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Structure-preserving weighted BDF2 methods for Anisotropic Cahn-Hilliard model: uniform/variable-time-steps

Meng Li, Jingjiang Bi, Nan Wang

TL;DR

This work addresses the challenge of simulating anisotropic Cahn–Hilliard dynamics with stable, mass-conserving time integration. It develops uniform- and variable-time-step weighted and shifted BDF2 schemes (WSBDF2) that integrate the scalar auxiliary variable (SAV) approach with stabilization to achieve energy stability for both linear and Willmore regularized models. The authors prove energy stability and mass conservation for the uniform schemes via G-stability and a separate analysis for variable steps, and introduce efficient solution strategies that preserve the problem's structure. Numerical experiments in 1D and 2D validate stability, accuracy, and the expected regularization- and anisotropy-induced behaviors, demonstrating the practical effectiveness of the proposed methods for structure-preserving simulations of anisotropic CH dynamics.

Abstract

In this paper, we innovatively develop uniform/variable-time-step weighted and shifted BDF2 (WSBDF2) methods for the anisotropic Cahn-Hilliard (CH) model, combining the scalar auxiliary variable (SAV) approach with two types of stabilized techniques. Using the concept of $G$-stability, the uniform-time-step WSBDF2 method is theoretically proved to be energy-stable. Due to the inapplicability of the relevant G-stability properties, another technique is adopted in this work to demonstrate the energy stability of the variable-time-step WSBDF2 method. In addition, the two numerical schemes are all mass-conservative.Finally, numerous numerical simulations are presented to demonstrate the stability and accuracy of these schemes.

Structure-preserving weighted BDF2 methods for Anisotropic Cahn-Hilliard model: uniform/variable-time-steps

TL;DR

This work addresses the challenge of simulating anisotropic Cahn–Hilliard dynamics with stable, mass-conserving time integration. It develops uniform- and variable-time-step weighted and shifted BDF2 schemes (WSBDF2) that integrate the scalar auxiliary variable (SAV) approach with stabilization to achieve energy stability for both linear and Willmore regularized models. The authors prove energy stability and mass conservation for the uniform schemes via G-stability and a separate analysis for variable steps, and introduce efficient solution strategies that preserve the problem's structure. Numerical experiments in 1D and 2D validate stability, accuracy, and the expected regularization- and anisotropy-induced behaviors, demonstrating the practical effectiveness of the proposed methods for structure-preserving simulations of anisotropic CH dynamics.

Abstract

In this paper, we innovatively develop uniform/variable-time-step weighted and shifted BDF2 (WSBDF2) methods for the anisotropic Cahn-Hilliard (CH) model, combining the scalar auxiliary variable (SAV) approach with two types of stabilized techniques. Using the concept of -stability, the uniform-time-step WSBDF2 method is theoretically proved to be energy-stable. Due to the inapplicability of the relevant G-stability properties, another technique is adopted in this work to demonstrate the energy stability of the variable-time-step WSBDF2 method. In addition, the two numerical schemes are all mass-conservative.Finally, numerous numerical simulations are presented to demonstrate the stability and accuracy of these schemes.
Paper Structure (13 sections, 10 theorems, 95 equations, 12 figures)

This paper contains 13 sections, 10 theorems, 95 equations, 12 figures.

Table of Contents

  1. Introduction
  2. Anisotropic Cahn–Hilliard model equation and its energy law
  3. Numerical schemes on uniform temporal mesh
  4. Linear regularization model
  5. Willmore regularization model
  6. Numerical scheme on nonuniform temporal mesh
  7. Linear regularization model
  8. Willmore regularization model
  9. Numerical simulations
  10. Numerical simulations in 1D
  11. Numerical simulations in 2D
  12. Conclusions
  13. Acknowledgements

Key Result

Lemma 3.1

\newlabellem:lem3.1 For any given sequence $\{u^n\}$, it holds where $\alpha_0=\alpha_2=(\theta(2\theta-1))^{\frac{1}{2}}$ and $\alpha_1=-(\theta(2\theta-1))^{\frac{1}{2}}$.

Figures (12)

  • Figure 5.1: $L^2$-norm errors for the uniform-time-step method with different $\theta$: $\theta=0.5$ (first column), $\theta=0.75$ (second column), $\theta=1$ (last column), and with different $\alpha$: $\alpha = 0$ (first row), $\alpha=0.05$ (second row) and $\alpha = 0.3$ (last row). The other parameters are selected by \ref{['eqn:parameters']}.
  • Figure 5.2: $L^2$-norm errors for the variable-time-step method with different $\theta$: $\theta=0.5$ (first column), $\theta=0.75$ (second column), $\theta=1$ (last column), and with different $\alpha$: $\alpha = 0$ (first row), $\alpha=0.05$ (second row) and $\alpha = 0.3$ (last row). The other parameters are selected by \ref{['eqn:parameters']}.
  • Figure 5.3: The relative error of mass and the energy evolutions for $\mathcal{U}_L$-method with different $\theta$: $\theta=0.5$ (first column), $\theta=0.75$ (second column), $\theta=1$ (last column). The initial condition is chosen as \ref{['eqn:initial-condition_2_1']} and the other parameters are selected by \ref{['eqn:parameters']}. (a) The relative error of mass with $\tau=1e-3$. (b) The modified energy \ref{['eqn:UE_linear']} with $\tau=1e-3$.
  • Figure 5.4: The relative error of mass and the energy evolutions for $\mathcal{V}_L$-method with different $\theta$: $\theta=0.5$ (first column), $\theta=0.75$ (second column), $\theta=1$ (last column). The initial condition is chosen as \ref{['eqn:initial-condition_2_1']} and the other parameters are selected by \ref{['eqn:parameters']}. (a) The relative error of mass with $\tau_{max}=1.0165e-4$. (b) The modified energy \ref{['eqn:VE_linear']} with $\tau_{max}=1.0165e-4$.
  • Figure 5.5: The relative error of mass and the energy evolutions for $\mathcal{U}_L$-method with different $\theta$: $\theta=0.5$ (first column), $\theta=0.75$ (second column), $\theta=1$ (last column). The random initial condition \ref{['eqn:initial-condition_2_2']} is chosen and the other parameters are selected by \ref{['eqn:parameters']}. (a) The relative error of mass with $\tau=1e-3$. (b) The modified energy \ref{['eqn:VE_linear']} with $\tau=1e-3$.
  • ...and 7 more figures

Theorems & Definitions (19)

  • Lemma 3.1
  • proof
  • Remark 3.1
  • Theorem 3.1
  • proof
  • Theorem 3.2
  • proof
  • Remark 3.2
  • Theorem 3.3
  • Theorem 3.4
  • ...and 9 more