We approach the adaptive implicit method (AIM) in a new way that is based on the implicit matrix equation. The first step is to recover from the implicit equations the underlying capacitance and flow coefficients. Then, from these, we compute switching criteria that determine whether a given gridblock is to be treated implicitly or in IMPES fashion. These criteria are closely related to the CFL condition and offer insight in certain instances in which a computation is stable despite having a CFL greater than one. We also use the capacitance and flow coefficients to construct the IMPES pressure equation at all gridblocks.
We solve the resulting mixed-implicit matrix equation using an iterative method based on a two-stage preconditioner. The first preconditioning stage is to solve the IMPES matrix equation for pressure at all gridblocks. The second stage is solution of the full set of equations on the implicit gridblocks only.
Limited computational results indicate that, though the adaptive implicit procedure works, it is not efficient. It obtains only an approximation of the implicit matrix equation solution, leading to an increase in the number of Newton iterations required. The two-stage preconditioner for solving the resulting mixed-implicit matrix equation appears to work well, but further evaluation of it is needed.