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

TL;DR

This paper introduces an energy-stable, mass-conservative discontinuous Galerkin method for simulating the compressible Navier--Stokes--Allen--Cahn system, achieving high spatial and temporal accuracy while preserving energy dissipation.

Contribution

It presents a novel fully-discrete DG scheme that ensures energy stability and mass conservation for the NSAC system, with higher-order spatial and second-order temporal accuracy.

Findings

Numerical experiments confirm energy stability and mass conservation.

The method achieves higher-order spatial accuracy and second-order temporal accuracy.

The approach is applicable to simulating two-phase fluid flows with energy dissipation.

Abstract

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.