In this paper, we construct and test a class of linear numerical schemes for the Functionalized Cahn-Hilliard (FCH) equation with a symmetric double-well potential function by using a stabilized scalar auxiliary variable (SAV) method. To get a fair assessment of these new SAV-type schemes, we compare output with numerical solutions obtained by the classical, fully-implicit BDF1 and BDF2 schemes. We prove the unconditional unique solvability of the SAV systems and demonstrate the detailed steps used for finding the solutions to these systems. Two sixth-order constant-coefficient linear equations need to be solved at each time step for every SAV scheme. We also provide a theoretical analysis of the unconditional modified-energy stability for the schemes using the usual tools. The Fourier pseudo-spectral method is used as the spatial discretization. Several numerical tests are performed to verify the theoretical analyses and to compute some interesting problems that are physically relevant. Simulations of phase separation in 2D and 3D show the schemes can capture the correct qualitative dynamical behavior and, at the same time, the original physical FCH free energy is dissipated. The classical BDF1 (backward Euler) and BDF2 fully implicit methods, which have significantly smaller local truncation errors (LTEs), are used to repeat several numerical calculations and give a more objective measure of the accuracy and efficiency of the SAV schemes. To keep things simple and fair, for this preliminary battery of comparison tests, we use only fixed, uniform time step sizes. In this setting, the SAV schemes often have an advantage in terms of computational efficiency, being up to three times faster in CPU time when a relatively large time step size is used. However, when accuracy is counted in the measures of computational efficiency, the classical BDF methods often perform better than the linear SAV methods, with an advantage of up to three digits of precision. If the final target of a computation is a relatively high global accuracy, then the method with the least computational time to achieve that accuracy is very often classical BDF2. But, these conclusions are not universal; some test results are subtle and ambiguous. In any case, while SAV methods can be constructed in such a way that they are both energy stable and accurate, they are, however, not always a good choice in practical, real-world computations, because their large LTEs can severely limit their true efficiency.

在本文中，我们使用稳定的标量辅助变量（SAV）方法构造并测试了具有对称双阱势函数的功能化Cahn-Hilliard（FCH）方程的一类线性数值方案。为了公平地评估这些新的SAV型方案，我们将输出与通过经典的，完全隐式的BDF1和BDF2方案获得的数值解进行比较。我们证明了SAV系统的无条件唯一可解性，并说明了用于查找这些系统解决方案的详细步骤。对于每种SAV方案，每个时间步都需要求解两个六阶常系数线性方程。我们还使用常规工具对方案的无条件修正能量稳定性进行了理论分析。傅里叶伪谱方法被用作空间离散化。进行了一些数值测试，以验证理论分析并计算一些与物理相关的有趣问题。在2D和3D中进行相分离的仿真表明，该方案可以捕获正确的定性动力学行为，同时可以消散原始的物理FCH自由能。具有显着较小的本地截断误差（LTE）的经典BDF1（向后欧拉）和BDF2全隐式方法用于重复多次数值计算，并更客观地衡量SAV方案的准确性和效率。为了使事情简单明了，对于这组初步的比较测试，我们仅使用固定，统一的时间步长。在这种情况下，SAV方案通常在计算效率方面具有优势，当使用相对较大的时间步长时，CPU时间快三倍。但是，当在计算效率的度量中计算准确性时，传统的BDF方法通常比线性SAV方法表现更好，其优点是精度高达三位数。如果计算的最终目标是相对较高的全局精度，那么使用最少的计算时间即可达到该精度的方法通常是经典BDF2。但是，这些结论并不普遍。一些测试结果是含糊不清的。无论如何，尽管可以以既稳定又准确的方式构造SAV方法，但是它们在实际的实际计算中并不总是一个好的选择，因为它们的大型LTE可能严重限制其真实性能。效率。