A second order fully-discrete linear energy stable scheme for a binary compressible viscous fluid model

We present a linear, second order fully discrete numerical scheme on a staggered grid for a thermodynamically consistent hydrodynamic phase field model of binary compressible fluid flows, derived from the generalized Onsager Principle. The hydrodynamic model possesses not only the variational structure in its constitutive equation, but also warrants the mass, linear momentum conservation as well as energy dissipation. We first reformulate the model using the energy quadratization method into an equivalent form and then discretize the reformulated model to obtain a semidiscrete partial differential equation system using the Crank-Nicolson method in time. The semi-discrete numerical scheme preserves the mass conservation and energy dissipation law in time. Then, we discretize the semi-discrete PDE system on a staggered grid in space to arrive at a fully discrete scheme using 2nd order finite difference methods, which respects a discrete energy dissipation law. We prove the unique solvability of the linear system resulting from the fully discrete scheme. Mesh refinements and numerical examples on phase separation due to spinodal decomposition in binary polymeric fluids and interface evolution in the gas-liquid mixture are presented to show the convergence property and the usefulness of the new scheme in applications, respectively.