Chebyshev Spectral Methods and the Lane-Emden Problem

Abstract

The three-dimensional spherical polytropic Lane-Emden problem is yrr + (2/r)yr + y m = 0, y(0) = 1, yr(0) = 0 where m∈ (0,5) is a constant parameter. The domain is r∈ (0,ξ) where ξ is the first root of y(r). We recast this as a non- linear eigenproblem, with three boundary conditions and ξ as the eigenvalue allow- ing imposition of the extra boundary condition, by making the change of coordinate x≡ r/ξ: yxx + (2/x)yx +ξ 2 y m = 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 a n ∼ constant/n 2m+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 inte- ger. The numerical algorithm is so simple that the complete code (in Maple) is given as a one page table. AMS subject classifications: 65L10, 65D05, 85A15