The Asymptotic Eigenvalues of First-Order Spectral Differentiation Matrices

Jue Wang^1*, Fabian Waleffe²

¹Department of Mathematics, Union College, Schenectady, USA.
²Department of Mathematics, University of Wisconsin-Madison, Madison, USA.

DOI: 10.4236/jamp.2014.25022                

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.

Keywords

Asymptotic Analysis, Spectral Approximations, Jacobi Polynomials, Collocation, Eigenvalues

Conflicts of Interest

The authors declare no conflicts of interest.

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