Logo
ScienceStack
Table of Contents
Fetching ...
Sign In

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

Lukas Ostrowski, Christian Rohde

TL;DR

This work addresses numerically solving the isothermal compressible Navier–Stokes–Allen–Cahn (NSAC) system for two-phase flow by developing a fully discrete, energy-stable discontinuous Galerkin method. It builds a mixed first-order reformulation and uses Crank–Nicolson time stepping to achieve second-order accuracy in time while preserving mass and ensuring discrete energy dissipation through carefully designed inter-element fluxes. The method demonstrates unconditional energy stability and convergence in space and time, and numerical experiments validate energy decay and the ability to capture topological changes like droplet merging. The approach provides a robust framework for accurate, thermodynamically consistent simulations of compressible two-phase flow with diffuse interfaces, with potential extensions to temperature effects and more complex multi-phase settings.

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.

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

TL;DR

This work addresses numerically solving the isothermal compressible Navier–Stokes–Allen–Cahn (NSAC) system for two-phase flow by developing a fully discrete, energy-stable discontinuous Galerkin method. It builds a mixed first-order reformulation and uses Crank–Nicolson time stepping to achieve second-order accuracy in time while preserving mass and ensuring discrete energy dissipation through carefully designed inter-element fluxes. The method demonstrates unconditional energy stability and convergence in space and time, and numerical experiments validate energy decay and the ability to capture topological changes like droplet merging. The approach provides a robust framework for accurate, thermodynamically consistent simulations of compressible two-phase flow with diffuse interfaces, with potential extensions to temperature effects and more complex multi-phase settings.

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.
Paper Structure (18 sections, 6 theorems, 73 equations, 3 figures, 6 tables)

This paper contains 18 sections, 6 theorems, 73 equations, 3 figures, 6 tables.

Table of Contents

  1. Introduction
  2. A Phase-Field Approach to Compressible Two-Phase Flow
  3. The Compressible Navier--Stokes--Allen--Cahn System
  4. A Mixed Formulation of the NSAC System and Energy Stability
  5. Proof:
  6. An Energy-Stable dG Method for the NSAC System
  7. Proof:
  8. Proof:
  9. A Time Semi-Discrete Scheme for the NSAC System
  10. Proof:
  11. A Fully-Discrete dG Method for the NSAC System
  12. Proof:
  13. Numerical Experiments
  14. Convergence Studies
  15. Order of Convergence in Space
  16. ...and 3 more sections

Key Result

Theorem 2.3

Let $\boldsymbol{\mathbf{U}}= (\rho,\boldsymbol{\mathbf{v}},\varphi, \mu,\tau,\boldsymbol{\mathbf{\sigma}})$ be a weak solution to the initial boundary value problem for eq:p2:mixed. Then for all $t \in (0,T)$, the following energy inequality holds:

Figures (3)

  • Figure 1: (a) SI approach with time-dependent liquid/vapour bulk domains $\Omega_{L/V}=\Omega_{L/V}(t)$ and separating interface $\Gamma(t)$, (b) DI approach with ${\mathcal{O}}(\gamma)$-interfacial width.
  • Figure 2: Density $\rho$ at times $t=0$, $t=0.2$, and $t=2$ for the merging of two droplets.
  • Figure 3: Time evolution of the discrete energy $E= \hat{E}^{\Delta t}_h$ with different values for the mobility $\eta$. As expected, the energy decays faster with $\eta$ increasing.

Theorems & Definitions (13)

  • Remark 2.1
  • Definition 2.2: Weak solution of the mixed formulation \ref{['eq:p2:mixed']}
  • Definition 2.2: Weak solution of the mixed formulation \ref{['eq:p2:mixed']}
  • Theorem 2.3: Energy stability of the weak solution
  • Definition 3.1: General spatially semi-discrete dG method
  • Remark 3.2
  • Proposition 3.3: Elementwise integration
  • Theorem 3.4: Semi-discrete energy
  • Corollary 3.5: Stabilized semi-discrete dG approximation
  • Definition 3.6: Second-order time discretization
  • ...and 3 more