SIAM Journal on Scientific Computing, Volume 42, Issue 4, Page A2514-A2536, January 2020.
We present several essential improvements to the powerful scalar auxiliary variable (SAV) approach. Firstly, by using the introduced scalar variable to control both the nonlinear and the explicit linear terms, we are able to reduce the number of linear equations with constant coefficients to be solved at each time step from two to one, so the computational cost of the new SAV approach is essentially half of the original SAV approach while keeping all its other advantages. This technique is also extended to the multiple SAV approach. Secondly, instead of discretizing the dynamical equation for the auxiliary variable, we use a first-order approximation of the energy balance equation, which allows us to construct high-order unconditionally energy-stable SAV schemes with uniform and, more importantly, variable time step sizes, enabling us to construct, for the first time, high-order unconditionally stable adaptive time-stepping backward differentiation formula schemes. Representative numerical examples are provided to demonstrate the improved efficiency and accuracy of the proposed method.

中文翻译：

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一种高效，精确的梯度流新标量辅助变量方法

SIAM科学计算杂志，第42卷，第4期，第A2514-A2536页，2020年1月。
我们提出了对强大的标量辅助变量（SAV）方法的一些基本改进。首先，通过使用引入的标量变量来控制非线性项和显式线性项，我们能够将每个时间步长要求解的具有恒定系数的线性方程的数量从2减少到1，因此计算量新的SAV方法实质上是原始SAV方法的一半，同时保留了所有其他优点。该技术还扩展到了多个SAV方法。其次，我们不使用辅助变量的动力学方程来离散化，而是使用能量平衡方程的一阶近似，这使我们能够构造具有均匀且更重要的是可变时间步长的高阶无条件能量稳定SAV方案尺寸，使我们能够首次构建高阶无条件稳定的自适应时间步长后向微分公式方案。提供了具有代表性的数值示例，以证明所提出方法的改进效率和准确性。