Arbitrary High-Order Maximum Principle-Preserving and Energy Dissipating Schemes for Gradient Flows

TL;DR

This work tackles the challenge of designing time-stepping schemes for gradient flows that are arbitrarily high-order in time while strictly preserving energy dissipation and the maximum bound principle (MBP). It introduces a predictor–corrector–corrector (PCC) framework based on Karush–Kuhn–Tucker (KKT) conditions, expanding the original gradient-flow system and enforcing energy and bound constraints through two Lagrange multipliers, $\eta$ and $\lambda$. The framework is shown to be solvable and structure-preserving, with an efficient compatible ETDRK-based implementation (ETDRK-PC) that can omit one correction step under energy-stability assumptions, and a convergence analysis establishing optimal rates under standard conditions. Spatial discretizations (Fourier spectral and finite difference) are shown to retain the energy-dissipation and MBP properties at the discrete level, and extensive numerical experiments on Allen–Cahn and Cahn–Hilliard models demonstrate robust energy decay, MBP preservation, and high accuracy, including avoidance of nonphysical oscillations. Overall, the PCC approach provides a flexible, high-order, structure-preserving framework for gradient flows with broad applicability to phase-field models.

Abstract

For gradient flows, the existing structure-preserving schemes are difficult to achieve arbitrary high-order accuracy in time while preserving maximum-principle (MBP) and energy dissipating simultaneously. In this paper, we develop a new framework for constructing structure-preserving schemes which shall preserve those nice properties. By introducing KKT-conditions for energy dissipating and bound-preserving, we rewrite the original gradient flow into an expanded and coupled system. We shall utilize a novel predictor-corrector-corrector framework, termed the PCC method, which consists of a prediction from any numerical scheme to the user's favor, followed by two correction steps designed to enforce energy stability and MBP, respectively. We take the exponential time differencing Runge-Kutta scheme (ETDRK) as an example and establish the unique solvability and robust error analysis for our new framework. Extensive numerical experiments are provided to validate the efficiency and accuracy of our new approach. Enough numerical comparisons with the existing popular schemes are shown that our structure-preserving schemes can avoid numerical oscillations and capture the exact evolution of energy.

Figures (13)

• Figure 1: (Example \ref{['Ex 1']}) Time evolution of energy $\mathcal{E}[\phi^{n}]$ of the numerical solution $\phi^{n}$ with $\tau = 0.01$ and $S=2/\varepsilon^2$.
• Figure 1: (Example \ref{['Ex 1']}) Spatial error of proposed schemes under $M\times M$ uniform grid.
• Figure 2: (Example \ref{['Ex 1']}) The upper bounds of the numerical solution $\phi^{n}$ with $\tau = 0.01$ and $S=2/\varepsilon^2$.
• Figure 2: (Example \ref{['Ex 1']}) Contour plots of the numerical solution $\phi^n$ at $t = 0$, $0.1$, $0.2$, $0.5$ with $\tau = 0.01$. 1st row: U-ETDRK3; 2nd row: U-ETDRK3-PCC.
• Figure 3: (Example \ref{['Ex 1']}) Time evolution of energy $\mathcal{E}[\phi^{n}]$, and the upper and lower bounds of the numerical solution $\phi^{n}$ from U-ETDRK and U-ETDRK-PCC schemes with $\tau = 0.01$ and $S=1/\varepsilon^2$.

Theorems & Definitions (21)

• Theorem 2.1: Existence and uniqueness
• Lemma 2.2: zorich2016mathematical, Section 10.7, Implicit function theorem
• Theorem 2.3: Energy stability and MBP preservation
• Lemma 2.4: gilbarg1977elliptic, Section 7.4, Theorem 7.8
• Theorem 2.5: Properties of PCC'
• Lemma 2.6: du2021maximum
• Lemma 2.7
• Proof 2
• Remark 3.1
• Lemma 3.2: fu2024higher