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A new decoupled unconditionally stable scheme and its optimal error analysis for the Cahn-Hilliard-Navier-Stokes equations

Haijun Gao, Xi Li, Minfu Feng

TL;DR

Addresses the numerical solution of the CHNS system by developing a decoupled, first-order, fully discrete scheme that preserves unconditional energy stability via SAV and a pressure-correction projection. The method explicitly treats the CH velocity to decouple CH from NS, and uses Ritz quasi-projections to achieve optimal $L^2$ error estimates for the fully discrete scheme. Theoretical results include unconditional stability and optimal error bounds, validated by numerical experiments for convergence and interface dynamics. The work provides an efficient, stable framework for simulating two-phase incompressible flows with provable accuracy guarantees.

Abstract

We construct a decoupled, first-order, fully discrete, and unconditionally energy stable scheme for the Cahn-Hilliard-Navier-Stokes equations. The scheme is divided into two main parts. The first part involves the calculation of the Cahn-Hilliard equations, and the other part is calculating the Navier-Stokes equations subsequently by utilizing the phase field and chemical potential values obtained from the above step. Specifically, the velocity in the Cahn-Hilliard equation is discretized explicitly at the discrete time level, which enables the computation of the Cahn-Hilliard equations is fully decoupled from that of Navier-Stokes equations. Furthermore, the pressure-correction projection method, in conjunction with the scalar auxiliary variable approach not only enables the discrete scheme to satisfy unconditional energy stability, but also allows the convective term in the Navier-Stokes equations to be treated explicitly. We subsequently prove that the time semi-discrete scheme is unconditionally stable and analyze the optimal error estimates for the fully discrete scheme. Finally, several numerical experiments validate the theoretical results.

A new decoupled unconditionally stable scheme and its optimal error analysis for the Cahn-Hilliard-Navier-Stokes equations

TL;DR

Addresses the numerical solution of the CHNS system by developing a decoupled, first-order, fully discrete scheme that preserves unconditional energy stability via SAV and a pressure-correction projection. The method explicitly treats the CH velocity to decouple CH from NS, and uses Ritz quasi-projections to achieve optimal error estimates for the fully discrete scheme. Theoretical results include unconditional stability and optimal error bounds, validated by numerical experiments for convergence and interface dynamics. The work provides an efficient, stable framework for simulating two-phase incompressible flows with provable accuracy guarantees.

Abstract

We construct a decoupled, first-order, fully discrete, and unconditionally energy stable scheme for the Cahn-Hilliard-Navier-Stokes equations. The scheme is divided into two main parts. The first part involves the calculation of the Cahn-Hilliard equations, and the other part is calculating the Navier-Stokes equations subsequently by utilizing the phase field and chemical potential values obtained from the above step. Specifically, the velocity in the Cahn-Hilliard equation is discretized explicitly at the discrete time level, which enables the computation of the Cahn-Hilliard equations is fully decoupled from that of Navier-Stokes equations. Furthermore, the pressure-correction projection method, in conjunction with the scalar auxiliary variable approach not only enables the discrete scheme to satisfy unconditional energy stability, but also allows the convective term in the Navier-Stokes equations to be treated explicitly. We subsequently prove that the time semi-discrete scheme is unconditionally stable and analyze the optimal error estimates for the fully discrete scheme. Finally, several numerical experiments validate the theoretical results.
Paper Structure (19 sections, 16 theorems, 204 equations, 3 figures, 2 tables)

This paper contains 19 sections, 16 theorems, 204 equations, 3 figures, 2 tables.

Table of Contents

  1. Introduction
  2. Preliminaries
  3. Some basic notations
  4. Ritz and Stokes quasi-projections
  5. The numerical scheme
  6. SAV method
  7. The first order semi-discretization scheme
  8. Unconditional energy stability
  9. Fully discrete scheme
  10. Error analysis
  11. Estimates for $e_\phi^{n+1}$ and $e_\mu^{n+1}$
  12. Estimate for $e_\mathbf{u}^{n+1}$
  13. $L^2$ norm estimates for $e_\phi^{n+1}$ and $e_\mu^{n+1}$
  14. Estimates for $e_p^{n+1}$
  15. Numerical experiments
  16. ...and 4 more sections

Key Result

Lemma 2.1

For the Ritz quasi-projection $\Pi_h$, it holds that

Figures (3)

  • Figure 1: Snapshots of phase function at difference times from left to right row by row with $t=0, 5\!\times\!10^{-6}, 10^{-5}, 5\!\times\!10^{-5}, 5\!\times\!10^{-4}, 10^{-3}$, respectively.
  • Figure 2: Snapshots of phase function at difference times from left to right row by row with $t=0, 0.001, 0.03, 0.08, 0.3, 1$, respectively.
  • Figure 3: Evolution of the modified energy, left: example \ref{['example_DESF']}; right: example \ref{['example_DSSF']} .

Theorems & Definitions (30)

  • Lemma 2.1: 2023_CaiWentao_Optimal_L2_error_estimates_of_unconditionally_stable_finite_element_schemes_for_the_Cahn_Hilliard_Navier_Stokes_system
  • Lemma 2.2: 2023_CaiWentao_Optimal_L2_error_estimates_of_unconditionally_stable_finite_element_schemes_for_the_Cahn_Hilliard_Navier_Stokes_system
  • Lemma 2.3: 2023_CaiWentao_Optimal_L2_error_estimates_of_unconditionally_stable_finite_element_schemes_for_the_Cahn_Hilliard_Navier_Stokes_system
  • Lemma 2.4: 2008_Brenner_Susanne_C_The_mathematical_theory_of_finite_element_methods
  • Lemma 2.5: 2007_HeYinnian_SunWeiwei_Stability_and_convergence_of_the_Crank_Nicolson_Adams_Bashforth_scheme_for_the_time_dependent_Navier_Stokes_equations
  • Remark 3.1
  • Remark 3.2
  • Theorem 3.1
  • proof
  • Remark 3.3
  • ...and 20 more