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Unconditionally Stable, Variable Step DLN Methods for the Allen-Cahn Active Fluid Model: A Divergence-free Preserving Approach

Nan Zheng, Wenlong Pei, Qingguang Guan, Wenju Zhao

TL;DR

This work develops an unconditionally stable, divergence-free mixed finite element framework for the Allen-Cahn active fluid system by introducing auxiliary variables to reduce the fourth-order problem to a second-order form and to preserve incompressibility. The spatial discretization is paired with a variable-step DLN time integrator, and the authors prove that the fully discrete scheme satisfies a discrete energy dissipation law on arbitrary time grids. An adaptive time-stepping strategy based on minimum dissipation further enhances computational efficiency. Comprehensive 2D and 3D numerical experiments demonstrate accurate convergence, robust phase-field dynamics including spinodal decomposition and mean-curvature motion, and substantial gains from time adaptivity. The approach provides a practical, stable toolkit for simulating complex phase-field–fluid interactions in active matter systems.

Abstract

This paper addresses the divergence-free mixed finite element method (FEM) for nonlinear fourth-order Allen-Cahn phase field coupled active fluid equations. By introducing an auxiliary variable $w = Δu$, the original fourth-order problem is converted into a system of second-order equations, thereby easing the regularity constraints imposed on standard $H^2$-comforming finite element spaces. To further refine the formulation, an additional auxiliary variable $ξ$, analogous to the pressure, is introduced, resulting in a mixed finite element scheme that preserves the divergence-free condition in $which = Δu$ inherited from the model. A fully discrete scheme is then established by combining the spatial approximation by the divergence-free mixed finite element method with the variable-step Dahlquist-Liniger-Nevanlinna (DLN) time integrator. The boundedness of the scheme is rigorously derived under suitable regularity assumptions. Additionally, an adaptive time-stepping strategy based on the minimum dissipation criterion is carried out to enhance computational efficiency. Several numerical experiments validate the theoretical findings and demonstrate the method's effectiveness and accuracy in simulating complex active fluid dynamics.

Unconditionally Stable, Variable Step DLN Methods for the Allen-Cahn Active Fluid Model: A Divergence-free Preserving Approach

TL;DR

This work develops an unconditionally stable, divergence-free mixed finite element framework for the Allen-Cahn active fluid system by introducing auxiliary variables to reduce the fourth-order problem to a second-order form and to preserve incompressibility. The spatial discretization is paired with a variable-step DLN time integrator, and the authors prove that the fully discrete scheme satisfies a discrete energy dissipation law on arbitrary time grids. An adaptive time-stepping strategy based on minimum dissipation further enhances computational efficiency. Comprehensive 2D and 3D numerical experiments demonstrate accurate convergence, robust phase-field dynamics including spinodal decomposition and mean-curvature motion, and substantial gains from time adaptivity. The approach provides a practical, stable toolkit for simulating complex phase-field–fluid interactions in active matter systems.

Abstract

This paper addresses the divergence-free mixed finite element method (FEM) for nonlinear fourth-order Allen-Cahn phase field coupled active fluid equations. By introducing an auxiliary variable , the original fourth-order problem is converted into a system of second-order equations, thereby easing the regularity constraints imposed on standard -comforming finite element spaces. To further refine the formulation, an additional auxiliary variable , analogous to the pressure, is introduced, resulting in a mixed finite element scheme that preserves the divergence-free condition in inherited from the model. A fully discrete scheme is then established by combining the spatial approximation by the divergence-free mixed finite element method with the variable-step Dahlquist-Liniger-Nevanlinna (DLN) time integrator. The boundedness of the scheme is rigorously derived under suitable regularity assumptions. Additionally, an adaptive time-stepping strategy based on the minimum dissipation criterion is carried out to enhance computational efficiency. Several numerical experiments validate the theoretical findings and demonstrate the method's effectiveness and accuracy in simulating complex active fluid dynamics.
Paper Structure (14 sections, 2 theorems, 44 equations, 48 figures, 4 tables, 1 algorithm)

This paper contains 14 sections, 2 theorems, 44 equations, 48 figures, 4 tables, 1 algorithm.

Table of Contents

  1. Introduction
  2. Notation and preliminaries
  3. Spatial and temporal discretization
  4. Spatial discretization
  5. Fully discrete DLN scheme
  6. Discrete energy dissipation law
  7. Numerical results
  8. Convergence Test
  9. Phase-field spinodal decomposition and fluid self-organization in active fluid
  10. Phase-field shape relaxation and fluid self-organization in active fluid
  11. Phase-field dynamics of circular bubble shrinking and fluid self-organization in active fluid
  12. Three-dimensional phase-field motion by mean curvature
  13. Time adaptive test
  14. Conclusion

Key Result

Lemma 4.2

\newlabelG-stable-lemma For any sequence $\{ v_{n} \}_{n=0}^{M} \subset [L^2(D)]^2$, the following $G$-stability identity holds holds for all $n = 1,2, \cdots, M-1$ and any fixed $\theta \in [0,1]$. Here $\{a_{\ell}^{(n)}\}_{\ell=0}^2$ in G-stable are

Figures (48)

  • Figure 5.2: Energy dissipation in the phase-field spinodal decomposition and fluid self-organization.
  • Figure : $t=0$
  • Figure : $t=0$
  • Figure : $t=0$
  • Figure : $t=0$
  • ...and 43 more figures

Theorems & Definitions (8)

  • Remark 1
  • Remark 2
  • Definition 4.1
  • Lemma 4.2
  • proof
  • Definition 4.3
  • Theorem 4.4: Discrete Energy dissipation Law
  • proof