Astronomy and Astrophysics – Astrophysics – General Relativity and Quantum Cosmology
Scientific paper
2000-03-17
J.Comput.Phys. 170 (2001) 231-260
Astronomy and Astrophysics
Astrophysics
General Relativity and Quantum Cosmology
Minor revisions. 31 pages, 13 figures, accepted for publication J. Comp. Phys
Scientific paper
10.1006/jcph.2001.6734
We present a spectral method for solving elliptic equations which arise in general relativity, namely three-dimensional scalar Poisson equations, as well as generalized vectorial Poisson equations of the type $\Delta \vec{N} + \lambda \vec{\nabla}(\vec{\nabla}\cdot \vec{N}) = \vec{S}$ with $\lambda \not= -1$. The source can extend in all the Euclidean space ${\bf R}^3$, provided it decays at least as $r^{-3}$. A multi-domain approach is used, along with spherical coordinates $(r,\theta,\phi)$. In each domain, Chebyshev polynomials (in $r$ or $1/r$) and spherical harmonics (in $\theta$ and $\phi$) expansions are used. If the source decays as $r^{-k}$ the error of the numerical solution is shown to decrease at least as $N^{-2(k-2)}$, where $N$ is the number of Chebyshev coefficients. The error is even evanescent, i.e. decreases as $\exp(-N)$, if the source does not contain any spherical harmonics of index $l\geq k -3$ (scalar case) or $l\geq k-5$ (vectorial case).
Bonazzola Silvano
Gourgoulhon Eric
Grandclement Philippe
Marck Jean-Alain
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