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Convergence analysis of pseudospectral collocation methods for a singular differential equation
Huang, Weizhang Ma, Heping Sun, Weizwei
Huang, Weizhang
Ma, Heping
Sun, Weizwei
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Abstract
Solutions of partial differential equations with coordinate singularities often have special behavior near the singularities, which forces them to be smooth. Special treatment for these coordinate singularities is necessary in spectral approximations in order to avoid degradation of accuracy and efficiency. It has been observed numerically in the past that, for a scheme to attain high accuracy, it is unnecessary to impose all the pole conditions, the constraints representing the special solution behavior near singularities. In this paper we provide a theoretical justification for this observation. Specifically, we consider an existing approach, which uses a pole condition as the boundary condition at a singularity and solves the reformulated boundary value problem with a commonly used Gauss--Lobatto collocation scheme. Spectral convergence of the Legendre and Chebyshev collocation methods is obtained for a singular differential equation arising from polar and cylindrical geometries.
Description
This is the published version, also available here: http://dx.doi.org/10.1137/S0036142902381024.
Date
2003-01-02
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Society for Industrial and Applied Mathematics
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Keywords
coordinate singularity, convergence, spectral collection method
Citation
Huang, Weizhang., Ma, Heping., Sun, Weiwei. "Convergence analysis of pseudospectral collocation methods for a singular differential equation." SIAM J. Numer. Anal., 41(6), 2333–2349. (17 pages). http://dx.doi.org/10.1137/S0036142902381024.