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

Jerome Novak (1) , Jean-Louis Cornou (1) , Nicolas Vasset (1)

1 LUTH (UMR_8102) - Laboratoire Univers et Théories

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.

Mots clés: Divergence-free evolution; Spherical harmonics; General relativity

Domaines: Relativité Générale et Cosmologie Quantique [gr-qc]

Jerome Novak, Jean-Louis Cornou, Nicolas Vasset. A spectral method for the wave equation of divergence-free vectors and symmetric tensors inside a sphere. Journal of Computational Physics, 2010, 229 (2), pp.399-414. ⟨10.1016/j.jcp.2009.09.033⟩. ⟨hal-00383481v2⟩