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Original energy dissipation preserving corrections of integrating factor Runge-Kutta methods for gradient flow problems

Hong-lin Liao, Xuping Wang, Cao Wen

Abstract

Explicit integrating factor Runge-Kutta methods are attractive and popular in developing high-order maximum bound principle preserving time-stepping schemes for Allen-Cahn type gradient flows. However, they always suffer from the non-preservation of steady-state solution and original energy dissipation law. To overcome these disadvantages, some new integrating factor methods are developed by using two classes of difference correction, including the telescopic correction and nonlinear-term translation correction, enforcing the preservation of steady-state solution. Then the original energy dissipation properties of the new methods are examined by using the associated differential forms and the differentiation matrices. As applications, some new integrating factor Runge-Kutta methods up to third-order maintaining the original energy dissipation law are constructed by applying the difference correction strategies to some popular explicit integrating factor methods in the literature. Extensive numerical experiments are presented to support our theory and to demonstrate the improved performance of new methods.

Original energy dissipation preserving corrections of integrating factor Runge-Kutta methods for gradient flow problems

Abstract

Explicit integrating factor Runge-Kutta methods are attractive and popular in developing high-order maximum bound principle preserving time-stepping schemes for Allen-Cahn type gradient flows. However, they always suffer from the non-preservation of steady-state solution and original energy dissipation law. To overcome these disadvantages, some new integrating factor methods are developed by using two classes of difference correction, including the telescopic correction and nonlinear-term translation correction, enforcing the preservation of steady-state solution. Then the original energy dissipation properties of the new methods are examined by using the associated differential forms and the differentiation matrices. As applications, some new integrating factor Runge-Kutta methods up to third-order maintaining the original energy dissipation law are constructed by applying the difference correction strategies to some popular explicit integrating factor methods in the literature. Extensive numerical experiments are presented to support our theory and to demonstrate the improved performance of new methods.
Paper Structure (23 sections, 9 theorems, 95 equations, 22 figures)

This paper contains 23 sections, 9 theorems, 95 equations, 22 figures.

Table of Contents

  1. Introduction
  2. Difference correction of IF1 scheme
  3. IF1 method
  4. Two corrections of IF1 scheme
  5. Telescopic-type correction
  6. Nonlinear-term translation correction
  7. Tests of corrected IF1 schemes
  8. A unified framework and corrections of IF2 methods
  9. A unified theoretical framework for difference corrections
  10. Corrections of Heun's IF2 method
  11. Corrections of Ralston's IF2 method
  12. Tests of corrected IF2 schemes
  13. Extensions to high-order IF methods
  14. Corrections of Heun's IF3 method
  15. Corrections of Ralston's IF3 method
  16. ...and 8 more sections

Key Result

Lemma 2.1

Liao:2024arxiv If the nonlinear function $g$ is Lipschitz-continuous with a constant $\ell_{g}>0$ and the stabilized parameter $\kappa\ge2\ell_g$, then where the energy $E[\,\cdot\,]$ is defined in eq: continuous energy.

Figures (22)

  • Figure 1: Errors of corrected IF1 schemes and energy dissipation of IF1 scheme.
  • Figure 2: Comparisons of IF1 and corrected IF1 schemes for $\tau=0.05$.
  • Figure 3: Comparisons of IF1 and corrected IF1 schemes for $\tau=0.5$.
  • Figure 4: Correction coefficients of two corrected IF1 schemes.
  • Figure 5: Correction coefficients of two corrected IF2-Heun schemes.
  • ...and 17 more figures

Theorems & Definitions (15)

  • Lemma 2.1
  • Lemma 2.2
  • Theorem 2.1
  • proof
  • Theorem 2.2
  • Example 1
  • Theorem 3.1
  • Proposition 3.1
  • proof
  • Theorem 3.2
  • ...and 5 more