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Volume 4, Issue 2
Chebyshev Spectral Methods and the Lane-Emden Problem

John P. Boyd

Numer. Math. Theor. Meth. Appl., 4 (2011), pp. 142-157.

Published online: 2011-04

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  • Abstract

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 $m \in [0, 5]$ is a constant parameter. The domain is $r \in [0, \xi]$ where $\xi$ is the first root of $y(r)$. We recast this as a nonlinear eigenproblem, with three boundary conditions and $\xi$ as the eigenvalue allowing imposition of the extra boundary condition, by making the change of coordinate $x \equiv r/\xi$: $y_{xx}+(2/x) y_{x}+ \xi^{2} y^{m}=0, y(0)=1, y_{x}(0)=0,$ $y(1)=0$. 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 $a_{n} \sim constant/n^{2m+5}$ as $n \rightarrow \infty$. 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.

  • AMS Subject Headings

65L10, 65D05, 85A15

  • Copyright

COPYRIGHT: © Global Science Press

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@Article{NMTMA-4-142, author = {}, title = {Chebyshev Spectral Methods and the Lane-Emden Problem}, journal = {Numerical Mathematics: Theory, Methods and Applications}, year = {2011}, volume = {4}, number = {2}, pages = {142--157}, abstract = {

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 $m \in [0, 5]$ is a constant parameter. The domain is $r \in [0, \xi]$ where $\xi$ is the first root of $y(r)$. We recast this as a nonlinear eigenproblem, with three boundary conditions and $\xi$ as the eigenvalue allowing imposition of the extra boundary condition, by making the change of coordinate $x \equiv r/\xi$: $y_{xx}+(2/x) y_{x}+ \xi^{2} y^{m}=0, y(0)=1, y_{x}(0)=0,$ $y(1)=0$. 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 $a_{n} \sim constant/n^{2m+5}$ as $n \rightarrow \infty$. 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.

}, issn = {2079-7338}, doi = {https://doi.org/10.4208/nmtma.2011.42s.2}, url = {http://global-sci.org/intro/article_detail/nmtma/5962.html} }
TY - JOUR T1 - Chebyshev Spectral Methods and the Lane-Emden Problem JO - Numerical Mathematics: Theory, Methods and Applications VL - 2 SP - 142 EP - 157 PY - 2011 DA - 2011/04 SN - 4 DO - http://doi.org/10.4208/nmtma.2011.42s.2 UR - https://global-sci.org/intro/article_detail/nmtma/5962.html KW - Lane-Emden, Chebyshev polynomial, pseudospectral. AB -

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 $m \in [0, 5]$ is a constant parameter. The domain is $r \in [0, \xi]$ where $\xi$ is the first root of $y(r)$. We recast this as a nonlinear eigenproblem, with three boundary conditions and $\xi$ as the eigenvalue allowing imposition of the extra boundary condition, by making the change of coordinate $x \equiv r/\xi$: $y_{xx}+(2/x) y_{x}+ \xi^{2} y^{m}=0, y(0)=1, y_{x}(0)=0,$ $y(1)=0$. 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 $a_{n} \sim constant/n^{2m+5}$ as $n \rightarrow \infty$. 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.

John P. Boyd. (2020). Chebyshev Spectral Methods and the Lane-Emden Problem. Numerical Mathematics: Theory, Methods and Applications. 4 (2). 142-157. doi:10.4208/nmtma.2011.42s.2
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