Energy diminishing implicit-explicit Runge--Kutta methods for gradient flows

TL;DR

This paper introduces high-order implicit-explicit Runge--Kutta methods that unconditionally preserve energy dissipation in gradient flows, using a stabilization technique and a new framework for energy stability verification.

Contribution

It develops the first linear high-order single-step schemes that guarantee unconditional energy stability for general gradient flows and introduces a new energy-preserving IMEX-RK scheme.

Findings

Methods successfully preserve energy dissipation without time-step restrictions.

Numerical examples confirm stability and accuracy of the proposed schemes.

A new four-stage third-order IMEX-RK scheme reduces energy.

Abstract

This study focuses on the development and analysis of a group of high-order implicit-explicit (IMEX) Runge--Kutta (RK) methods that are suitable for discretizing gradient flows with nonlinearity that is Lipschitz continuous. We demonstrate that these IMEX-RK methods can preserve the original energy dissipation property without any restrictions on the time-step size, thanks to a stabilization technique. The stabilization constants are solely dependent on the minimal eigenvalues that result from the Butcher tables of the IMEX-RKs. Furthermore, we establish a simple framework that can determine whether an IMEX-RK scheme is capable of preserving the original energy dissipation property or not. We also present a heuristic convergence analysis based on the truncation errors. This is the first research to prove that a linear high-order single-step scheme can ensure the original energy stability unconditionally for general gradient flows. Additionally, we provide several high-order IMEX-RK schemes that satisfy the established framework. Notably, we discovered a new four-stage third-order IMEX-RK scheme that reduces energy. Finally, we provide numerical examples to demonstrate the stability and accuracy properties of the proposed methods.