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

TL;DR

This paper develops new explicit integrating factor Runge-Kutta methods that preserve steady-state solutions and energy dissipation laws for gradient flow problems, improving stability and accuracy.

Contribution

It introduces difference correction strategies to existing methods, enabling energy dissipation preservation up to third-order accuracy.

Findings

New methods preserve energy dissipation law.

Methods demonstrate improved stability in numerical experiments.

Third-order methods outperform existing schemes.

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.