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

TL;DR

The paper tackles the challenge of ensuring original energy stability for gradient-flow discretizations while preserving linearity. It introduces the improved SAV (iSAV) scheme, which replaces the prior scalar with the true functional $r[\phi^n]$ and adds a stabilization term, yielding a discrete energy law that guarantees $\frac{1}{\tau}(\mathcal{E}[\phi^{n+1}] - \mathcal{E}[\phi^n]) \le -\|\mathcal{G}^{1/2}\mu^{n+1}\|^2$. A rigorous first-order convergence result for the iSAV-BE scheme is established under suitable assumptions, with detailed a priori estimates and a proof structure based on truncation error analysis and discrete Gronwall. The paper also discusses high-order extensions via iSAV-BDF and supports the theory with comprehensive 2D numerical experiments on double-well and Flory–Huggins potentials, showing superior original-energy stability and competitive accuracy relative to SAV. The findings suggest that iSAV provides a practical, robust framework for linear, energy-stable gradient-flow simulations with potential for higher-order generalizations.

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.

Figures (10)

• Figure 1: (Example \ref{['Ex 2']}): Time evolutions of the energy in SAV and iSAV. (a): BE schemes with $\tau = 0.01$; (b): BE schemes with $\tau = 0.001$; (c): BDF schemes with $\tau = 0.001$.
• Figure 2: (Example \ref{['Ex 2']}): Time evolution of $\mathcal{E}[\phi^{n}]-\mathcal{E}[\phi^{n-1}]+\tau\| \mathcal{G}^{\frac{1}{2}}\mu^{n} \|^2$ in SAV-BE and iSAV-BE with $\tau = 0.001$. (a) $\varepsilon=0.1$; (b) $\varepsilon=0.04$; (c) $\varepsilon=0.01$.
• Figure 3: (Example \ref{['Ex 2']}): Time evolution of $\mathcal{E}_{2}[\phi^{n},\phi^{n-1}]-\mathcal{E}_{2}[\phi^{n-1},\phi^{n-2}]+\tau\| \mathcal{G}^{\frac{1}{2}}\mu^{n} \|^2$ of SAV-BDF and iSAV-BDF with $\tau = 0.001$. (a) $\varepsilon=0.1$; (b) $\varepsilon=0.04$; (c) $\varepsilon=0.01$.
• Figure 4: (Example \ref{['Ex 2']}): Difference between $r[\phi^{n}]$ and $r^{n}$ or $\tilde{r}^{n}$ in SAV or iSAV. (a): BE schemes with $\tau = 0.001$; (b): BDF schemes with $\tau = 0.001$.
• Figure 5: (Example \ref{['Ex 2']}) Contour plots of the numerical solution $\phi^n$ at $t = 0.01$, $0.02$, $0.1$ and $0.7$ with $\tau = 0.01$. Top: SAV-BE; Down: iSAV-BE.

Theorems & Definitions (21)

• Theorem 2.1: Original energy stability
• proof
• Remark 1
• Remark 2
• Remark 3
• Remark 4
• Theorem 3.1: Convergence and error bound
• Corollary 3.2
• Remark 5
• Lemma 3.3: Shen2018ConvergenceSAV