Approximations of Sturm-Liouville eigenvalues using boundary value methods
type: Item
instance of: scholarly article
title: Approximations of Sturm-Liouville eigenvalues using boundary value methods
zbMATH Open document ID: 0877.65056
DOI: 10.1016/S0168-9274(96)00073-6
author: Paolo Ghelardoni
published in: Applied Numerical Mathematics
publication date: 1997-12-10T00:00:00Z
review text: This paper is concerned with the numerical calculation of eigenvalues in Sturm-Liouville boundary value problems. As it is well known there are many numerical methods which provide good approximations to the first harmonics but often they are not satisfactory when the number of harmonics becomes large. The author proposes an approach to the last problem, when the boundary value problem is written in Liouville's normal form, combining an asymptotic technique of error correction of the eigenvalues [cf. \textit{A. L. Andrew} and \textit{J. W. Paine}, Numer. Math. 47, 289-300 (1985; Zbl 0554.65060)] with the discretization of the boundary value problem by means of multistep methods which use first- and second-order derivatives (sometimes referred to as Obrechkoff multistep methods). Some numerical experiments are presented to compare the efficiency of the proposed method with other similar methods based on Numerov's formula or Simpson's rule.
Mathematics Subject Classification ID: 65L15
Mathematics Subject Classification ID: 34L15
Mathematics Subject Classification ID: 65L06
zbMATH DE Number: 1022210
zbMATH Keywords: eigenvalues
zbMATH Keywords: Sturm-Liouville boundary value problems
zbMATH Keywords: harmonics
zbMATH Keywords: error correction
zbMATH Keywords: Obrechkoff multistep methods
zbMATH Keywords: numerical experiments
zbMATH Keywords: Numerov formula
zbMATH Keywords: Simpson rule
reviewed by: Manuel Calvo
MaRDI profile type: Publication
cites work: Boundary value methods based on Adams-type methods
cites work: Stability of some boundary value methods for the solution of initial value problems
cites work: Q3805818
cites work: Correction of finite element eigenvalues for problems with natural or periodic boundary conditions
cites work: Correction of finite difference eigenvalues of periodic Sturm-Liouville problems
cites work: Q4839863
cites work: Correction of Numerov's eigenvalue estimates
cites work: Correction of finite element estimates for Sturm-Liouville eigenvalues
cites work: High-order multistep methods for boundary value problems
cites work: Q4393432
cites work: Stability of some boundary value methods for IVPs
cites work: Q5719651
cites work: Q5674335
cites work: Stability and convergence of boundary value methods for solving ODE
cites work: On the correction of finite difference eigenvalue approximations for Sturm-Liouville problems
cites work: A modified numerov method for higher sturm-liouville eigenvalues
cites work: Accurate computation of higher Sturm-Liouville eigenvalues
cites work: On a correction of Numerov-like eigenvalue approximations for Sturm- Liouville problems
cites work: A finite-element estimate with trigonometric hat functions for Sturm- Liouville eigenvalues
cites work: A parallel approach to the modified Numerov-like eigenvalue determination for Sturm-Liouville problems
description: scientific article; zbMATH DE number 1022210