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Average energy dissipation rates of additive implicit-explicit Runge-Kutta methods for gradient flow problems

Hong-lin Liao, Xuping Wang, Cao Wen

Abstract

A unified theoretical framework is suggested to examine the energy dissipation properties at all stages of additive implicit-explicit Runge-Kutta (IERK) methods up to fourth-order accuracy for gradient flow problems. We construct some parameterized IERK methods by applying the so-called first same as last method, that is, the diagonally implicit Runge-Kutta method with the explicit first stage and stiffly-accurate assumption for the linear stiff term, and applying the explicit Runge-Kutta method for the nonlinear term. The main part of the novel framework is to construct the differential forms and the associated differentiation matrices of IERK methods by using the difference coefficients of method and the so-called discrete orthogonal convolution kernels. As the main result, we prove that an IERK method can preserve the original energy dissipation law unconditionally if the associated differentiation matrix is positive semi-definite. The recent indicator, namely average energy dissipation rate, is also adopted for these multi-stage methods to evaluate the overall energy dissipation rate of an IERK method such that one can choose proper parameters in some parameterized IERK methods. It is found that the selection of method parameters in the IERK methods is at least as important as the selection of different IERK methods. Extensive numerical experiments are also included to support our theory.

Average energy dissipation rates of additive implicit-explicit Runge-Kutta methods for gradient flow problems

Abstract

A unified theoretical framework is suggested to examine the energy dissipation properties at all stages of additive implicit-explicit Runge-Kutta (IERK) methods up to fourth-order accuracy for gradient flow problems. We construct some parameterized IERK methods by applying the so-called first same as last method, that is, the diagonally implicit Runge-Kutta method with the explicit first stage and stiffly-accurate assumption for the linear stiff term, and applying the explicit Runge-Kutta method for the nonlinear term. The main part of the novel framework is to construct the differential forms and the associated differentiation matrices of IERK methods by using the difference coefficients of method and the so-called discrete orthogonal convolution kernels. As the main result, we prove that an IERK method can preserve the original energy dissipation law unconditionally if the associated differentiation matrix is positive semi-definite. The recent indicator, namely average energy dissipation rate, is also adopted for these multi-stage methods to evaluate the overall energy dissipation rate of an IERK method such that one can choose proper parameters in some parameterized IERK methods. It is found that the selection of method parameters in the IERK methods is at least as important as the selection of different IERK methods. Extensive numerical experiments are also included to support our theory.
Paper Structure (17 sections, 8 theorems, 84 equations, 13 figures, 3 tables)

This paper contains 17 sections, 8 theorems, 84 equations, 13 figures, 3 tables.

Table of Contents

  1. Introduction
  2. Stage energy laws of IERK methods
  3. Our theoretical framework
  4. Average dissipation rate
  5. Discrete energy laws of IERK2 methods
  6. Lobatto-type IERK2 methods
  7. Radau-type IERK2 methods
  8. Tests of IERK2 methods
  9. Discrete energy laws of IERK3 methods
  10. IERK3 methods with four stages
  11. IERK3 methods with five stages
  12. Radau-type IERK3 methods with five stages
  13. Tests for IERK3 methods
  14. Discrete energy laws of IERK4-A methods
  15. IERK4-A methods
  16. ...and 2 more sections

Key Result

Lemma 2.1

LiaoWang:2024arxiv If $g_h$ is Lipschitz-continuous with a constant $\ell_{g}>0$ and $\kappa\ge2\ell_g$, then where the energy $E$ is defined in problem: gradient flows.

Figures (13)

  • Figure 1: Solution errors of IERK2 methods with different method parameters.
  • Figure 2: Final solution and energy behaviors of IERK2-1 methods \ref{['Scheme: IERK2-two parameters']} with $c_2=1$.
  • Figure 3: Energy behaviors of $a_{33}$-parameterized IERK2-2 methods \ref{['Scheme: IERK2-one parameter']}.
  • Figure 4: Energy behaviors of IERK2-Radau methods \ref{['Scheme: IERK2-Radau-one parameter']}.
  • Figure 5: Energy behaviors of IERK2-1 method for $c_2=1$ and $a_{33}=\frac{1}{2}$, IERK2-2 method for $a_{33}=\tfrac{1+\sqrt{2}}{4}\approx 0.6036$, and IERK2-Radau method for $c_{2}=\frac{3}{2}$.
  • ...and 8 more figures

Theorems & Definitions (10)

  • Lemma 2.1
  • Lemma 2.2
  • Theorem 3.1
  • Theorem 3.2
  • Example 1
  • Example 2
  • Theorem 4.1
  • Theorem 4.2
  • Theorem 4.3
  • Theorem 5.1