Chebyshev Spectral Methods and the Lane-Emden Problem

The three-dimensional spherical polytropic Lane-Emden problem is y(rr) + (2/r)y(r) + y(m) = 0, y(0) = 1, y(r)(0) = 0 where m is an element of [0,5] is a constant parameter. The domain is r is an element of [0, xi] where xi is the first root of y(r). We recast this as a non-linear eigenproblem, with three boundary conditions and xi as the eigenvalue allowing imposition of the extra boundary condition, by making the change of coordinate x equivalent to r/xi: y(xx) + (2/x)y(x) + xi(2)y(m) = 0, y(0) = 1, y(x) (0) = 0, y(1) = 0. We find that a Newton-Kantorovich iteration always converges from an m-independent starting point y((0)) (x) = cos([pi/2]x), xi((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) similar to constant/n(2m+5) as n -> infinity. 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.