Logo
ScienceStack
Table of Contents
Fetching ...
Sign In

Dynamic-stabilization-based linear schemes for the Allen-Cahn equation with degenerate mobility: MBP and energy stability

Hongfei Fu, Dianming Hou, Zhonghua Qiao, Bingyin Zhang

TL;DR

This work addresses the Allen–Cahn equation with general, including degenerate, mobility by introducing a dynamic stabilization framework that yields MBP preservation and energy stability at the discrete level. It develops a first-order DsBE scheme and a second-order DsCN scheme that, through a novel prediction–correction approach and a cut-off preprocessing step, require only a single linear solve per time step while maintaining unconditional mobility-robust MBP and energy dissipation. The authors prove maximum-norm error estimates for both schemes and establish mobility-robust energy stability despite degeneracy in the mobility function. Numerical experiments across constant and degenerate mobilities, including 3D simulations, validate the theoretical results and demonstrate improved efficiency via adaptive time stepping and robust long-time dynamics under degenerate mobility.

Abstract

In this paper, we investigate linear first- and second-order numerical schemes for the Allen--Cahn equation with a general (possibly degenerate) mobility. Compared with existing numerical methods, our schemes employ a novel dynamic stabilization approach that guarantees unconditional preservation of the maximum bound principle (MBP) and energy stability. A key advance is that the discrete energy stability remains valid even in the presence of degenerate mobility-a property we refer to as mobility robustness. Rigorous maximum-norm error estimates are also established. In particular, for the second-order scheme, we introduce a new prediction strategy with a cut-off preprocessing procedure on the extrapolation solution, and only one linear system needs to be solved per time level. Representative numerical examples are provided to validate the theoretical findings and performance of the proposed schemes.

Dynamic-stabilization-based linear schemes for the Allen-Cahn equation with degenerate mobility: MBP and energy stability

TL;DR

This work addresses the Allen–Cahn equation with general, including degenerate, mobility by introducing a dynamic stabilization framework that yields MBP preservation and energy stability at the discrete level. It develops a first-order DsBE scheme and a second-order DsCN scheme that, through a novel prediction–correction approach and a cut-off preprocessing step, require only a single linear solve per time step while maintaining unconditional mobility-robust MBP and energy dissipation. The authors prove maximum-norm error estimates for both schemes and establish mobility-robust energy stability despite degeneracy in the mobility function. Numerical experiments across constant and degenerate mobilities, including 3D simulations, validate the theoretical results and demonstrate improved efficiency via adaptive time stepping and robust long-time dynamics under degenerate mobility.

Abstract

In this paper, we investigate linear first- and second-order numerical schemes for the Allen--Cahn equation with a general (possibly degenerate) mobility. Compared with existing numerical methods, our schemes employ a novel dynamic stabilization approach that guarantees unconditional preservation of the maximum bound principle (MBP) and energy stability. A key advance is that the discrete energy stability remains valid even in the presence of degenerate mobility-a property we refer to as mobility robustness. Rigorous maximum-norm error estimates are also established. In particular, for the second-order scheme, we introduce a new prediction strategy with a cut-off preprocessing procedure on the extrapolation solution, and only one linear system needs to be solved per time level. Representative numerical examples are provided to validate the theoretical findings and performance of the proposed schemes.
Paper Structure (18 sections, 7 theorems, 78 equations, 11 figures, 3 tables)

This paper contains 18 sections, 7 theorems, 78 equations, 11 figures, 3 tables.

Table of Contents

  1. Introduction
  2. First-order stabilized backward Euler scheme
  3. Dynamically stabilized backward Euler scheme
  4. The MBP and energy dissipation
  5. Maximum-norm error estimate
  6. Second-order stabilized Crank-Nicolson scheme
  7. Step 1
  8. Step 2
  9. Discrete MBP
  10. Maximum-norm error estimate
  11. Discrete energy stability
  12. Numerical experiments
  13. Temporal convergence
  14. MBP preservation and energy stability
  15. Application of adaptive time-stepping strategy
  16. ...and 3 more sections

Key Result

Lemma 2.3

If $\kappa \geq \left\|f^{\prime}\right\|_{C[-1, 1]} = 2$ holds for some positive constant $\kappa$, then we have $|f( \xi ) + \kappa \xi | \leq \kappa$ for any $\xi \in[-1, 1]$.

Figures (11)

  • Figure 1: The maximum norm (top) and energy (bottom) of simulated solutions computed by the DsBE scheme for different mobilities.
  • Figure 2: The maximum norm (top) and energy (bottom) of simulated solutions computed by the DsCN scheme with $S_{2} = \left( S_{1}/4 + \varepsilon^{2}/h^2 \right)^{2}$ for different mobilities.
  • Figure 3: The dynamic snapshots of the numerical solution $\phi$ at different time instants obtained by the DsCN scheme with uniform (top, $\tau = 0.25$; bottom, $\tau = 0.025$) and adaptive (middle) time stepsizes.
  • Figure 4: The maximum norm (left), energy (middle), and stepsizes (right) for the DsCN scheme.
  • Figure 5: Plot of $\mathcal{ M }(\phi)$ on the interval $[-1,1]$.
  • ...and 6 more figures

Theorems & Definitions (21)

  • Remark 2.1
  • Remark 2.2
  • Lemma 2.3: SIREV_Du_2021
  • Theorem 2.4: MBP of the DsBE scheme
  • proof
  • Theorem 2.5: Energy dissipation of the DsBE scheme
  • proof
  • Theorem 2.6: Error estimate of the DsBE scheme
  • proof
  • Remark 3.1
  • ...and 11 more