Alternating-Direction Implicit Iterative Methods

Abstract

In the previous chapter, we considered various block iterative methods, such as the line and 2-line successive overrelaxation iterative methods, which as pratical methods involved the direct (or implicit) solution of particular lower-order matrix equations. Also, as a standard for comparison of the asymptotic rates of convergence of these methods, we considered the numerical solution of the Dirichet problem for the unit square with uniform mesh spacings, calling this the model problem. This knowledge of block methods, as well as representative asymptotic rates of convergence for the model problem for the iterative methods considered thus far, serves as the basis for the introduction of the alternating-direction implicit iterative methods, due to Peaceman and Rachford (1955) and Douglas and Rachford (1956).