Solution Procedure
The Eqns. (28.3) and (28.4) can be evaluated in an implicit manner. For example, in Eq. (28.3)  and all the neighbors,  and  are unknowns and they can be expressed as a system of equations by substitution of  to  and  to  . This will involve solution of a penta-diagonal matrix. Patankar (1980) splits the evolution procedure. At the first place, he writes Eqns. (28.3) and (28.4) as tridiagonal systems along each  grid line and solves them using Thomas algorithm. In a subsequent step, Eqns. (28.3) and (28.4) are written as tridiagonal systems along each  grid line  and solve them using Thomas algorithm. This is equivalent to implicit evaluation using ADI scheme. Now, if we substract Eq. (28.3) from (27.7), we shall obtain
In a similar manner, Eq. (28.4) is substracted from Eq. (28.2) to produce a correction equation for 
In order to make the link between  and  explicit, Eq. (29.1) can be reduced to
where 
An equivalent expression can be obtained for  as
where 
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