The ADI (alternate directions implicit) method is widely used for the numerical solution of multidimensional parabolic PDE (partial differential equations). . Although the method is known for a long time and is well described in the text-books, its practical realizations sometimes appear to be inaccurate . The inaccuracy arises every time when one neglects the correct account of the so-called intermediate boundary conditions. This neglect can become the cause of instabilities even when the used ADI-scheme is known to be unconditionally stable in the frame of von Neumann spectral analysis technique . The procedures of a correct account of the intermediate boundary conditions (for the Peaceman-Rachford ADI-scheme), are described in [4, 5, 6].
Below, we are going to consider the correct account of the intermediate boundary conditions for the Douglas-Rachford ADI-scheme [3, 7]: