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Maximum bound principle and original energy dissipation of arbitrarily high-order rescaled exponential time differencing Runge-Kutta schemes for Allen--Cahn equations

Chaoyu Quan, Xiaoming Wang, Pinzhong Zheng, Zhi Zhou

TL;DR

This work tackles the challenge of designing high-order time-stepping schemes for the Allen–Cahn equation that preserve the original energy dissipation and the maximum bound principle (MBP). It develops arbitrarily high-order exponential time differencing Runge–Kutta (ETDRK) schemes and introduces a rescaling post-processing technique that enforces MBP unconditionally while maintaining the same convergence order, and provides energy-dissipation guarantees with explicit step-size constraints. The authors establish a convergence theory showing $\|u_r^n-u(t_n)\|_{\infty} = O(\tau^r)$ under suitable regularity, and demonstrate, through numerical experiments, that MBP can be preserved unconditionally and the original energy decreases, even beyond theoretical limits. The results offer a practically valuable framework for reliable, high-accuracy simulations of gradient-flow phase-field models, with potential extensions to other nonlinear parabolic systems.

Abstract

The energy dissipation law and the maximum bound principle are two critical physical properties of the Allen--Cahn equations. While many existing time-stepping methods are known to preserve the energy dissipation law, most apply to a modified form of energy. In this work, we demonstrate that, when the nonlinear term of the Allen--Cahn equation is Lipschitz continuous, a class of arbitrarily high-order exponential time differencing Runge--Kutta (ETDRK) schemes preserve the original energy dissipation property, under a mild step-size constraint. Additionally, we guarantee the Lipschitz condition on the nonlinear term by applying a rescaling post-processing technique, which ensures that the numerical solution unconditionally satisfies the maximum bound principle. Consequently, our proposed schemes maintain both the original energy dissipation law and the maximum bound principle and can achieve arbitrarily high-order accuracy. We also establish an optimal error estimate for the proposed schemes. Some numerical experiments are carried out to verify our theoretical results.

Maximum bound principle and original energy dissipation of arbitrarily high-order rescaled exponential time differencing Runge-Kutta schemes for Allen--Cahn equations

TL;DR

This work tackles the challenge of designing high-order time-stepping schemes for the Allen–Cahn equation that preserve the original energy dissipation and the maximum bound principle (MBP). It develops arbitrarily high-order exponential time differencing Runge–Kutta (ETDRK) schemes and introduces a rescaling post-processing technique that enforces MBP unconditionally while maintaining the same convergence order, and provides energy-dissipation guarantees with explicit step-size constraints. The authors establish a convergence theory showing under suitable regularity, and demonstrate, through numerical experiments, that MBP can be preserved unconditionally and the original energy decreases, even beyond theoretical limits. The results offer a practically valuable framework for reliable, high-accuracy simulations of gradient-flow phase-field models, with potential extensions to other nonlinear parabolic systems.

Abstract

The energy dissipation law and the maximum bound principle are two critical physical properties of the Allen--Cahn equations. While many existing time-stepping methods are known to preserve the energy dissipation law, most apply to a modified form of energy. In this work, we demonstrate that, when the nonlinear term of the Allen--Cahn equation is Lipschitz continuous, a class of arbitrarily high-order exponential time differencing Runge--Kutta (ETDRK) schemes preserve the original energy dissipation property, under a mild step-size constraint. Additionally, we guarantee the Lipschitz condition on the nonlinear term by applying a rescaling post-processing technique, which ensures that the numerical solution unconditionally satisfies the maximum bound principle. Consequently, our proposed schemes maintain both the original energy dissipation law and the maximum bound principle and can achieve arbitrarily high-order accuracy. We also establish an optimal error estimate for the proposed schemes. Some numerical experiments are carried out to verify our theoretical results.
Paper Structure (11 sections, 6 theorems, 109 equations, 3 figures, 3 tables)

This paper contains 11 sections, 6 theorems, 109 equations, 3 figures, 3 tables.

Table of Contents

  1. Introduction
  2. Original energy dissipation of ETDRK methods
  3. ETDRK methods with rescaling technique
  4. Rescaling technique
  5. Energy dissipation of ETDRK methods with rescaling technique
  6. Convergence analysis
  7. Numerical experiments
  8. Convergence in time
  9. Unconditional preservation of the MBP
  10. Original energy dissipation law
  11. Conclusions

Key Result

Lemma 2.1

\newlabellemma:psi_k bound0 For any $t>0$, consider the negative-definite operator $t\mathcal{L}_{\kappa}=t(\varepsilon^{2}\Delta-\kappa \mathcal{I})$ with homogeneous Neumann boundary condition. For any integer $k \ge 0$ and $\lambda \in (0,1)$, the following inequality holds: where $\|\cdot\|$ is short for $\|\cdot\|_{L^2(\Omega)}$.

Figures (3)

  • Figure 1: Comparison of the cut-off (left) and rescaling (right) techniques.
  • Figure 1: Evolutions of the maximum norms of the numerical solutions for $r=3,5$ and $7$, respectively (left to right).
  • Figure 2: Comparison of original energy with different schemes but same time steps for $\tau=0.2$, $0.1$ and $0.01$, respectively (left to right).

Theorems & Definitions (15)

  • Lemma 2.1
  • Proof 1
  • Theorem 2.2
  • Proof 2
  • Remark 2.3
  • Lemma 3.1: DU-JU-LI-QIAO2021ETD
  • Theorem 3.2
  • Proof 3
  • Remark 3.3
  • Remark 3.4
  • ...and 5 more