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


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.


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


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


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

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1. Straughan, B.,  Stability of Wave Motion in Porous Media, Springer Science+Business Media: New York, NY, USA, 2008.
2. Trefethen, L.N., Birkisson, A.; Driscoll, T.A., Exploring ODEs; SIAM: Philadelphia, PA, USA, 2018.
3. Pruess, S., Fulton, C.T., Mathematical Software for Sturm-Liouville Problem. ACM Trans. Math. Softw. 1993, 19, 360–376.
4. Pruess, S., Fulton, C.T., Xie, Y., An Asymptotic Numerical Method for a Class of Singular Sturm-Liouville Problems. ACM SIAM J. Numer. Anal. 1995, 32, 1658–1676.
5. Marletta, M., Pryce, J.D., LCNO Sturm-Liouville problems. Computational difficulties and examples. Numer. Math. 1995, 69, 303–320.
6. Pryce, J.D., Marletta, M., A new multi-purpose software package for Schrödinger and Sturm–Liouville computations. Comput. Phys. Comm. 1991, 62, 42–54.
7. Bailey, P.B., Everitt, W.N., Zettl, A., Computing Eigenvalues of Singular Sturm-Liouville Problems. Results Math. 1991, 20, 391–423.
8. Bailey, P.B., Garbow, B., Kaper, H., Zettl, A., Algorithm 700: A FORTRAN software package for Sturm-Liouville problems. ACM Trans. Math. Softw. 1991, 17, 500–501.
9. Ledoux, V.,Van Daele, M., Vanden Berghe, G., MATSLISE: A MATLAB Package for the Numerical Solution of Sturm-Liouville and Schrödinger Equations. ACM Trans. Math. Softw. 2005, 31, 532–554.
10. Baeyens, T., Van Daele, M., The Fast and Accurate Computation of Eigenvalues and Eigenfunctions of Time-Independent One-Dimensional Schrödinger Equations. Comput. Phys. Commun. 2021, 258, 107568, doi:10.1016/j.cpc.2020.107568.
11. Abbasb, Y.S., Shirzadi, A., A new application of the homotopy analysis method: Solving the Sturm—Liouville problems. Commun. Nonlinear. Sci. Numer. Simulat. 2011, 16, 112—126.
12. Perera, U., Böckmann, C., Solutions of Direct and Inverse Even-Order Sturm-Liouville Problems Using Magnus Expansion. Mathematics 2019, 7, 544, doi:10.3390/math7060544.
13. Gheorghiu, C.I., Spectral Methods for Non-Standard Eigenvalue Problems: Fluid and Structural Mechanics and Beyond; Springer: Heidelberg, Germany, 2014.
14. Greenberg, G., Marletta, M., Oscillation Theory and Numerical Solution of Sixth Order Sturm-Liouville Problems. SIAM J. Numer. Anal. 1998 35, 2070–2098.
15. Everitt, W.N., A Catalogue of Sturm-Liouville Differential Equations. In Sturm-Liouville theory: Past and Present; Amrein, W.O., Hinz, A.M., Hinz, D.B., Eds.; Birkhäuser Verlag: Basel, Switzerland, 2005; pp. 271–331.
16. Driscoll, T.A., Bornemann, F., Trefethen, L.N., The CHEBOP System for Automatic Solution of Differential Equations. BIT 2008, 48, 701–723.
17. Driscoll, T.A., Hale, N., Trefethen, L.N., Chebfun Guide; Pafnuty Publications: Oxford, UK, 2014.
18. Driscoll, T.A., Hale, N., Trefethen, L.N., Chebfun-Numerical Computing with Functions. Available online: http://www.chebfun. org (accessed on 15 November 2019).
19. Gheorghiu, C.I., Spectral Collocation Solutions to Problems on Unbounded Domains; Casa Cartii de ¸Stiinta Publishing House: ClujNapoca, Romania, 2018.
20. Weideman, J.A.C.; Reddy, S.C., A MATLAB Differentiation Matrix Suite. ACM Trans. Math. Softw. 2000, 26, 465–519.
21. Gheorghiu, C.I., Pop, I.S., A Modified Chebyshev-Tau Method for a Hydrodynamic Stability Problem. In Proceedings of the International Conference on Approximation and Optimization; Stancu, D.D., Coman, G., Breckner, W.W., Blaga, P., Eds.; Transilvania Press: Cluj-Napoca, Romania, 1996; Volume II, pp. 119–126.
22. Boyd, J.P., Traps and Snares in Eigenvalue Calculations with Application to Pseudospectral Computations of Ocean Tides in a Basin Bounded by Meridians. J. Comput. Phys. 1996, 126, 11–20.
23. Gheorghiu, C.I,. Accurate Spectral Collocation Computation of High Order Eigenvalues for Singular Schrödinger Equations. Computation 2021, 9, 2. doi:10.3390/computation9010002.
24. Boyd, J.P., Chebyshev and Fourier Spectral Methods; Dover Publications: New York, NY, USA, 2000; pp. 127–158.
25. Huang, W.,  Sloan, D.M., The Pseudospectral Method for Solving Differential Eigenvalue Problems. J. Comput. Phys. 1994, 111, 399–409.
26. Straughan, B., The Energy Method, Stability, and Nonlinear Convection; Springer: New York, NY, USA, 1992; pp. 218–222.
27. Lesnic, D., Attili, S., An Efficient Method for Sixth-order Sturm-Liouville Problems. Int. J. Sci. Technol. 2007, 2, 109–114.
28. Gheorghiu, C.I., Rommes, J., Application of the Jacobi-Davidson method to accurate analysis of singular linear hydrodynamic stability problems. Int. J. Numer. Meth. Fluids 2013, 71, 358–369. doi:10.1002/fld.3669.
29. Gardner, D.R., Trogdon, S.A., Douglass, R.W., A Modified Spectral Tau Method That Eliminates Spurious Eigenvalues. J. Comput. Phys. 1989, 80, 137–167.
30. Fornberg, B., A Practical Guide to Pseudospectral Methods; Cambridge University Press: Cambridge, UK, 1998; pp. 89–90.
31. McFadden, G.B.; Murray, B.T.; Boisvert, R.F., Elimination of Spurious Eigenvalues in the Chebyshev Tau Spectral Method. J. Comput. Phys. 1990, 91, 228–239.
32. Mai-Duy, N., An effective spectral collocation method for the direct solution of high-order ODEs. Commun. Numer. Methods Eng. 2006, 22, 627–642.
33. Zhao, S., Wei, G.W., Xiang, Y., DSC analysis of free-edged beams by an iteratively matched boundary method. J. Sound. Vib. 2005, 284, 487–493.

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