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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Finite-difference approximation of the boundary conditions of the second and third order for the nonlinear heat conduction equation

Let us consider non-linear heat conduction equation

, (1)

, (2)

with the initial condition , and boundary conditions
, (3)

At . If , condition (3) is the boundary condition of the second order, but if , it is the boundary condition of the third order.

Let us introduce analytical grids in space and time:, , , .

Then, define grid functions , , , , , , , .

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