[M-GI30] Computational sciences on the universe, galaxies, stars, planets and their environments

[MGI30-11] Attempts for direct solution of flow fields of 3-D mantle convection on coarse Yin-Yang grids

*Masanori Kameyama^1,2, Yusuke Uemura¹ (1.Geodynamics Research Center, Ehime University, 2.Center for Mathematical Science and Advanced Technology, Japan Agency for Marine-Earth Science and Technology)

Keywords:mantle convection, numerical experiments, multigrid method, Yin-Yang grid, 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 Yin-Yang grids, 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 radial and the horizontal directions. The fundamental differential equations are discretized by the finite volume method in the 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) grid systems. The "ersatz" basis functions are obtained from the eigenproblems of the matrices describing the discretized differential operators (Laplacian) on the spherical surface spanning the Yin and Yang grids. We confirmed that the "ersatz" bases derived thus can represent the horizontal distributions of field variables in a quite continuous and consistent manner over the entire sphere, which enables us to reduce the "mismatch" between the Yin and Yang during the interpolations required when applying the boundary conditions in the horizontal directions. Our experience showed that the present technique requires 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 spherical shell geometry, we also found that the new direct solver can yield very "well-behaved" solutions giving an appropriate accuracy at the coarsest grid level as well as a sufficient consistency between the Yin and Yang, which helps reduce the overall computational costs of our multigrid solver.