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A unified framework on the original energy laws of three effective classes of Runge-Kutta methods for phase field crystal type models

Xuping Wang, Xuan Zhao, Hong-lin Liao

TL;DR

This paper addresses the challenge of preserving the original energy dissipation law for gradient-flow phase-field models when using Runge-Kutta time integrators. It introduces a unified discrete differential-form framework with DOC kernels and induction to prove energy dissipation for three RK classes (IERK, EERK, CIFRK) under the condition that the differentiation matrix is positive definite. By applying Fourier-based spatial discretization and deriving uniform bounds on stage solutions, the authors provide explicit lower bounds λ_es for various schemes, enabling practical verification of energy stability across numerous methods. The results advance the understanding of internal nonlinear stability for dissipative semilinear parabolic problems and offer guidance for selecting stable, efficient time integrators for SH and PFC dynamics.

Abstract

The main theoretical obstacle to establish the original energy dissipation laws of Runge-Kutta methods for phase-field equations is to verify the maximum norm boundedness of the stage solutions without assuming global Lipschitz continuity of the nonlinear bulk. We present a unified theoretical framework for the energy stability of three effective classes of Runge-Kutta methods, including the additive implicit-explicit Runge-Kutta, explicit exponential Runge-Kutta and corrected integrating factor Runge-Kutta methods, for the Swift-Hohenberg and phase field crystal models. By the standard discrete energy argument, it is proven that the three classes of Runge-Kutta methods preserve the original energy dissipation laws if the associated differentiation matrices are positive definite. Our main tools include the differential form with the associated differentiation matrix, the discrete orthogonal convolution kernels and the principle of mathematical induction. Many existing Runge-Kutta methods in the literature are revisited by evaluating the lower bound on the minimum eigenvalues of the associated differentiation matrices. Our theoretical approach paves a new way for the internal nonlinear stability of Runge-Kutta methods for dissipative semilinear parabolic problems.

A unified framework on the original energy laws of three effective classes of Runge-Kutta methods for phase field crystal type models

TL;DR

This paper addresses the challenge of preserving the original energy dissipation law for gradient-flow phase-field models when using Runge-Kutta time integrators. It introduces a unified discrete differential-form framework with DOC kernels and induction to prove energy dissipation for three RK classes (IERK, EERK, CIFRK) under the condition that the differentiation matrix is positive definite. By applying Fourier-based spatial discretization and deriving uniform bounds on stage solutions, the authors provide explicit lower bounds λ_es for various schemes, enabling practical verification of energy stability across numerous methods. The results advance the understanding of internal nonlinear stability for dissipative semilinear parabolic problems and offer guidance for selecting stable, efficient time integrators for SH and PFC dynamics.

Abstract

The main theoretical obstacle to establish the original energy dissipation laws of Runge-Kutta methods for phase-field equations is to verify the maximum norm boundedness of the stage solutions without assuming global Lipschitz continuity of the nonlinear bulk. We present a unified theoretical framework for the energy stability of three effective classes of Runge-Kutta methods, including the additive implicit-explicit Runge-Kutta, explicit exponential Runge-Kutta and corrected integrating factor Runge-Kutta methods, for the Swift-Hohenberg and phase field crystal models. By the standard discrete energy argument, it is proven that the three classes of Runge-Kutta methods preserve the original energy dissipation laws if the associated differentiation matrices are positive definite. Our main tools include the differential form with the associated differentiation matrix, the discrete orthogonal convolution kernels and the principle of mathematical induction. Many existing Runge-Kutta methods in the literature are revisited by evaluating the lower bound on the minimum eigenvalues of the associated differentiation matrices. Our theoretical approach paves a new way for the internal nonlinear stability of Runge-Kutta methods for dissipative semilinear parabolic problems.

Paper Structure

This paper contains 17 sections, 4 theorems, 89 equations, 4 figures.

Table of Contents

  1. Introduction
  2. A concise framework for energy stability
  3. Spatial approximation
  4. Preliminary lemmas and an equivalent form
  5. Proof of Theorem \ref{['theorem: energy stability']} for $M_h=-I$
  6. Proof of Theorem \ref{['theorem: energy stability']} for $M_h=\Delta_h$
  7. Lower bounds $\lambda_{\mathrm{es}}$ for IERK, EERK and CIFRK methods
  8. IERK methods
  9. Second-order IERK methods
  10. Third-order IERK methods
  11. EERK methods
  12. Second-order EERK methods
  13. Third-order EERK methods
  14. CIFRK methods
  15. Second-order CIFRK methods
  16. ...and 2 more sections

Key Result

Theorem 1.1

\newlabeltheorem: energy stability0 Assume that the nonlinear bulk $f$ is continuously differentiable and the differentiation matrix $D(z)$ is positive definite for any $z < 0$. There exist two positive constants $\tau_0$ and ${{\mathcal{C}}}_f$ such that when the time-step size $\tau\le \tau_0$ a for $n\ge1$ and $1\le k\le s$. Moreover, it holds that

Figures (4)

  • Figure 1: Minimum eigenvalues of the second-order EERK methods.
  • Figure 2: Minimum eigenvalues of the third-order EERK methods.
  • Figure 3: Minimum eigenvalues of the second-order CIFRK schemes.
  • Figure 4: Minimum eigenvalues of the third-order CIFRK methods.

Theorems & Definitions (9)

  • Theorem 1.1
  • Lemma 2.1
  • Proof 1
  • Lemma 2.2
  • Proof 2
  • Lemma 2.3
  • Proof 3
  • Proof 4
  • Proof 5