Improved scalar auxiliary variable schemes for original energy stability of gradient flows

TL;DR

This paper introduces an improved scalar auxiliary variable (iSAV) scheme for gradient flows that maintains linearity and guarantees the stability of the original energy, with proven convergence and extensive numerical validation.

Contribution

The paper proposes an enhanced SAV scheme that ensures original energy stability while preserving linearity, with rigorous convergence analysis and high-order extension discussions.

Findings

Proven convergence and optimal error bounds for iSAV

Numerical experiments validate robustness and energy stability

Comparative analysis shows advantages over existing methods

Abstract

Scalar auxiliary variable (SAV) methods are a class of linear schemes for solving gradient flows that are known for the stability of a `modified' energy. In this paper, we propose an improved SAV (iSAV) scheme that not only retains the complete linearity but also ensures rigorously the stability of the original energy. The convergence and optimal error bound are rigorously established for the iSAV scheme and discussions are made for its high-order extension. Extensive numerical experiments are done to validate the convergence, robustness and energy stability of iSAV, and some comparisons are made.