Shallow water model on a modified Icosahedral geodesic grid by using spring dynamics

We develop a shallow water model on an icosahedral geodesic grid with several grid modifications. Discretizations of differential operators in the equations are based on the finite volume method, so that the global integrations of transported quantities are numerically conserved. Ordinarily, the standard grid is obtained by recursive grid division starting from the lowest order icosahedral grid. From the viewpoint of numerical accuracy of operators, we propose to relocate the variable-defined grid points from the standard positions to the gravitational centers of control volumes. From the other viewpoint of numerical stability, we modify the standard grid configuration by employing the spring dynamics, namely, the standard grid points are connected by appropriate springs, which move grid points until the dynamical system calms down. We find that the latter modification dramatically reduces the grid-noise in the numerical integration of equations. The reason for this is that the geometrical quantities of control volume such as its area and distortion of its shape exhibit the monotonic distribution on the sphere. By the combination of the two modifications, we can integrate the equations both with high accuracy and stability. The performance of our model is examined by the standard test cases of shallow water model proposed by D. L. Williamson et al. (1992, J. Comput. Phys. 102, 211). To investigate the convergence properties against resolution, we perform simulations from grid division level 4 (approximately 4.5degrees x 4.5degrees grids) to 7 (approximately 0.56degrees x 0.56degrees grids). The obtained results clearly indicate the advantage of use of our modified grid over the standard grid for the numerical accuracy and stability. (C) 2001 Elsevier Science.