A novel high-order linearly implicit and energy-stable additive Runge-Kutta methods for gradient flow models

TL;DR

This paper develops new high-order, energy-stable additive Runge-Kutta schemes for gradient flows using the scalar auxiliary variable approach, improving stability, efficiency, and applicability over existing methods.

Contribution

It introduces a general framework for constructing high-order, energy-stable schemes combining SAV and ARK methods, addressing limitations of previous approaches.

Findings

Schemes are proven to be energy-stable and convergent.

Numerical experiments confirm stability and efficiency.

New schemes outperform existing SAV-RK methods in stability.

Abstract

This paper introduces a novel paradigm for constructing linearly implicit and high-order unconditionally energy-stable schemes for general gradient flows, utilizing the scalar auxiliary variable (SAV) approach and the additive Runge-Kutta (ARK) methods. We provide a rigorous proof of energy stability, unique solvability, and convergence. The proposed schemes generalizes some recently developed high-order, energy-stable schemes and address their shortcomings. On the one other hand, the proposed schemes can incorporate existing SAV-RK type methods after judiciously selecting the Butcher tables of ARK methods \cite{sav_li,sav_nlsw}. The order of a SAV-RKPC method can thus be confirmed theoretically by the order conditions of the corresponding ARK method. Several new schemes are constructed based on our framework, which perform to be more stable than existing SAV-RK type methods. On the other hand, the proposed schemes do not limit to a specific form of the nonlinear part of the free energy and can achieve high order with fewer intermediate stages compared to the convex splitting ARK methods \cite{csrk}. Numerical experiments demonstrate stability and efficiency of proposed schemes.