Application of Direct Poisson Solvers to Time-Dependent Numerical Cloud Models

Abstract

Direct methods for solving the discrete Poisson equation on various types of regions are generally computationally faster than popular iterative techniques, but this depends somewhat on the initial solution provided the iterative method. If the region is non-rectangular, the direct Poisson solver requires at least twice the computation time for a corresponding rectangular region, and in time-dependent simulations the time required for an iterative method may not exceed that for the direct, since solution values from a previous time step may be good initial solution for the next. In this paper we briefly describe the direct methods and we apply them and an iterative method to several time-dependent cloud models to solve Poisson's equation for the streamfunction or electrical potential. For rectangular regions the direct method was clearly more efficient. For a mountain cumuli model an iterative method was faster in two or three cases. However, this was due in part to the fact that the iterative method was programmed in machine language, whereas the direct method was programmed in FORTRAN. Machine language codes for the direct methods would make them more than competitive with the iterative methods and other considerations such as reduced storage requirements and greater precision of the solution for the direct method lead to the conclusion that the direct methods described herein are superior to iterative methods.