An Energy-Stable Discontinuous Galerkin Method for the Compressible Navier--Stokes--Allen--Cahn System

We consider a Navier--Stokes--Allen--Cahn (NSAC) system that governs the compressible motion of a viscous, immiscible two-phase fluid at constant temperature. Weak solutions of the NSAC system dissipate an appropriate energy functional. Based on an equivalent re-formulation of the NSAC system we propose a fully-discrete discontinuous Galerkin (dG) discretization that is mass-conservative, energy-stable, and provides higher-order accuracy in space and second-order accuracy in time. The approach relies on the approach in \cite{Giesselmann2015a} and a special splitting discretization of the derivatives of the free energy function within the Crank-Nicolson time-stepping. Numerical experiments confirm the analytical statements and show the applicability of the approach.