TITLE:
The Asymptotic Eigenvalues of First-Order Spectral Differentiation Matrices
AUTHORS:
Jue Wang, Fabian Waleffe
KEYWORDS:
Asymptotic Analysis, Spectral Approximations, Jacobi Polynomials, Collocation, Eigenvalues
JOURNAL NAME:
Journal of Applied Mathematics and Physics,
Vol.2 No.5,
April
25,
2014
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 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.