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

JOURNAL OF COMPUTATIONAL PHYSICS

卷:298

Article

Plestenjak, Bor^1,2  Gheorghiu, Calin I.³  Hochstenbach, Michiel E.⁴ 

[1] Univ Ljubljana, IMFM, Jadranska 19, SI-1000 Ljubljana, Slovenia

[2] Univ Ljubljana, Dept Math, SI-1000 Ljubljana, Slovenia

[3] Romanian Acad, T Popoviciu Inst Numer Anal, Cluj Napoca 400110, Romania

[4] Eindhoven Univ Technol, Dept Math & Comp Sci, NL-5600 MB Eindhoven, Netherlands

关键词: 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;   

DOI  :  10.1016/j.jcp.2015.06.015

来源: Elsevier

【 摘 要 】

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.

【 授权许可】

Free