The optimal error estimate that depending only on the polynomial degree of $ \varepsilon^{-1}$ is established for the temporal semi-discrete scheme of the Cahn-Hilliard equation, which is based on the scalar auxiliary variable (SAV) formulation. The key to our analysis is to convert the structure of the SAV time-stepping scheme back to a form compatible with the original format of the Cahn-Hilliard equation, which makes it feasible to use spectral estimates to handle the nonlinear term. Based on the transformation of the SAV numerical scheme, the optimal error estimate for the temporal semi-discrete scheme which depends only on the low polynomial order of $\varepsilon^{-1}$ instead of the exponential order, is derived by using mathematical induction, spectral arguments, and the superconvergence properties of some nonlinear terms. Numerical examples are provided to illustrate the discrete energy decay property and validate our theoretical convergence analysis.

翻译：最佳误差估计是,Cahn-Hilliard 等式的时半分解方案,其依据是星际辅助变量(SAV)的配方。我们分析的关键是将SAV时间步进方案的结构转换回一个与Cahn-Hilliard 等式原格式相容的形式,使利用光谱估计来处理非线性术语成为可行。根据SAV数字方案的转换,仅依赖美元/瓦列普西隆-1美元这一低多位数的时半分解方案的最佳误差估计,而不是指数性序,是使用数学感应、光谱参数和一些非线性术语的超级相容特性得出的。提供了数字示例,以说明离散能量衰减特性,并验证我们的理论趋同分析。