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Comparison of Structure Preserving Schemes for the Cahn-Hilliard-Navier-Stokes Equations with Degenerate Mobility and Adaptive Mesh Refinement

Jimmy Kornelije Gunnarsson, Robert Klöfkorn

TL;DR

This work evaluates structure-preserving numerical schemes for the Cahn-Hilliard-Navier-Stokes system with degenerate mobility and adaptive mesh refinement. It compares decoupled IMEX-DG formulations (SIPG-SWIPG- variants) and auxiliary-variable approaches (ASU) against standard FEM schemes, focusing on energy dissipation, mass conservation, and bound preservation. The results show that DG-based schemes with limiters and the SWIP-L formulation achieve strong bound preservation and mass conservation, while manageably enforcing energy dissipation; adaptive mesh refinement enhances accuracy near interfaces. The findings guide the choice of schemes for multi-phase flow simulations and suggest pathways to extend the methods to grid- and space-adaptive high-order DG implementations.

Abstract

The Cahn-Hilliard-Navier-Stokes (CHNS) system utilizes a diffusive phase-field for interface tracking of multi-phase fluid flows. Recently structure preserving methods for CHNS have moved into focus to construct numerical schemes that, for example, are mass conservative or obey initial bounds of the phase-field variable. In this work decoupled implicit-explicit formulations based on the Discontinuous Galerkin (DG) methodology are considered and compared to existing schemes from the literature. For the fluid flow a standard continuous Galerkin approach is applied. An adaptive conforming grid is utilized to further draw computational focus on the interface regions, while coarser meshes are utilized around pure phases. All presented methods are compared against each other in terms of bound preservation, mass conservation, and energy dissipation for different examples found in the literature, including a classical rising droplet problem.

Comparison of Structure Preserving Schemes for the Cahn-Hilliard-Navier-Stokes Equations with Degenerate Mobility and Adaptive Mesh Refinement

TL;DR

This work evaluates structure-preserving numerical schemes for the Cahn-Hilliard-Navier-Stokes system with degenerate mobility and adaptive mesh refinement. It compares decoupled IMEX-DG formulations (SIPG-SWIPG- variants) and auxiliary-variable approaches (ASU) against standard FEM schemes, focusing on energy dissipation, mass conservation, and bound preservation. The results show that DG-based schemes with limiters and the SWIP-L formulation achieve strong bound preservation and mass conservation, while manageably enforcing energy dissipation; adaptive mesh refinement enhances accuracy near interfaces. The findings guide the choice of schemes for multi-phase flow simulations and suggest pathways to extend the methods to grid- and space-adaptive high-order DG implementations.

Abstract

The Cahn-Hilliard-Navier-Stokes (CHNS) system utilizes a diffusive phase-field for interface tracking of multi-phase fluid flows. Recently structure preserving methods for CHNS have moved into focus to construct numerical schemes that, for example, are mass conservative or obey initial bounds of the phase-field variable. In this work decoupled implicit-explicit formulations based on the Discontinuous Galerkin (DG) methodology are considered and compared to existing schemes from the literature. For the fluid flow a standard continuous Galerkin approach is applied. An adaptive conforming grid is utilized to further draw computational focus on the interface regions, while coarser meshes are utilized around pure phases. All presented methods are compared against each other in terms of bound preservation, mass conservation, and energy dissipation for different examples found in the literature, including a classical rising droplet problem.
Paper Structure (27 sections, 10 theorems, 109 equations, 19 figures, 5 tables, 3 algorithms)

This paper contains 27 sections, 10 theorems, 109 equations, 19 figures, 5 tables, 3 algorithms.

Table of Contents

  1. Introduction
  2. Mathematical Model
  3. Governing equations
  4. Physical laws
  5. Discretization
  6. Notation
  7. Discontinuous Galerkin formulation
  8. The Acosta-Soba upwinding scheme
  9. Time discretization
  10. Scaling limiter
  11. Standard Finite-Element Schemes
  12. The Incremental Pressure Correction Scheme
  13. Algorithm
  14. Numerics
  15. Software implementation
  16. ...and 12 more sections

Key Result

Theorem 2.1

The function $\boldsymbol{J}$ corresponds to the diffusion of the conserved physical mass $m_{\rho}$ in time as a consequence of the Eqs.eq:chdyn---eq:ch1 if $\boldsymbol{n} \cdot \boldsymbol{u}|_{\partial \Omega} = 0$.

Figures (19)

  • Figure 1: An example of a scaling limiter for $\psi = 1.1\tanh(10(x-0.5))$.
  • Figure 2: SWIP-L: Evolution of the phase-field $\psi_h$ at different time steps for the finest grid.
  • Figure 3: ASU: Evolution of the phase-field $\psi_h$ at different time steps for the finest grid.
  • Figure 4: Mass conservation comparison between DG and FEM-based schemes. For FEM-C we see a clear deviation from the initial mass.
  • Figure 5: Energy dissipation comparison between DG and FEM-based schemes.
  • ...and 14 more figures

Theorems & Definitions (42)

  • Theorem 2.1: Equivalence to physical mass diffusion
  • proof
  • Remark 2.1: Pressure boundary conditions
  • Remark 2.2
  • Definition 2.1: Phase-field mass
  • Remark 2.3: Mass conservation
  • Theorem 2.2: Energy dissipation
  • proof
  • Theorem 2.3: Bound preservation, Elliott:2000
  • proof
  • ...and 32 more