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High-Order and Energy-Stable Implicit-Explicit Relaxation Runge-Kutta Schemes for Gradient Flows

Yuxiu Cheng, Kun Wang, Kai Yang

TL;DR

This paper addresses the challenge of achieving high-order time accuracy while preserving unconditional energy dissipation in gradient-flow-based phase-field models, namely the Allen-Cahn and Cahn-Hilliard equations. It develops a class of high-order implicit-explicit relaxation Runge-Kutta (IMEX RRK) schemes built on the scalar auxiliary variable (SAV) reformulation, incorporating a relaxation coefficient to enforce a discrete energy dissipation law. The authors prove energy stability for the SAV-based IMEX RRK schemes, analyze the relaxation coefficient and truncation error, and validate the theory through numerical experiments on single- and multi-component systems, demonstrating correct convergence orders and efficient long-time simulations. The methods combine SAV stabilization, diagonally IMEX RK structure, and relaxation to balance accuracy and stability, with successful extensions to vector-valued gradient flows. The approach offers a robust framework for energy-stable, high-precision time stepping in phase-field simulations with potential broad applicability to multi-component problems.

Abstract

In this paper, we propose a class of high-order and energy-stable implicit-explicit relaxation Runge-Kutta (IMEX RRK) schemes for solving the phase-field gradient flow models. By incorporating the scalar auxiliary variable (SAV) method, the original equations are reformulated into equivalent forms, and the modified energy is introduced. Then, based on the reformulated equations, we propose a kind of IMEX RRK methods, which are rigorously proved to preserve the energy dissipation law and achieve high-order accuracy for both Allen-Cahn and Cahn-Hilliard equations. Numerical experiments are conducted to validate the theoretical results, including the accuracy of the approximate solution and the efficiency of the proposed scheme. Furthermore, the schemes are extended to multi-component gradient flows, with the vector-valued Allen-Cahn equations serving as a representative example.

High-Order and Energy-Stable Implicit-Explicit Relaxation Runge-Kutta Schemes for Gradient Flows

TL;DR

This paper addresses the challenge of achieving high-order time accuracy while preserving unconditional energy dissipation in gradient-flow-based phase-field models, namely the Allen-Cahn and Cahn-Hilliard equations. It develops a class of high-order implicit-explicit relaxation Runge-Kutta (IMEX RRK) schemes built on the scalar auxiliary variable (SAV) reformulation, incorporating a relaxation coefficient to enforce a discrete energy dissipation law. The authors prove energy stability for the SAV-based IMEX RRK schemes, analyze the relaxation coefficient and truncation error, and validate the theory through numerical experiments on single- and multi-component systems, demonstrating correct convergence orders and efficient long-time simulations. The methods combine SAV stabilization, diagonally IMEX RK structure, and relaxation to balance accuracy and stability, with successful extensions to vector-valued gradient flows. The approach offers a robust framework for energy-stable, high-precision time stepping in phase-field simulations with potential broad applicability to multi-component problems.

Abstract

In this paper, we propose a class of high-order and energy-stable implicit-explicit relaxation Runge-Kutta (IMEX RRK) schemes for solving the phase-field gradient flow models. By incorporating the scalar auxiliary variable (SAV) method, the original equations are reformulated into equivalent forms, and the modified energy is introduced. Then, based on the reformulated equations, we propose a kind of IMEX RRK methods, which are rigorously proved to preserve the energy dissipation law and achieve high-order accuracy for both Allen-Cahn and Cahn-Hilliard equations. Numerical experiments are conducted to validate the theoretical results, including the accuracy of the approximate solution and the efficiency of the proposed scheme. Furthermore, the schemes are extended to multi-component gradient flows, with the vector-valued Allen-Cahn equations serving as a representative example.

Paper Structure

This paper contains 11 sections, 5 theorems, 67 equations, 12 figures, 4 tables.

Table of Contents

  1. Introduction
  2. IMEX RRK-SAV schemes
  3. SAV formulation
  4. Diagonally IMEX RK schemes
  5. IMEX RRK-SAV schemes
  6. Energy dissipation law
  7. Error analysis
  8. Estimate of the relaxation coefficient
  9. Analysis of the truncation error
  10. Numerical tests
  11. Conclusions

Key Result

Theorem 2.1

If $\gamma_n > 0$ and $b_j = \bar{b}_j \ge 0$ for $j=1,\cdots,s$, then, IMEX RRK methods imex rrk-sav preserve the discrete energy decay property, namely

Figures (12)

  • Figure 1: $\left\|\gamma _n-1\right\|_{\ell^{\infty}}$ for some relaxation (RT) methods (AC equation)
  • Figure 2: $\left\|G_n(1)\right\|_{\ell^{\infty}}$ for some relaxation (RT) methods (AC equation)
  • Figure 3: Discrete energy evolution (AC equation)
  • Figure 4: Solution snapshots for phase separation in the AC equation
  • Figure 5: $\left\|\gamma _n-1\right\|_{\ell^{\infty}}$ for some relaxation (RT) methods (CH equation)
  • ...and 7 more figures

Theorems & Definitions (14)

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