Fully Discretized Energy Stable Schemes for Hydrodynamic Equations Governing Two-Phase Viscous Fluid Flows

We develop systematically a numerical approximation strategy to discretize a hydrodynamic phase field model for a binary fluid mixture of two immiscible viscous fluids, derived using the generalized Onsager principle that warrants not only the variational structure but also the energy dissipation property. We first discretize the governing equations in space to arrive at a semi-discretized, time-dependent ordinary differential and algebraic equation (DAE) system in which a corresponding discrete energy dissipation law is preserved. Then, we discretize the DAE system in time to obtain a fully discretized system using a structure preserving finite difference method like the Crank–Nicolson method, which satisfies a fully discretized energy dissipation law. Alternatively, we solve the first order DAE system using the integration factor method after the algebraic equation is solved firstly. The integration factor method, which treats the linear, spatial derivative terms explicitly. Finally, two numerical examples are presented to compare the efficiency and accuracy of the two proposed methods.