2009/05/31 by Jerome Novak, J. Novák, Jean-Louis Cornou +2
Earth and Planetary Sciences · Engineering · Mathematics · Physics and Astronomy · #Cartesian tensor #Differential equation #Exact solutions in general relativity #Geometry #Geophysics and Gravity Measurements #Magnetic field #Mathematical analysis #Mathematics #Ordinary differential equation #Physics #Pulsars and Gravitational Waves Research #Quantum mechanics #Scalar (mathematics) #Solenoidal vector field #Spherical harmonics #Superconducting Materials and Applications #Symmetric tensor #Tensor (intrinsic definition) #Tensor density #Tensor field #Vector Laplacian #Vector field #Vector potential #Vector spherical harmonics #Zonal spherical harmonics #gr-qc
paper · pdf · doi:10.1016/j.jcp.2009.09.033
published as J.Comput.Phys.229:399-414,2010 · 25 pages, 4 figures, to appear in Journal of Computational Physics
openalex publication_date 2009/10/05 · arxiv created 2009/11/18 · openalex created_date 2025/10/10 · openalex updated_date 2026/08/05
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.