Usually the most time consuming process in the execution of the BPM simulation is the solution of the linear system in Equation 24. Sometimes, it is possible to speed this calculation by manipulating the formulation so that only simple linear systems result. There are other formulations of the problem which permit the operator to be split into two parts, one which involves derivatives in the x  direction only, and the other having derivatives in y only. In the first step, the operator with x  derivatives is used. The operator with x derivatives uses simple finite difference equations, therefore no more than the next neighbor in the mesh need be used. The resulting linear system is tridiagonal; it has matrix elements in the main diagonal and the two adjacent diagonals.

The tridiagonal system is solved much more quickly than the general linear system. In the next step, the operator involves derivatives with only y . By changing the orientation of the problem (alternate direction), another tridiagonal system is obtained. This changing of the sense of direction to get tridiagonal systems in solving the implicit part of the propagation step is the origin of the name Alternating Direction Implicit. OptiBPM uses ADI whenever possible, to obtain the fastest algorithm [32].