This paper continues to study linear and unconditionally modified-energy stable (abbreviated as SAV-GL) schemes for the gradient flows. The schemes are built on the SAV technique and the general linear time discretizations (GLTD) as well as the extrapolation for the nonlinear term. Different from [44], the GLTDs with three parameters discussed here are not necessarily algebraically stable. Some algebraic identities are derived by using the method of undetermined coefficients and further used to establish the modified-energy inequalities for the unconditional modified-energy stability of the semi-discrete-in-time SAV-GL schemes. It is worth emphasizing that those algebraic identities or energy inequalities are not necessarily unique for some choices of three parameters in the GLTDs. Numerical experiments on the Allen-Cahn, the Cahn-Hilliard and the phase field crystal models with the periodic boundary conditions are conducted to validate the unconditional modified-energy stability of the SAV-GL schemes, where the Fourier pseudo-spectral method is employed in space with the zero-padding to eliminate the aliasing error and the time stepsizes for ensuring the original-energy decay are estimated by using the stability regions of our SAV-GL schemes for the test equation. The resulting time stepsize constraints for the SAV-GL schemes are almost consistent with the numerical results on the above gradient flow models.

翻译：本文继续研究梯度流问题的线性无条件修正能量稳定格式（简称SAV-GL格式）。该格式基于标量辅助变量（SAV）技术、一般线性时间离散格式（GLTD）以及非线性项的外推方法构建。与文献[44]不同的是，本文讨论的带三个参数的GLTD不必满足代数稳定性。通过待定系数法推导出若干代数恒等式，并进一步用于建立半离散时间SAV-GL格式的无条件修正能量稳定不等式。值得强调的是，对于GLTD中某些三参数选择，这些代数恒等式或能量不等式并非唯一确定。通过对周期边界条件下的Allen-Cahn方程、Cahn-Hilliard方程及相场晶体模型进行数值实验，验证了SAV-GL格式的无条件修正能量稳定性。空间离散采用傅里叶伪谱方法，通过补零消除混叠误差；同时利用SAV-GL格式在测试方程上的稳定性区域，预估保证原始能量衰减的时间步长。所得到的SAV-GL格式时间步长约束与上述梯度流模型的数值结果基本一致。