Alternate directions implicit scheme and the intermediate boundary conditions. Third type boundary conditions.

This second Note is dedicated to the discussion of the case of 3-rd type boundary conditions posed on the edges of the modeling region and will present an outline of the algorithm for the account of both the Dirichlet and 3-rd type boundary conditions on different edges.

Before passing to the discussion of that scheme, a few words should be said about another ADI scheme – namely 2D Peaceman – Rachford finite difference scheme. The correct account of the intermediate boundary conditions for both Dirichlet and 3-rd order types has been thoroughly discussed in [1]. One essential difference between the 3D Douglas – Rachford and 2D Peaceman – Rachford schemes is that in the latter scheme the spatial coordinates enter symmetrically, such that the second equation (in 2D Peaceman – Rachford) contains finite difference representations of both spatial derivatives. This fact leads to a cumbersome infer of relation between the values of the unknown function on different edges – for details see [1].

Surprisingly, although the Douglas – Rachford scheme is designed for a 3D spatial region, it is easier to treat the 3-rd order boundary conditions for it correctly due to the “non-symmetrical” entrance of the spatial coordinates in three equations – see Eq. (1) – (3):

, (1)
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Alternate directions implicit scheme and the intermediate boundary conditions

The ADI (alternate directions implicit) method is widely used for the numerical solution of multidimensional parabolic PDE (partial differential equations). [1]. 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 [2]. 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 [3]. 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]:

, (1)
, (2)
, (3)
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