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A high-order accurate unconditionally stable bound-preserving numerical scheme for the Cahn-Hilliard-Navier-Stokes equations

Yali Gao, Daozhi Han, Sayantan Sarkar

TL;DR

This work develops a high-order, bound-preserving, unconditionally energy-stable finite element scheme for the Cahn-Hilliard-Navier-Stokes system with the Flory-Huggins potential. It employs $Q_k$ mass-lumped elements on rectangles, Crank-Nicolson time stepping with convex-concave splitting, and a pressure-correction step, together with a discrete $L^1$ bound on the singular potential to ensure $\phi\in(-1,1)$. The authors prove unique solvability and a discrete energy law, and validate the method through convergence tests and simulations of spinodal decomposition, rotational flow, lid-driven cavity, and Rayleigh–Taylor instability, demonstrating robustness and accuracy. The approach advances high-order, energy-stable simulations of binary fluids with singular potentials and is extendable to other phase-field models on polygonal domains.

Abstract

A high-order numerical method is developed for solving the Cahn-Hilliard-Navier-Stokes equations with the Flory-Huggins potential. The scheme is based on the $Q_k$ finite element with mass lumping on rectangular grids, the second-order convex splitting method, and the pressure correction method. The unique solvability, unconditional stability, and bound-preserving properties are rigorously established. The key to bound-preservation is the discrete $L^1$ estimate of the singular potential. Ample numerical experiments are performed to validate the desired properties of the proposed numerical scheme.

A high-order accurate unconditionally stable bound-preserving numerical scheme for the Cahn-Hilliard-Navier-Stokes equations

TL;DR

This work develops a high-order, bound-preserving, unconditionally energy-stable finite element scheme for the Cahn-Hilliard-Navier-Stokes system with the Flory-Huggins potential. It employs mass-lumped elements on rectangles, Crank-Nicolson time stepping with convex-concave splitting, and a pressure-correction step, together with a discrete bound on the singular potential to ensure . The authors prove unique solvability and a discrete energy law, and validate the method through convergence tests and simulations of spinodal decomposition, rotational flow, lid-driven cavity, and Rayleigh–Taylor instability, demonstrating robustness and accuracy. The approach advances high-order, energy-stable simulations of binary fluids with singular potentials and is extendable to other phase-field models on polygonal domains.

Abstract

A high-order numerical method is developed for solving the Cahn-Hilliard-Navier-Stokes equations with the Flory-Huggins potential. The scheme is based on the finite element with mass lumping on rectangular grids, the second-order convex splitting method, and the pressure correction method. The unique solvability, unconditional stability, and bound-preserving properties are rigorously established. The key to bound-preservation is the discrete estimate of the singular potential. Ample numerical experiments are performed to validate the desired properties of the proposed numerical scheme.
Paper Structure (9 sections, 2 theorems, 64 equations, 9 figures, 4 tables)

This paper contains 9 sections, 2 theorems, 64 equations, 9 figures, 4 tables.

Table of Contents

  1. Introduction
  2. The numerical scheme
  3. Numerical experiments
  4. Convergence and accuracy
  5. Energy dissipation
  6. Rotational flow
  7. Lid-driven cavity flow
  8. Rayleigh-Taylor instability
  9. Conclusions

Key Result

Lemma 2.1

Suppose $|\phi_h^n(\bm{x}_i)|\leq 1-\delta, \forall \bm{x}_i \in Z_0$ for sufficiently small $\delta>0$. Then there exist constants $C_1, C_2>0$ independent of $\delta$ such that

Figures (9)

  • Figure 1: Spectral finite elements for quadrilateral element with degree $k=1,2,3$ in two dimensions, $\circ$ represents Gauss-Lobatto node.
  • Figure 2: Taylor-Hood element ${\bm{Q}}_k$--$Q_{k-1}$ element, $\circ$ denotes the velocity, $\times$ denotes the pressure.
  • Figure 3: The evolution of phase variable for coarsening process.
  • Figure 4: Evolution of the discrete energy and mass for coarsening process.
  • Figure 5: Evolution of the maximum value (left) and mininum value (right) of the phase variable.
  • ...and 4 more figures

Theorems & Definitions (3)

  • Lemma 2.1
  • Theorem 2.1
  • proof