A Fast Algorithm for Solving the Poisson Equation on a Nested Grid

Abstract

We present a numerical method for solving the Poisson equation on a nested grid. A nested grid consists of uniform grids having different grid spacing and is designed to cover the space closer to the center with a finer grid. Thus, our numerical method is suitable for computing the gravity of a centrally condensed object. It consists of two parts: the difference scheme for the Poisson equation on the nested grid and the multigrid iteration algorithm. It has three advantages: accuracy, fast convergence, and scalability. First, it computes the gravitational potential of a close binary accurately up to the quadrupole moment, even when the binary is resolved only in the fine grids. Second, the residual decreases by a factor of 300 or more with each iteration. We confirmed experimentally that the iteration always converges to the exact solution of the difference equation. Third, the computation load of the iteration is proportional to the total number of the cells in the nested grid. Thus, our method gives a good solution at a minimum expense when the nested grid is large. The difference scheme is applicable also to adaptive mesh refinement, in which cells of different sizes are used to cover a domain of computation.