High-efficiency and positivity-preserving stabilized SAV methods for gradient flows

TL;DR

This paper introduces new stabilized SAV schemes for gradient flows that are energy stable, positivity-preserving, more efficient, and maintain accuracy, with rigorous analysis and numerical validation.

Contribution

The paper develops novel first- and second-order stabilized SAV schemes that reduce computational cost and ensure positivity and energy stability for gradient flows.

Findings

Nearly half computational cost reduction compared to baseline SAV.

Schemes preserve positivity and energy dissipation unconditionally.

Numerical examples confirm accuracy and efficiency.

Abstract

The scalar auxiliary variable (SAV)-type methods are very popular techniques for solving various nonlinear dissipative systems. Compared to the semi-implicit method, the baseline SAV method can keep a modified energy dissipation law but doubles the computational cost. The general SAV approach does not add additional computation but needs to solve a semi-implicit solution in advance, which may potentially compromise the accuracy and stability. In this paper, we construct a novel first- and second-order unconditional energy stable and positivity-preserving stabilized SAV (PS-SAV) schemes for $L^2$ and $H^{-1}$ gradient flows. The constructed schemes can reduce nearly half computational cost of the baseline SAV method and preserve its accuracy and stability simultaneously. Meanwhile, the introduced auxiliary variable is always positive while the baseline SAV cannot guarantee this positivity-preserving property. Unconditionally energy dissipation laws are derived for the proposed numerical schemes. We also establish a rigorous error analysis of the first-order scheme for the Allen-Cahn type equation in $l^{\infty}(0,T; H^1(\Omega) ) $ norm. In addition we propose an energy optimization technique to optimize the modified energy close to the original energy. Several interesting numerical examples are presented to demonstrate the accuracy and effectiveness of the proposed methods.