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

Hong-lin Liao, Xuping Wang

Abstract

We propose a unified theoretical framework to examine the energy dissipation properties at all stages of explicit exponential Runge-Kutta (EERK) methods for gradient flow problems. The main part of the novel framework is to construct the differential form of EERK method by using the difference coefficients of method and the so-called discrete orthogonal convolution kernels. As the main result, we prove that an EERK method can preserve the original energy dissipation law unconditionally if the associated differentiation matrix is positive semi-definite. A simple indicator, namely average dissipation rate, is also introduced for these multi-stage methods to evaluate the overall energy dissipation rate of an EERK method such that one can choose proper parameters in some parameterized EERK methods or compare different kinds of EERK methods. Some existing EERK methods in the literature are evaluated from the perspective of preserving the original energy dissipation law and the energy dissipation rate. Some numerical examples are also included to support our theory.

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

Abstract

We propose a unified theoretical framework to examine the energy dissipation properties at all stages of explicit exponential Runge-Kutta (EERK) methods for gradient flow problems. The main part of the novel framework is to construct the differential form of EERK method by using the difference coefficients of method and the so-called discrete orthogonal convolution kernels. As the main result, we prove that an EERK method can preserve the original energy dissipation law unconditionally if the associated differentiation matrix is positive semi-definite. A simple indicator, namely average dissipation rate, is also introduced for these multi-stage methods to evaluate the overall energy dissipation rate of an EERK method such that one can choose proper parameters in some parameterized EERK methods or compare different kinds of EERK methods. Some existing EERK methods in the literature are evaluated from the perspective of preserving the original energy dissipation law and the energy dissipation rate. Some numerical examples are also included to support our theory.
Paper Structure (23 sections, 16 theorems, 111 equations, 17 figures, 4 tables)

This paper contains 23 sections, 16 theorems, 111 equations, 17 figures, 4 tables.

Table of Contents

  1. Introduction
  2. Stage energy laws of EERK methods
  3. General class of EERK methods
  4. Our theoretical framework
  5. Averaged dissipation rate
  6. Simple case: ETD1
  7. Discrete energy laws of second-order methods
  8. EERK2 method
  9. Weak variants of EERK2 method
  10. Remarks for the three-stage EERK method
  11. Discrete energy laws of third-order methods
  12. Simplified procedure and two simple cases
  13. One-parameter EERK3 methods
  14. Two-parameter EERK3 methods
  15. The case $c_2=1$ with $c_3\in(0,1]$
  16. ...and 8 more sections

Key Result

Lemma 2.1

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 (17)

  • Figure 1: Curves of comparison functions $g^*_{21}$ and $g^*_{22}$.
  • Figure 2: Averaged dissipation rates $\mathcal{R}^{(2)}(c_2,z)$ and $\mathcal{R}^{(2,w)}(c_2,z)$ for different abscissas $c_2$.
  • Figure 3: Dissipation rate comparisons of EERK2, EERK2-w and EERK2-S methods.
  • Figure 4: Leading principal minors (LPM) of associated differential matrices.
  • Figure 5: Averaged dissipation rates of EERK3-1 and EERK3-2 methods.
  • ...and 12 more figures

Theorems & Definitions (27)

  • Lemma 2.1
  • proof
  • Lemma 2.2
  • Remark 1
  • Corollary 2.1
  • Proposition 3.1
  • proof
  • Corollary 3.1
  • Proposition 3.2
  • proof
  • ...and 17 more