Peaceman-Rachford iterative method

Here, the matrices H and V are symmetric and positive definite matrices, which are each, after suitable permutation of indices, tridiagonal matrices. The matrix S is a non-negative diagonal matrix. Recalling that tridiagonal matrix equations are efficiently solved by the Gauss elimination method, we consider now the Peaceman-Rachford iterative method [27], a particular variant of the lAD methods, which is defined by... 

For purposes for comparison, we again consider the numerical solution of the Dirichlet problem for the unit square with uniform mesh spacings A = jp. For this problem, it can be shown that (4.26) is valid, and that the average rate of convergence R[T p. ] of the Peaceman-Rachford iterative method, with suitably selected parameters pu is [9 56]... 

The alternating direction implicit (ADI) method (Peaceman and Rachford, 1955) is a partially implicit method. The equation is rearranged so that one coordinate may be solved implicitly using the Thomas algorithm whilst the others are treated explicitly. If this is done alternately, each coordinate has a share of the implicit iterations and the efficiency (Gavaghan and Rollett, 1990) as well as the stability is improved. The method was used by Heinze for microdisc simulations (Heinze, 1981 Heinze and Storzbach, 1986) and has subsequently been adopted by others (Taylor et al, 1990 Fisher et al., 1997). 

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