Accuracy of the general peaceman and Rachford alternating direction implicit method

Abstract

In this paper, the truncation error of the general Peaceman and Rachford alternating direction implicit (ADI) method in a N-dimensional space was derived by using the Taylor-series-expansion approach. The formulation of this truncation error is TE= 2−N 2N 2 ∑ i=l N ∑ j=l N ∂ 4θ ∂η 2 iη 2 j δτ+O(δτ 2)+ ∑ i=l N O(δη 2 i which indicates that the method is second order accurate in both time and space for two dimensions but reduces to first order accuracy in time for higher dimensions. However the second order time accuracy is recovered when N → ∞. The worst case corresponding to the maximum value of (2-N) 2N 2 occurs at N = 4.