# OPTIMAL $$\rho$$ PARAMETER FOR THE ADI ITERATION FOR THE SEPARABLE DIFFUSION EQUATION IN THREE DIMENSIONS

• Ma, Sang-Back (Hanyang.University Computer. Science Department )
• Published : 1995.01.01

#### Abstract

The ADI method was introduced by Peaceman and Rachford [6] in 1955, to solve the discretized boundary value problems for elliptic and parabolic PDEs. The finite difference discretization of the model elliptic problem $$(1) -\Delta u = f, \Omega = [0, 1] \times [0, 1]$$ $$u = 0 on \delta \Omega$$ with 5-point centered finite difference discretization, with n +2 mesh-points in the x - direction and m + 2 points in the y direction, leads to the solution of a linear system of equations of the form $$(2) Au = b$$ where A is a matrix of dimension $N = n \times m$. Without loss of generality and for the sake of simplicity, we will assume for the remainder of this paper that m = n, so that $N = n^2$.