Development of Numerical Methods for Geodynamo and Mantle Convection Simulations

2. Development of New Spherical Grid: Yin-Yang Grid Since the finite difference method enables us to make highly optimized programs for massively parallel computers, we exploit the possibility of the finite difference method for simulations in spherical shell geometry with radius r (ri ≤ r ≤ ro), colatitude θ (0 ≤ θ ≤ π), and longitude φ (0 ≤ φ < 2π). Because there is no grid mesh that is orthogonal all over the spherical surface and, at the same time, free of coordinate singularity or grid convergence, we decompose the spherical surface into subregions. The decomposition, or dissection, enables us to cover each subregion by a grid system that is individually orthogonal and singularity-free. This divide-and-rule approach has been used with good success in the computational aerodynamics that incorporates complex geometry of aircraft's body with wings/stores/blades. The dissection of the computational domain generates internal border or boundary between the subregions. There are two different approaches to handle the internal boundaries. One is the patched grid method10) and the other is the overset grid method3). In the patched grid approach, the subdomains contact one another without any overlap on their borders. In the overset grid method, on the other hand, the By combining the ideas of pseudo-compressibility and local time stepping together with multigrid method, we have developed a new numerical algorithm for mantle convection simulation with strongly variable viscosity, which is a characteristic feature of mantle material, and implemented the algorithm to a mantle convection simulation code in box geometry. We have also devised a new grid system for fluid simulations in spherical shell geometry. This new spherical grid, named Yin-Yang grid, is based on the overset (or Chimera) grid methodology. The Yin-Yang grid is applied to geodynamo and mantle convection simulations in spherical shells with good success.