The Asymptotic Eigenvalues of First-Order Spectral Differentiation Matrices
Jue Wang, Fabian Waleffe · Journal of Applied Mathematics and Physics · 2014
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.