoustic and Electromagnetic Scattering Theory. 2nd Edition, Springer.

[19]   Doha, E.H. (2002) On the Coefficients of Eifferentiated Expansions and Derivatives of Jacobi Polynomials. J. Phys. A: Math. Gen., 35, 3467-3478.