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A pseudospectral matrix method for time-dependent tensor fields on a spherical shell

Bernd Bruegmann
Theoretical Physics Institute, University of Jena, 07743 Jena, Germany
arXiv:1104.3408 [physics.comp-ph] (18 Apr 2011)

@article{2011arXiv1104.3408B,

   author={Bruegmann}, B.},

   title={“{A pseudospectral matrix method for time-dependent tensor fields on a spherical shell}”},

   journal={ArXiv e-prints},

   archivePrefix={“arXiv”},

   eprint={1104.3408},

   primaryClass={“physics.comp-ph”},

   keywords={Physics – Computational Physics, General Relativity and Quantum Cosmology},

   year={2011},

   month={apr},

   adsurl={http://adsabs.harvard.edu/abs/2011arXiv1104.3408B},

   adsnote={Provided by the SAO/NASA Astrophysics Data System}

}

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We construct a pseudospectral method for the solution of time-dependent, non-linear partial differential equations on a three-dimensional spherical shell. The problem we address is the treatment of tensor fields on the sphere. As a test case we consider the evolution of a single black hole in numerical general relativity. A natural strategy would be the expansion in tensor spherical harmonics in spherical coordinates. Instead, we consider the simpler and potentially more efficient possibility of a double Fourier expansion on the sphere for tensors in Cartesian coordinates. As usual for the double Fourier method, we employ a filter to address time-step limitations and certain stability issues. We find that a tensor filter based on spin-weighted spherical harmonics is successful, while two simplified, non-spin-weighted filters do not lead to stable evolutions. The derivatives and the filter are implemented by matrix multiplication for efficiency. A key technical point is the construction of a matrix multiplication method for the spin-weighted spherical harmonic filter. As example for the efficient parallelization of the double Fourier, spin-weighted filter method we discuss an implementation on a GPU, which achieves a speed-up of up to a factor of 20 compared to a single core CPU implementation.
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