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Maximum bound preservation of exponential integrators for Allen-Cahn equations

Chaoyu Quan, Pingzhong Zheng, Zhi Zhou

TL;DR

The paper tackles preserving the maximum bound in numerical solutions of the Allen--Cahn equation while achieving arbitrarily high temporal accuracy. It introduces MBP exponential integrators that stabilize the nonlinear term using a parameter $\kappa$ and employ Gauss--Legendre or left Gauss--Radau quadrature to ensure underestimation of the nonlinear integral, enabling unconditional MBP without any postprocessing. A rigorous $L^2$ error analysis shows the $k$th-order EI$k$ schemes converge with $O(\tau^{k})$ accuracy, and numerical experiments validate both convergence and MBP preservation, while illustrating that inappropriate quadrature can violate the maximum principle. The approach provides a robust, high-order, MBP time-stepping framework for phase-field simulations, with practical impact on long-time stability and physical fidelity of Allen--Cahn dynamics.

Abstract

We develop and analyze a class of arbitrarily high-order, maximum bound preserving time-stepping schemes for solving Allen-Cahn equations. These schemes are constructed within the iterative framework of exponential integrators, combined with carefully chosen numerical quadrature rules, including the Gauss-Legendre quadrature rule and the left Gauss-Radau quadrature rule. Notably, the proposed schemes are rigorously proven to unconditionally preserve the maximum bound without requiring any additional postprocessing techniques, while simultaneously achieving arbitrarily high-order temporal accuracy. A thorough error analysis in the $L^2$ norm is provided. Numerical experiments validate the theoretical results, demonstrate the effectiveness of the proposed methods, and highlight that an inappropriate choice of quadrature rules may violate the maximum bound principle, leading to incorrect dynamics.

Maximum bound preservation of exponential integrators for Allen-Cahn equations

TL;DR

The paper tackles preserving the maximum bound in numerical solutions of the Allen--Cahn equation while achieving arbitrarily high temporal accuracy. It introduces MBP exponential integrators that stabilize the nonlinear term using a parameter and employ Gauss--Legendre or left Gauss--Radau quadrature to ensure underestimation of the nonlinear integral, enabling unconditional MBP without any postprocessing. A rigorous error analysis shows the th-order EI schemes converge with accuracy, and numerical experiments validate both convergence and MBP preservation, while illustrating that inappropriate quadrature can violate the maximum principle. The approach provides a robust, high-order, MBP time-stepping framework for phase-field simulations, with practical impact on long-time stability and physical fidelity of Allen--Cahn dynamics.

Abstract

We develop and analyze a class of arbitrarily high-order, maximum bound preserving time-stepping schemes for solving Allen-Cahn equations. These schemes are constructed within the iterative framework of exponential integrators, combined with carefully chosen numerical quadrature rules, including the Gauss-Legendre quadrature rule and the left Gauss-Radau quadrature rule. Notably, the proposed schemes are rigorously proven to unconditionally preserve the maximum bound without requiring any additional postprocessing techniques, while simultaneously achieving arbitrarily high-order temporal accuracy. A thorough error analysis in the norm is provided. Numerical experiments validate the theoretical results, demonstrate the effectiveness of the proposed methods, and highlight that an inappropriate choice of quadrature rules may violate the maximum bound principle, leading to incorrect dynamics.

Paper Structure

This paper contains 8 sections, 4 theorems, 66 equations, 5 figures, 2 tables.

Table of Contents

  1. Introduction
  2. Preliminaries
  3. MBP exponential integrators
  4. Error analysis
  5. Numerical experiments
  6. Convergence tests
  7. Maximum bound preservation tests
  8. Concluding remarks

Key Result

Lemma 2.1

\newlabellemma:gauss_quadrature0 For a function $g$ defined on $[a,b]$, the optimal quadrature rule of degree $2J+K-1$ for numerical integration is given by where $\{t_j\}_{j=1}^{J} \subset (a,b)$ are inner nodes, and $\{z_k\}_{k=1}^{K} \subset \{a,b\}$ are boundary nodes. The corresponding weights $\{w_j\}_{j=1}^{J}$ and $\{v_k\}_{k=1}^{K}$ are positive. The remainder term $R_{J,K}(g)$ vanishes

Figures (5)

  • Figure 1: Evolution of the energy and supremum norm of numerical solutions for \ref{['eq:AC_log']}, computed using the EI2 to EI5 schemes with left Gauss–Radau quadrature (time step size $\tau=0.1$).
  • Figure 2: Evolution of the energy and supremum norm of numerical solutions for \ref{['eq:AC_log']}, computed using the EI2 to EI5 schemes with left Gauss–Radau quadrature (time step size $\tau=0.01$).
  • Figure 3: Solution snapshots of EI2 to EI4 using left/right Gauss–Radau rules with $\tau = 10^{-3}$, and EI5 (reference) using the left Gauss–Radau rule with $\tau = 10^{-4}$ for solving \ref{['eq:AC_log_2']}. (The solutions of right EI2 and right EI3 blow up.)
  • Figure 4: Evolution of the supremum norm of numerical solutions computed by the EI2 and EI3 using right Gauss--Radau rule with $\tau = 10^{-3}$ for solving \ref{['eq:AC_log_2']}.
  • Figure 5: Evolution of the energy and supremum norm of numerical solutions the computed by EI2 to EI4 schemes using the left Gauss--Radau rule ($\tau = 10^{-3}$), EI4 using the right Gauss--Radau rule ($\tau = 10^{-3}$), and EI5 (reference) using the left Gauss--Radau rule ($\tau = 10^{-4}$) for solving \ref{['eq:AC_log_2']}.

Theorems & Definitions (9)

  • Lemma 2.1
  • Remark 2.2
  • Lemma 2.3
  • Proof 1: Proof of Lemma \ref{['lemma:gauss_quadrature']}
  • Theorem 3.1: discrete maximum principle
  • Proof 2
  • Remark 3.2
  • Theorem 4.1
  • Proof 3