Numerical solution of eigenvalue problems for linear boundary value ODEs
type: Item
instance of: scholarly article
title: Numerical solution of eigenvalue problems for linear boundary value ODEs
zbMATH Open document ID: 0726.65101
DOI: 10.1016/0021-9991(91)90226-B
author: S. Bramley
author: Luca Dieci
author: Robert D. Russell
published in: Journal of Computational Physics
publication date: 1991-01-01T00:00:00Z
review text: The authors consider the numerical solution of eigenvalue problems for linear ordinary differential equations, although some of the techniques discussed apply naturally to nonlinear differential equations and/or to problems where the eigenvalue dependence may be nonlinear. Actually all these problems, including the one considered here are nonlinear (i.e. at least quadratic in the eigenfunction/eigenvalue unknowns), although special techniques can be used when a linear differential operator is involved. Several different methods are considered for the case in which only a few eigenpairs are desired and interrelationships among these methods are shown. Both shooting, i.e., initial value problem based, and global, i.e., finite difference or collocation methods are discussed. A Riccati type, a continuous orthonormalization, and a collocation method are used to solve numerically the complex Orr-Sommerfeld eigenvalue problem for large Reynolds numbers. Unfortunately, none of the approaches mentioned here can guarantee that the sought after eigenpairs will be obtained. This, as any other nonlinear problem has a mind of itself, and the iterative methods used for their solutions will, in the best of cases, converge to a solution, and as the authors point out, sometimes to a spurious one generated by the discretization. The fact that very accurate initial guesses are required to guarantee convergence should be a warning flag. In recognizing this inherent difficulty the authors resort to continuation on Reynolds number, a time honored solution, which amounts to realizing that a reasonable way to try to reach a particular solution of a non- monotone problem (i.e. one with multiple solutions), is by considering the more complex problem of solution branch following and bifurcation. Of course, having taken that line of thought, it would have been reasonable to consider other software available for solving these much harder problems, like that of E. Doedel and R. Seydel.
reviewed by: Victor Pereyra
Mathematics Subject Classification ID: 65L15
Mathematics Subject Classification ID: 34L15
Mathematics Subject Classification ID: 34L30
zbMATH DE Number: 4199533
zbMATH Keywords: nonlinear eigenvalue problem
zbMATH Keywords: shooting method
zbMATH Keywords: eigenpairs
zbMATH Keywords: finite difference
zbMATH Keywords: collocation
zbMATH Keywords: complex Orr-Sommerfeld eigenvalue problem
zbMATH Keywords: convergence
zbMATH Keywords: solution branch following
zbMATH Keywords: bifurcation
MaRDI profile type: Publication
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full work available at URL: 0021-9991(91)90226-b
OpenAlex ID: W2055990716
description: scientific article; zbMATH DE number 4199533