A New Scalar Auxiliary Variable Approach for Gradient Flows

TL;DR

This work tackles numerical solution of gradient-flow systems by improving energy-stable, linear-solver-friendly schemes. It introduces the Constant Scalar Auxiliary Variable (CSAV) method, which replaces the SAV’s time-varying auxiliary with a constant $r(t)$ governed by a small stabilization ODE, enabling a single linear system with constant coefficients per time step and removing the need for the nonlinear free-energy to be bounded below. The authors prove unconditional energy stability for first- and second-order CSAV schemes (and their multi-term MSAV extension), and demonstrate through extensive simulations on Allen-Cahn, Cahn-Hilliard, MBE, and PFC-type models that CSAV achieves high-order accuracy with $r$ staying near 1 and energy dissipation closely tracking the true energy. These results provide a scalable, flexible framework for robust gradient-flow simulations across a wide range of phase-field models.

Abstract

The scalar auxiliary variable (SAV) approach is a highly efficient method widely used for solving gradient flow systems. This approach offers several advantages, including linearity, unconditional energy stability, and ease of implementation. By introducing scalar auxiliary variables, a modified system that is equivalent to the original system is constructed at the continuous level. However, during temporal discretization, computational errors can lead to a loss of equivalence and accuracy. In this paper, we introduce a new Constant Scalar Auxiliary Variable (CSAV) approach in which we derive an Ordinary Differential Equation (ODE) for the constant scalar auxiliary variable r. We also introduce a stabilization parameter (α) to improve the stability of the scheme by slowing down the dynamics of r. The CSAV approach provides additional benefits as well. We explicitly discretize the auxiliary variable in combination with the nonlinear term, enabling the solution of a single linear system with constant coefficients at each time step. This new approach also eliminates the need for assumptions about the free energy potential, removing the bounded-from-below restriction imposed by the nonlinear free energy potential in the original SAV approach. Finally, we validate the proposed method through extensive numerical simulations, demonstrating its effectiveness and accuracy.

Figures (18)

• Figure 1: (a) Convergence tests of the auxiliary variable $r$ at t=0.5 for different $\alpha$ in the Allen-Cahn equation. (b) Convergence tests of the auxiliary variable $r$ at t=0.121 for different $\alpha$ in the Cahn-Hilliard equation.
• Figure 2: (a) The time evolution of the energy functional for the different time step sizes $\Delta t=2\times 10^{-2}$, $1\times 10^{-2}$, $1\times 10^{-3}$ in Example 2. (b) The time evolution of the energy functional for the different $\alpha$ in Example 2.
• Figure 3: (a) The evolution of $r_{n}$ for different $\alpha$ in Example 2. (b) Snapshots of the phase variable $\phi$ are taken at t=0, 2, 4, 6 in clockwise sense in Example 2.
• Figure 4: (a) The comparisons of the CSAV modified energy and SAV modified energy in Example 3. (b) The comparisons of the coefficients in \ref{['eq:SAV_d2']} and \ref{['SeONuSCH2']} for Example 3.
• Figure 5: Snapshots of the phase variable $\phi$ at t=0, 0.01, 0.1, 1 for Example 3.