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

TL;DR

This paper develops a unified theoretical framework to analyze energy dissipation in explicit exponential Runge-Kutta methods for gradient flow problems, introducing an indicator for overall dissipation rate and verifying the theory with numerical examples.

Contribution

It introduces a novel framework using differential forms and discrete kernels to assess energy dissipation, and provides criteria for methods to preserve energy laws unconditionally.

Findings

EERK methods can preserve energy dissipation unconditionally with positive semi-definite matrices.

The average dissipation rate indicator effectively evaluates energy dissipation.

Numerical examples confirm the theoretical results.

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.