发表机构
School of Mathematics and Statistics, Henan University; Center for Applied Mathematics of Henan Province, Henan University（河南大学数学与统计学院; 河南省应用数学中心）
机构由 AI 辅助整理，请以论文原文为准。AI 中文总结
针对SAV方法离散后r可能变负的问题，通过将r_t重构为(r^2)_t的二次形式，保持平方根定义，实现无条件能量稳定的保正格式，且计算成本不变。
AI 中文摘要
标量辅助变量（SAV）方法通过用正标量 $r(t)=\sqrt{\mathcal{E}_{\mathcal{N}}[\phi]+C}>0$ 替换自由能的非线性部分，从而为梯度流生成线性、无条件能量稳定的格式。然而，在离散层面，标准的向后欧拉和Crank-Nicolson离散化无法保证计算出的 $r^{n+1}$ 保持正值，这与连续定义不一致，违背了平方根假设，并可能损害长期鲁棒性。尽管SAV方法已被广泛应用，但这一微妙而重要的问题却很少受到关注。我们首先通过推导时间步长的严格判据和充分条件来定量刻画这一失效现象，并构造一个显式反例，表明在实际相关参数下会发生符号丢失。与现有保正变体中修改 $r$ 的定义不同，我们保留平方根形式，并将离散演化从 $r_t$ 重构为 $(r^2)_t$，这将标量方程转化为具有严格负常数项的凸二次方程，总能产生唯一的正根。对于Crank-Nicolson格式，乘积形式离散化 $r^{n+1}r^n$ 保持了这种二次结构，而传统替代方案则不能。所得格式的计算成本与原SAV方法相同，并被证明无条件能量稳定。针对Cahn-Hilliard方程的数值实验证实了预期的正性、能量稳定性和收敛阶。
英文摘要
The scalar auxiliary variable (SAV) method replaces the nonlinear part of the free energy by a positive scalar $r(t)=\sqrt{\mathcal{E}_{_\mathcal{N}}[ϕ]+C}>0$, thereby yielding linear, unconditionally energy-stable schemes for gradient flows. At the discrete level, however, the standard backward Euler and Crank--Nicolson discretizations provide no guarantee that the computed $r^{n+1}$ remains positive, an inconsistency with the continuous definition that contradicts the square-root ansatz and may compromise long-time robustness. Although the SAV method has been widely applied, this subtle but consequential issue has received little attention. We first characterize this failure quantitatively by deriving a sharp criterion and a sufficient condition on the time step size, and construct an explicit counterexample showing that sign loss occurs for parameters of practical relevance. Rather than modifying the definition of $r$ as in existing positivity-preserving variants, we retain the square-root form and reformulate the discrete evolution from $r_t$ to $(r^{2})_t$, which converts the scalar equation into a convex quadratic with a strictly negative constant term, always yielding a unique positive root. For the Crank--Nicolson scheme, the product-form discretization $r^{n+1}r^{n}$ preserves this quadratic structure, while conventional alternatives do not. The resulting schemes incur the same computational cost as the original SAV method and are proved unconditionally energy-stable. Numerical experiments for the Cahn--Hilliard equation confirm the predicted positivity, energy stability, and convergence rates.
