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Chebyshev Spectral Methods and the Lane-Emden Problem

Published online by Cambridge University Press:  28 May 2015

John P. Boyd
John P. Boyd*
Affiliation:
Department of Atmospheric, Oceanic and Space Science, University of Michigan, 2455 Hayward Avenue, Ann Arbor, MI 48109-2143, USA
*
*Corresponding author.Email address:jpboyd@umich.edu
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Abstract

The three-dimensional spherical polytropic Lane-Emden problem is yrr + (2/r)yr + ym = 0, y(0) = 1, yr(0) = 0 where m ϵ [0,5] is a constant parameter. The domain is r ϵ [0, ξ] where ξ is the first root of y(r). We recast this as a nonlinear eigenproblem, with three boundary conditions and ξ as the eigenvalue allowing imposition of the extra boundary condition, by making the change of coordinate x ≡ r/ξ: yxx + (2/x)yx + ξ2ym = 0, y(0) = 1, yx(0) = 0, y(1) = 0. We find that a Newton-Kantorovich iteration always converges from an m-independent starting point y(0)(x) = cos([π/2]x), ξ(0) = 3. We apply a Chebyshev pseudospectral method to discretize x. The Lane-Emden equation has branch point singularities at the endpoint x = 1 whenever m is not an integer; we show that the Chebyshev coefficients are an ~ constant/n2m+5 as n → ∞. However, a Chebyshev truncation of N = 100 always gives at least ten decimal places of accuracy — much more accuracy when m is an integer. The numerical algorithm is so simple that the complete code (in Maple) is given as a one page table.

Keywords

65L1065D0585A15Lane-EmdenChebyshev polynomialpseudospectral
Type
Research Article
Information
Numerical Mathematics: Theory, Methods and Applications , Volume 4 , Issue 2 , May 2011 , pp. 142 - 157
Copyright
Copyright © Global Science Press Limited 2011

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