Spectral collocation for multiparameter eigenvalue problems arising from separable boundary value problems

Bor Plestenjak

doi:10.1016/j.jcp.2015.06.015

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Spectral collocation for multiparameter eigenvalue problems arising from separable boundary value problems

2015-10-01

Bor Plestenjak *

Călin-Ioan Gheorghiu

Michiel E. Hochstenbach

DOI：10.1016/j.jcp.2015.06.015

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Abstract

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In numerous science and engineering applications a partial differential equation has to be solved on some fairly regular domain that allows the use of the method of separation of variables. In several orthogonal coordinate systems separation of variables applied to the Helmholtz, Laplace, or Schrodinger equation leads to a multiparameter eigenvalue problem (MEP); important cases include Mathieu's system, Lame's system, and a system of spheroidal wave functions. Although multiparameter approaches are exploited occasionally to solve such equations numerically, MEPs remain less well known, and the variety of available numerical methods is not wide. The classical approach of discretizing the equations using standard finite differences leads to algebraic MEPs with large matrices, which are difficult to solve efficiently. The aim of this paper is to change this perspective. We show that by combining spectral collocation methods and new efficient numerical methods for algebraic MEPs it is possible to solve such problems both very efficiently and accurately. We improve on several previous results available in the literature, and also present a MATLAB toolbox for solving a wide range of problems. (C) 2015 Elsevier Inc. All rights reserved.

Keywords:

Helmholtz equation

Schrodinger equation

Separation of variables

Mathieu's system

Lame's system

Spectral methods

Chebyshev collocation

Laguerre collocation

Multiparameter eigenvalue problem

Two-parameter eigenvalue problem

Three-parameter eigenvalue problem

Sylvester equation

Bartels-Stewart method

Subspace methods

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