×

Correction of eigenvalues estimated by the Legendre-Gauss tau method. (English) Zbl 1277.65061

The Legendre-Gauss tau method is proved to be effective for the estimation of a large number of eigenvalues, even when the corresponding eigenfunctions feature a sharp oscillatory behavior.
The authors develop an exponentially fitted version of this method suitable for higher-order ordinary differential equations. Moreover, efficient formulas to correct the estimated eigenvalues are developed. Several numerical examples illustrate the obtained results and methods.

MSC:

65L15 Numerical solution of eigenvalue problems involving ordinary differential equations
65L70 Error bounds for numerical methods for ordinary differential equations
34L16 Numerical approximation of eigenvalues and of other parts of the spectrum of ordinary differential operators
PDFBibTeX XMLCite
Full Text: DOI

References:

[1] Anastassi, Z.A., Simos, T.E.: A family of exponentially-fitted Runge-Kutta methods with exponential order up to three for the numerical solution of the Schr¨odinger equation. J. Math. Chem. 41, 79-100 (2007) · Zbl 1125.81017 · doi:10.1007/s10910-006-9071-3
[2] Anderssen, R.S., De Hoog, F.R.: On the correction of finite difference eigenvalue approximations for Sturm-Liouville problems with general boundary conditions. BIT 24, 401-412 (1984) · Zbl 0552.65065 · doi:10.1007/BF01934900
[3] Canuto, C., Hussaini, M.Y., Quarteroni, A., Zang, T.A.: Spectral Methods Fundamentals in Single Domain. Springer, Berlin (2006) · Zbl 1093.76002
[4] El-Daou, M.K.: Spectral corrections for a class of eigenvalue problems. In: Proc. 6th IMT-GT ICMSA2010, Kuala Lumpur, pp. 131-143 (2010) · Zbl 1196.81127
[5] El-Daou, M.K.: Exponentially weighted Legendre-Gauss Tau methods for linear second-order differential equations. Comput. Math. Appl. 62, 51-64 (2011) · Zbl 1228.65108 · doi:10.1016/j.camwa.2011.04.045
[6] El-Daou, M.K., Al Matar, N.R.: An improved tau method for a class of Sturm-Liouville problems. Appl. Math. Comput. 216, 1923-1937 (2010) · Zbl 1220.65104 · doi:10.1016/j.amc.2010.03.022
[7] Ghelardoni, P., Gheri, G., Marletta, M.: Numerical solution of a λ-rational Sturm-Liouville problem. IMA J. Numer. Anal. 23, 29-53 (2003) · Zbl 1019.65055 · doi:10.1093/imanum/23.1.29
[8] Gheri, G., Aceto, L., Ghelardoni, P.: A polynomial characterization of spectral error estimation for Sturm-Liouville problems. In: NAOF, Gent Univ. (2008) · Zbl 1138.65063
[9] Ghelardoni, P., Gheri, G., Marletta, M.: A polynomial approach to the spectral corrections for Sturm-Liouville problems. J. Comput. Appl. Math. 185, 360-376 (2006) · Zbl 1092.65064 · doi:10.1016/j.cam.2005.03.016
[10] Gottlieb, D., Orszag, S.: Numerical Analysis of Spectral Methods: Theory and Applications. Series in Appl. Math. SIAM, Philadelphia (1977) · Zbl 0412.65058 · doi:10.1137/1.9781611970425
[11] Ixaru, L.Gr.: Numerical Methods for Differential Equations and Applications. Reidel, Dordrecht (1984) · Zbl 0543.65047
[12] Ixaru, L.Gr.: CP methods for Schrödinger equation. J. Comput. Appl. Math. 125, 347-357 (2000) · Zbl 0971.65067 · doi:10.1016/S0377-0427(00)00478-7
[13] Ixaru, L.Gr., Vanden Berghe, G.: Exponential Fitting. Kluwer Academic, Dordrecht (2004) · Zbl 1105.65082
[14] Lanczos, C.: Applied Analysis. Prentice-Hall, Englewood Cliffs (1956) · Zbl 0111.12403
[15] Ledoux, V., Rizea, M., Ixaru, L., Vanden Berghe, G., Van Daele, M.: Solution of the Schrödinger equation by a high order perturbation method based on linear reference potential. Comput. Phys. Commun. 175, 424-439 (2006) · Zbl 1196.81127 · doi:10.1016/j.cpc.2006.06.005
[16] Ledoux, V., Van Daele, M.: Solving Sturm-Liouville problems by piecewise perturbation methods, revisited. Comput. Phys. Commun. 181, 1335-1345 (2010) · Zbl 1219.65075 · doi:10.1016/j.cpc.2010.03.017
[17] Ortiz, E.L.: The Tau method. SIAM J. Numer. Anal. 6, 480-492 (1969) · Zbl 0195.45701 · doi:10.1137/0706044
[18] Paine, J.: Correction of Sturm-Liouville eigenvalue estimates. Math. Comput. 160, 415-420 (1982) · Zbl 0493.65034
[19] Pryce, J.D.: Numerical Solution of Sturm-Liouville Problems. Monographs on Numerical Analysis, Oxford University Press (1993) · Zbl 0795.65053
[20] Vanden Berghe, G., De Meyer, H.: On a correction of Numerov-like eigenvalue approximations for Sturm-Liouville problems. J. Comput. Appl. Math. 37, 179-186 (1991) · Zbl 0753.65071 · doi:10.1016/0377-0427(91)90116-2
[21] Vanden Berghe, G., De Meyer, H.: A finite element estimate with trigonometric hat functions for Sturm-Liouville eigenvalues. J. Comput. Appl. Math. 53, 389-396 (1994) · Zbl 0815.65099 · doi:10.1016/0377-0427(94)90066-3
[22] Vanden Berghe, G., Van Daele, M.: Symplectic exponentially-fitted four-stage Runge-Kutta methods of the Gauss type. Numer. Algorithms 56, 591-608 (2011) · Zbl 1215.65130 · doi:10.1007/s11075-010-9407-8
This reference list is based on information provided by the publisher or from digital mathematics libraries. Its items are heuristically matched to zbMATH identifiers and may contain data conversion errors. In some cases that data have been complemented/enhanced by data from zbMATH Open. This attempts to reflect the references listed in the original paper as accurately as possible without claiming completeness or a perfect matching.