Accurate spectral collocation solutions to 2nd-order Sturm–Liouville problems

Abstract

This work is about the use of some classical spectral collocation methods as well as with the new software system Chebfun in order to compute the eigenpairs of some high order Sturm–Liouville eigenproblems. The analysis is divided into two distinct directions. For problems with clamped boundary conditions, we use the preconditioning of the spectral collocation differentiation matrices and for hinged end boundary conditions the equation is transformed into a second order system and then the conventional ChC is applied. A challenging set of “hard” benchmark problems, for which usual numerical methods (FD, FE, shooting, etc.) encounter difficulties or even fail, are analyzed in order to evaluate the qualities and drawbacks of spectral methods. In order to separate ‘‘good” and “bad” (spurious) eigenvalues, we estimate the drift of the set of eigenvalues of interest with respect to the order of approximation N. This drift gives us a very precise indication of the accuracy with which the eigenvalues are computed, i.e., an automatic estimation and error control of the eigenvalue error. Two MATLAB codes models for spectral collocation (ChC and SiC) and another for Chebfun are provided. They outperform the old codes used so far and can be easily modified to solve other problems.

Authors

Calin-Ioan Gheorghiu
“Tiberiu Popoviciu” Institute of Numerical Analysis, Romanian Academy

Keywords

Sturm–Liouville; clamped; hinged boundary condition; spectral collocation; Chebfun; chebop; eigenpairs; preconditioning; drift; error control

References

see the expanding block below

PDF

final version

arXiv:2012.01341v1

Cite this paper as:

C.I. Gheorghiu, Accurate spectral collocation solutions to 2nd-order Sturm–Liouville problems, Symmetry, 13 (2021) 3, 385, doi: 10.3390/sym13030385

About this paper

Journal

Symmetry

Publisher Name

MDPI

Print ISSN

Not available yet.

Online ISSN

2073-8994

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