I recently had a paper accepted at the Journal of Computational Physics on removing timestep restrictions in the numerical solution of two-phase flow problems in porous media. The paper can be downloaded for free from the journal until August 23rd by following this link. The idea is to add some carefully constructed phase-field terms that guarantee Newton’s method will converge. This changes the physics a little bit, but a solution to the original equations can be found by reducing the magnitude of these terms to zero over a few Newton updates.

Progression of the homotopy solution during one timestep with a very large dt.