Test of Convergence Speed of Iterative Methods for Solving 2 and 3 Dimensional Elliptic-type Differential Equations

[摘要] For solving numerically the Elliptic-type differential equation, the speeds of convergence by various iterative methods are tested. The equations treated are the 2 and 3 dimensional Poisson and Helmholtz-type equations, and the boundary conditions are of the first and of the second kinds. The methods taken up are three Accelerated Liebmann (AL), the Residual Polynomial Generation (RPG) and the Alternating Direction Implicit (ADI) methods. First, the theoretical considerations are made on the optimum value of the overrelaxation coefficient for the AL method and that of the iteration parameter used in the ADI method. Then, taking four practical cases, these methods are tested in various ways using the computing machine IBM 704. The results obtained are that the speed of the ADI method is highest compared with those of other methods, but this method is somewhat complicated in treatment and unstable in convergence. The speed by the RPG method is very low, but it is effective to accelerate the slow method, for instance, the method solving the Helmholtz-type equation of negative coefficient proposed by Matsuno. The AL method is very simple and it needs the fewest memory field, and the computation speed is fairly high, so far as the optimum overrelaxation coefficient is adopted.

[学科分类] 大气科学