The Asymptotic Eigenvalues of First-Order Spectral Differentiation Matrices

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.

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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.

Conflicts of Interest

The authors declare no conflicts of interest.

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