Implicit techniques ADI

Lapidus and Finder (1982) describe a method which meets our requirements the alternating directions implicit or ADI method of Feaceman and Rachford (1955). Consider Fig. 8.2, and Eq. 8.32 to be simulated. At time T, we work all rows, using fully implicit discretisation (in the Laasonen 1949 sense) in the X-direction but explicit discretisation in the Y-direction. Then, at a given (i,j) coordinate, assuming again 6X = SY = H, 

This rearranges, for all j, into a tridiagonal system of equations for 

For each j-row, all Cf can be solved as for the system 5.32. Again, in order to have available old C j values when working the next j-row, the computed C values must be stored separately until the next j-row has been worked. 

At the next T-step, a similar procedure, but this time applied to i-columns at a time, is applied. That is, the discretisation is now Laasonen-implicit in the Y-direction and explicit in the X-direction. 

Jain (1984) describes the method as two half-steps in time, each with an interval of j6T, in the two modes. This amounts to the same thing. The method is stable for all X, is efficient and relatively easy to implement. It has been used by Heinze (1981) and Heinze and Stdrzbach (1986), in simulating diffusion problems at a microelectrode. 

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