Efficient Multigrid Poisson Equation Solvers in High Performance Computing for Global Weather Models

A Z-grid finite volume atmospheric model possesses several important numerical advantages, such as, the best wave dispersion relation, physical quantity conservation, and much less grid space computational modes. Since a Z-grid model is based on primitive equations of vorticity-divergence, it requires an efficient Poisson solver for converting vorticity and divergence fields to stream function and velocity potential. For future extremely high resolution numerical models, it is critical to implement an efficient Poisson solver under the state-of-the-art massive parallel or exascale computing environment. In this presentation, we will report our progress on implementations of several iterative methods, such as Jacobi, Gauss-Seidel, conjugate gradient or some combined or hybrid of Jacobi and Gauss-Seidel methods, under such high performance computing environment. We try to take advantages of the data independence of Jacobi iterations and the fast convergent Gauss-Seidel iterations, and arrange the computations on each computation node in order to achieve the best overall performance with a proper, optimal computation and communication ratio. Based on the study of these iterative methods, we create an efficient multigrid solver of the Poisson equation to support the implementation of this most critical computational component of our future Z-grid regional and global numerical weather prediction models.