A spectral method for the wave equation of divergence-free vectors and symmetric tensors inside a sphere

Novak, Jerome
Cornou, Jean-Louis
Vasset, Nicolas
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The wave equation for vectors and symmetric tensors in spherical coordinates is studied under the divergence-free constraint. We describe a numerical method, based on the spectral decomposition of vector/tensor components onto spherical harmonics, that allows for the evolution of only those scalar fields which correspond to the divergence-free degrees of freedom of the vector/tensor. The full vector/tensor field is recovered at each time-step from these two (in the vector case), or three (symmetric tensor case) scalar fields, through the solution of a first-order system of ordinary differential equations (ODE) for each spherical harmonic. The correspondence with the poloidal-toroidal decomposition is shown for the vector case. Numerical tests are presented using an explicit Chebyshev-tau method for the radial coordinate.
Comment: 25 pages, 4 figures, to appear in Journal of Computational Physics
General Relativity and Quantum Cosmology