[M-GI29] Computational sciences on the universe, galaxies, stars, planets and their environments

[MGI29-09] Attempts for direct solution of flow fields of 3-D mantle convection on coarse grids

*Masanori Kameyama^1,2 (1.Geodynamics Research Center, Ehime University, 2.Japan Agency for Marine-Earth Science and Technology)

Keywords:mantle convection, numerical experiments, multigrid method, direct solution

In this study we are developing a numerical technique for direct solution of flow fields (velocity and pressure) of 3-D mantle convection on coarse spatial resolutions, aiming at accelerating the solution at the coarsest grid level of the multigrid solver for this problem. The present technique is based on the separation of variables into the vertical (or radial) and the horizontal directions. The fundamental differential equations are discretized by the finite volume method in the vertical (radial) direction, while in the horizontal directions they are solved by a method which we call "Ersatz Spectral method". The heart of our technique lies in the numerical construction of the set of "ersatz" basis functions appropriate for the spectral expansion on the given coarse spatial resolutions. The "ersatz" basis functions are obtained from the eigenproblems of the matrices describing the discretized differential operators (Laplacian) in the horizontal planes. Our experience showed that the present technique requires very small computational costs, both in the numerical construction of "ersatz" basis functions and in the numerical solutions of differential equations, by virtue of the coarse grid. By incorporating the present technique to our mantle convection simulation programs in 3-D Cartesian box, we also found that the new direct solver can yield very "well-behaved" solutions associated with an appropriate accuracy at the coarsest grid level, which greatly helps reduce the overall computational costs of multigrid iterations. In the presentation, the possible application of the present technique to non-Cartesian cases (such as spherical geometries) will be also discussed.