A new class of efficient and robust energy stable schemes for gradient flows

TL;DR

This paper introduces a scalar auxiliary variable (SAV) method for gradient flows that creates efficient, robust, and energy stable numerical schemes capable of handling complex nonlinearities with ease and high accuracy.

Contribution

The paper presents a novel SAV approach that simplifies the construction of energy stable schemes for gradient flows, applicable to a wide range of nonlinear problems, and demonstrates superior efficiency and robustness.

Findings

Schemes are unconditionally second-order energy stable.

Higher-order BDF schemes (third or fourth order) are easily constructed and robust.

Numerical results show improved efficiency and physical property capture.

Abstract

We propose a new numerical technique to deal with nonlinear terms in gradient flows. By introducing a scalar auxiliary variable (SAV), we construct efficient and robust energy stable schemes for a large class of gradient flows. The SAV approach is not restricted to specific forms of the nonlinear part of the free energy, and only requires to solve {\it decoupled} linear equations with {\it constant coefficients}. We use this technique to deal with several challenging applications which can not be easily handled by existing approaches, and present convincing numerical results to show that our schemes are not only much more efficient and easy to implement, but can also better capture the physical properties in these models. Based on this SAV approach, we can construct unconditionally second-order energy stable schemes; and we can easily construct even third or fourth order BDF schemes, although not unconditionally stable, which are very robust in practice. In particular, when coupled with an adaptive time stepping strategy, the SAV approach can be extremely efficient and accurate.