Adaptive implicit methods seek an efficient middle ground between IMPES and fully implicit time stepping. The principal issue in implementation of these methods is the switching criterion that determines whether a given unknown is implicit or explicit. We develop and justify a criterion based on the Courant-Friedrichs-Lewy (CFL) stability condition. This could also be used for time-step control in IMPES models by indicating the maximum stable time step.
First, we analyze simple problems theoretically and find that adaptive implicit stability, like fully explicit stability, depends on a CFL condition. Next, we develop a switching criterion for a black-oil model and indicate how it could be extended to compositional simulation. The CFL-based criterion involves quantities that an implicit Jacobian must already compute, so its use costs virtually nothing. Each unknown has its own CFL number at every outer Newton iteration, so any combination of implicit and explicit unknowns is possible and a given unknown may switch back and forth through time.
While discussing the simple problems, we analyze and compare three adaptive implicit methods that have been suggested in the literature. One of the methods yields better answers by reducing numerical dispersion. Also, its CFL restriction for practical calculations is looser than those of the other methods, allowing it to make more unknowns explicit.