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.

翻译：标量辅助变量（SAV）方法是一种高效且广泛用于求解梯度流系统的方法。该方法具有线性、无条件能量稳定及易于实现等优点。通过引入标量辅助变量，可在连续层面构建一个与原系统等价的修正系统。然而，在时间离散化过程中，计算误差可能导致等价性与精度的损失。本文提出了一种新型常数标量辅助变量（CSAV）方法，其中我们推导了常数标量辅助变量 r 的常微分方程（ODE）。我们还引入了一个稳定化参数（{\alpha}），通过减缓 r 的动态变化来提升格式的稳定性。CSAV 方法还带来了额外优势：我们显式离散辅助变量并结合非线性项处理，使得每个时间步仅需求解一个具有常系数的线性系统。这一新方法同时取消了对自由能势的假设，消除了原 SAV 方法中非线性自由能势需满足下有界的限制。最后，我们通过大量数值模拟验证了所提方法的有效性与准确性。