That let us, using boundary conditions, evaluate a line settling coefficients P and Q, and solve in reverse order evaluating the variables values. Than if we change the index of (9.3), we have:

Replacing (9.4) in (9.2) and comparing the result with (9.3), we find the following expressions for P and Q:

Equations (9.5) and (9.6) are recursive relations that allow us, provided that we know P1 and Q1, to evaluate all values of P and Q. Considering the problem given by (9.1), and if we solve the problem from West to East face, we have:

Inspecting (9.5) and (9.6) it is easy to infer how we can evaluate P1 and Q1 .We observe in theses equations that the boundary volumes, i. e., the volumes of the West face of the calculation domain should not depend on variable on the left. If we are handling columns, and the volumes belong to the South face they should not depend on the variable below. Therefore, C1 must be zero and this results

We know that the approximation equation, for volumes of boundaries East and North, must not depend on the variable on the right or above. Therefore, Bm given by Eq. (9.7) and (9.9), must be zero. It follows from Eq. (9.3) that

And the index N indicates the volumes on the East boundary if the sweeping is done line-by-line or the volumes of North boundary if we are calculating column-by-column.