A New Scalar Auxiliary Variable Approach for Gradient Flows

TL;DR

This paper introduces a novel Constant Scalar Auxiliary Variable (CSAV) method for gradient flows, enhancing stability, accuracy, and efficiency by simplifying computations and removing restrictions of previous SAV approaches.

Contribution

The paper proposes a new CSAV approach with an ODE for the auxiliary variable and a stabilization parameter, improving stability and eliminating assumptions on free energy potentials.

Findings

Enables solving a single linear system per time step

Provides unconditional energy stability

Demonstrates high accuracy through numerical simulations

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 ({\alpha}) 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.