Higher order time integration methods for two-phase flow

Time integration methods that adapt in both the order of approximation and time step have been shown to provide efficient solutions to Richardsʹ equation. In this work, we extend the same method of lines approach to solve a set of two-phase flow formulations and address some mass conservation issues from the previous work. We analyze these formulations and the nonlinear systems that result from applying the integration methods, placing particular emphasis on their index, range of applicability, and mass conservation characteristics. We conduct numerical experiments to study the behavior of the numerical models for three test problems. We demonstrate that higher order integration in time is more efficient than standard low-order methods for a variety of practical grids and integration tolerances, that the adaptive scheme successfully varies the step size in response to changing conditions, and that mass balance can be maintained efficiently using variable-order integration and an appropriately chosen numerical model formulation.