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Chebyshev Spectral Methods and the Lane-Emden Problem
Published online by Cambridge University Press: 28 May 2015
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.
- 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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