Two space coordinates The ADI method

4 Two space coordinates The ADI method Given a rectangle R (a x1 b,c x2 d) and the two-dimensional diffusion equation 

Explicit different schemes show poor stability properties (Mitchell, 1969). In terms of the central difference operator, it may be shown that an accurate implicit equation is 

For sake of demonstration, let us assume that concentration on the edge is constant 

We now get a fairly good feeling that handling boundary conditions in two dimensions is significantly more difficult than those of one-dimensional problems. Flux conditions can be applied to the boundaries using the method of fictitious points. 

The concentration matrices will be shown as O fringed on each edge by the boundary values written in italic, i.e., as (ny +1) x (nx+ 1) = 6 x 5 matrices, and a symbols with overbar CJ will be used. Given the initial concentration matrix 

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