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Energy diminishing implicit-explicit Runge--Kutta methods for gradient flows

Zhaohui Fu, Tao Tang, Jiang Yang

TL;DR

This paper develops high-order implicit-explicit Runge-Kutta (IMEX-RK) methods for gradient flows with Lipschitz nonlinearities and proves that, via a stabilization framework, these schemes unconditionally dissipate the original energy. The stability analysis hinges on positive-definite matrices $H_0,H_1,H_2$ and stabilizers $( abla ext{alpha}, eta)$ tied to the RK tableau, with explicit conditions enabling unconditional energy decay for Allen--Cahn and Cahn--Hilliard systems. Concrete schemes are constructed, including a new four-stage third-order IMEX-RK that preserves energy, accompanied by a truncation-error-based convergence analysis. Numerical experiments on phase-field models validate energy stability, accuracy, and the stabilizers’ beneficial impact on both error and dynamics. The work advances the design of efficient, stable, high-order time integrators for gradient-flow dynamics with nonlinearities satisfying Lipschitz conditions, with potential extensions to higher-order schemes and other stability properties.

Abstract

This study focuses on the development and analysis of a group of high-order implicit-explicit (IMEX) Runge--Kutta (RK) methods that are suitable for discretizing gradient flows with nonlinearity that is Lipschitz continuous. We demonstrate that these IMEX-RK methods can preserve the original energy dissipation property without any restrictions on the time-step size, thanks to a stabilization technique. The stabilization constants are solely dependent on the minimal eigenvalues that result from the Butcher tables of the IMEX-RKs. Furthermore, we establish a simple framework that can determine whether an IMEX-RK scheme is capable of preserving the original energy dissipation property or not. We also present a heuristic convergence analysis based on the truncation errors. This is the first research to prove that a linear high-order single-step scheme can ensure the original energy stability unconditionally for general gradient flows. Additionally, we provide several high-order IMEX-RK schemes that satisfy the established framework. Notably, we discovered a new four-stage third-order IMEX-RK scheme that reduces energy. Finally, we provide numerical examples to demonstrate the stability and accuracy properties of the proposed methods.

Energy diminishing implicit-explicit Runge--Kutta methods for gradient flows

TL;DR

This paper develops high-order implicit-explicit Runge-Kutta (IMEX-RK) methods for gradient flows with Lipschitz nonlinearities and proves that, via a stabilization framework, these schemes unconditionally dissipate the original energy. The stability analysis hinges on positive-definite matrices and stabilizers tied to the RK tableau, with explicit conditions enabling unconditional energy decay for Allen--Cahn and Cahn--Hilliard systems. Concrete schemes are constructed, including a new four-stage third-order IMEX-RK that preserves energy, accompanied by a truncation-error-based convergence analysis. Numerical experiments on phase-field models validate energy stability, accuracy, and the stabilizers’ beneficial impact on both error and dynamics. The work advances the design of efficient, stable, high-order time integrators for gradient-flow dynamics with nonlinearities satisfying Lipschitz conditions, with potential extensions to higher-order schemes and other stability properties.

Abstract

This study focuses on the development and analysis of a group of high-order implicit-explicit (IMEX) Runge--Kutta (RK) methods that are suitable for discretizing gradient flows with nonlinearity that is Lipschitz continuous. We demonstrate that these IMEX-RK methods can preserve the original energy dissipation property without any restrictions on the time-step size, thanks to a stabilization technique. The stabilization constants are solely dependent on the minimal eigenvalues that result from the Butcher tables of the IMEX-RKs. Furthermore, we establish a simple framework that can determine whether an IMEX-RK scheme is capable of preserving the original energy dissipation property or not. We also present a heuristic convergence analysis based on the truncation errors. This is the first research to prove that a linear high-order single-step scheme can ensure the original energy stability unconditionally for general gradient flows. Additionally, we provide several high-order IMEX-RK schemes that satisfy the established framework. Notably, we discovered a new four-stage third-order IMEX-RK scheme that reduces energy. Finally, we provide numerical examples to demonstrate the stability and accuracy properties of the proposed methods.
Paper Structure (18 sections, 4 theorems, 71 equations, 6 figures, 1 table)

This paper contains 18 sections, 4 theorems, 71 equations, 6 figures, 1 table.

Table of Contents

  1. Introduction
  2. Preliminaries
  3. A stabilization technique
  4. Implicit-Explicit Runge-Kutta schemes
  5. Energy decreasing property
  6. Main Theorem
  7. Application to Allen--Cahn and Cahn--Hilliard equations
  8. Error Analysis
  9. Implicit-Explicit Runge-Kutta schemes
  10. Example 1: first-order IMEX
  11. Example 2: a second-order IMEX
  12. Third-order schemes
  13. Numerical Tests
  14. Accuracy Test
  15. Energy Evolution
  16. ...and 3 more sections

Key Result

Theorem 3.1

The IMEX-RK scheme (IMEXRKsys) unconditionally decreases the energy of the phase field model (sgf) if the following three matrices are positive-definite: Here we say a matrix $M$ is positive-definite when $\frac{1}{2} (M+M^T)$ is positive-definite. In (H0 - H2) $\alpha \text{ and } \beta$ are stabilizer constants, $E=\mathbf{1}_{s\times s}$, $E_{L}=\left(\mathbf{1}_{i\geq j}\right)_{s\times s}$ r

Figures (6)

  • Figure 1: Accuracy Tests for Allen--Cahn (left) and Cahn--Hilliard (right) models
  • Figure 2: Energy curves of numerical solutions for the Allen--Cahn equation
  • Figure 3: Energy curves of numerical solutions for the Cahn--Hilliard equation
  • Figure 4: Error-time step curves for the Allen--Cahn equation
  • Figure 5: Small and large time step solutions with and without stabilization at $T=0.1$
  • ...and 1 more figures

Theorems & Definitions (14)

  • Remark 2.1
  • Theorem 3.1
  • proof
  • Remark 3.1
  • Remark 3.2
  • Remark 3.3
  • Remark 3.4
  • Theorem 3.2
  • Theorem 3.3
  • proof
  • ...and 4 more