H. Harder and U. Hansen
Institute of Geophysics, University of Muenster, Muenster, Germany.
Presently, all existing numerical methods to simulate the geodynamo use a spectral approach. Although a spectral expansion in spherical harmonics avoids the well known pole problem, such an approach has certain drawbacks. An efficient calculation of non-linear terms requires a spectral transform method, which prevents an implicit implementation of these terms. In addition, spectral transformations require global communicatation, which makes these methods less suitable for massively parallel computation. To avoid these problems, we are currently developing a finite volume method to simulate the geodynamo. The governing equations are formulated in a cartesian frame of reference, but the discretisation is adapted to a spherical shell. The grid is generated by the projection of an inscribed cube to the spherical surface, followed by an orthogonalization of the grid. Topologically this method maps the spherical shell to six cubes. We use domain decomposition and standard message passing routines for a parallel implementation of the method. Instead of the common staggered arrangement of variables, a colocated approach is prefered, where all variables are defined at the volume centres. This approach allows to calculate vector unknowns, as velocity and magnetic field, in a cartesian frame of reference which greatly simplifies the numerical method. The problem of pressure decoupling, commonly associated with the use of a colocated arrangement, is overcome by the method of operator interpolation (Rhies and Chow, 1982). Special interpolation formula are used to calculate the normal velocities at the volume faces to satisfy the continuity equations. We will present results for various test problems relevant to the geodynamo. In particular, we will compare our method with established spectral approaches for standard benchmark problems defined at the last SEDI meeting (Christensen et al. 2002).