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Higher-Order Energy-Decreasing Exponential Time Differencing Runge-Kutta methods for Gradient Flows

Zhaohui Fu, Jie Shen, Jiang Yang

TL;DR

This work addresses the challenge of designing unconditionally energy-stable exponential time-differencing Runge-Kutta (ETDRK) schemes for gradient flows. It develops a general stabilization-based framework in which energy dissipation is guaranteed if a determinant condition $\Delta(z)$ is positive-definite for all negative $z$, with a stabilization parameter $\beta$ chosen to satisfy $\beta \ge C_L$. The authors demonstrate that common third- and fourth-order ETDRK schemes fail this stability criterion, and they construct two new third-order schemes that satisfy it, proving unconditional energy decay via a positive-definite $\Delta'$ analysis. They validate the approach through extensive numerical experiments on phase-field models (including Allen–Cahn, Cahn–Hilliard, MBE, and phase-field crystal), highlighting robust accuracy for large time steps and the potential of adaptive time stepping enabled by unconditional energy stability.

Abstract

In this paper, we develop a general framework for constructing higher-order, unconditionally energy-stable exponential time differencing Runge-Kutta methods applicable to a range of gradient flows. Specifically, we identify conditions sufficient for ETDRK schemes to maintain the original energy dissipation. Our analysis reveals that commonly used third-order and fourth-order ETDRK schemes fail to meet these conditions. To address this, we introduce new third-order ETDRK schemes, designed with appropriate stabilization, which satisfy these conditions and thus guarantee the unconditional energy decaying property. We conduct extensive numerical experiments with these new schemes to verify their accuracy, stability, behavior under large time steps, long-term evolution, and adaptive time stepping strategy across various gradient flows. This study is the first to examine the unconditional energy stability of high-order ETDRK methods, and we are optimistic that our framework will enable the development of ETDRK schemes beyond the third order that are unconditionally energy stable.

Higher-Order Energy-Decreasing Exponential Time Differencing Runge-Kutta methods for Gradient Flows

TL;DR

This work addresses the challenge of designing unconditionally energy-stable exponential time-differencing Runge-Kutta (ETDRK) schemes for gradient flows. It develops a general stabilization-based framework in which energy dissipation is guaranteed if a determinant condition is positive-definite for all negative , with a stabilization parameter chosen to satisfy . The authors demonstrate that common third- and fourth-order ETDRK schemes fail this stability criterion, and they construct two new third-order schemes that satisfy it, proving unconditional energy decay via a positive-definite analysis. They validate the approach through extensive numerical experiments on phase-field models (including Allen–Cahn, Cahn–Hilliard, MBE, and phase-field crystal), highlighting robust accuracy for large time steps and the potential of adaptive time stepping enabled by unconditional energy stability.

Abstract

In this paper, we develop a general framework for constructing higher-order, unconditionally energy-stable exponential time differencing Runge-Kutta methods applicable to a range of gradient flows. Specifically, we identify conditions sufficient for ETDRK schemes to maintain the original energy dissipation. Our analysis reveals that commonly used third-order and fourth-order ETDRK schemes fail to meet these conditions. To address this, we introduce new third-order ETDRK schemes, designed with appropriate stabilization, which satisfy these conditions and thus guarantee the unconditional energy decaying property. We conduct extensive numerical experiments with these new schemes to verify their accuracy, stability, behavior under large time steps, long-term evolution, and adaptive time stepping strategy across various gradient flows. This study is the first to examine the unconditional energy stability of high-order ETDRK methods, and we are optimistic that our framework will enable the development of ETDRK schemes beyond the third order that are unconditionally energy stable.
Paper Structure (17 sections, 6 theorems, 57 equations, 11 figures, 2 tables, 1 algorithm)

This paper contains 17 sections, 6 theorems, 57 equations, 11 figures, 2 tables, 1 algorithm.

Table of Contents

  1. Introduction
  2. Energy decreasing ETDRK methods for gradient flows
  3. Phase-field models
  4. ETDRK methods
  5. A general framework for energy stable ETDRK schemes
  6. Examples of energy decreasing ETDRK schemes
  7. First-order ETD scheme
  8. Second-order ETDRK scheme
  9. Third-order ETDRK schemes
  10. Fourth-order ETDRK schemes
  11. Numerical Experiments
  12. Convergence tests
  13. Large time step tests
  14. Energy curves with different time steps and stabilization constants
  15. Long time behavior of the phase field crystal model
  16. ...and 2 more sections

Key Result

Lemma 2.1

Consider a positive-definite operator $L=\beta I+\mathcal{L}$, and let $f$ be an analytic function whose domain includes the spectrum of $L$, i.e., the values $\{f(\lambda_i)\}_{ i\in \mathcal{N}}$ exist, where $\{\lambda_i\}_{i\in\mathcal{N}}$ are the eigenvalues of $L$. Then, the eigenvalues of $f

Figures (11)

  • Figure 1: Positive-definiteness test: Determinants of leading principal minors
  • Figure 2: Eigenvalues of $\frac{1}{2}(\Delta+\Delta^T)$ for the three ETDRK3 schemes
  • Figure 3: Eigenvalues of ETDRK4 schemes from Cox and Mathhews ETDstiff, and Krogstad Gintfac
  • Figure 4: Error-$\tau$ figure for the AC (left) and CH (right)
  • Figure 5: Error-$\tau$ figure for the phase-field crystal model
  • ...and 6 more figures

Theorems & Definitions (14)

  • Lemma 2.1
  • Theorem 2.1
  • proof
  • Remark 2.1
  • Remark 2.2
  • Lemma 2.2
  • proof
  • Remark 2.3
  • Corollary 3.1
  • proof
  • ...and 4 more