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Analysis of the Energy Stability for Stabilized Semi-implicit Schemes of the Functionalized Cahn-Hilliard Mass-conserving Gradient Flow Equation
Journal of Scientific Computing ( IF 2.5 ) Pub Date : 2021-03-12 , DOI: 10.1007/s10915-021-01430-1
Chenhui Zhang , Jie Ouyang , Xiaodong Wang , Yong Chai , Mengxia Ma

A stabilized semi-implicit scheme was designed in [3] to solve the Functionalized Cahn-Hilliard (CCH) equation, but there is a lack of theoretical analysis of the energy stability. In this paper, we generalize this scheme to solve the general FCH mass-conserving gradient flow (FCH-MCGF) equation and show the theoretical analysis results about the unique solvability and energy stability. We successfully prove that this scheme is uniquely solvable and energy stable in theory by rewriting the double-well potential function to satisfy the Lipschitz-type condition. The range of stabilization parameters is theoretically given as well. In addition, another similar energy stable scheme is proposed, which slightly widens the range of stabilization parameters in theory and has almost the same precision as the previous one. Both the detailed numerical procedure and the selection of stabilization parameters are presented. Finally, several numerical experiments are performed for the FCH-MCGF equation based on these schemes. Specially, the adaptive time step size is considered in the scheme for the simulations of the phase separation in 2D and 3D, since any time step size can be used according to our theoretical results. Numerical results show that these schemes are energy stable and the large time step size indeed can be used in computations. Moreover, by comprehensive comparisons of stability and accuracy among the stabilized semi-implicit scheme, the convex splitting scheme, and the fully implicit scheme, we conclude that the performance of the stabilized semi-implicit scheme is the best, and the convex splitting scheme performs better than the fully implicit scheme.



中文翻译:

功能化Cahn-Hilliard守恒质量梯度流方程稳定半隐式方案的能量稳定性分析

在[3]中设计了一个稳定的半隐式方案来求解功能化的Cahn-Hilliard(CCH)方程,但是缺乏对能量稳定性的理论分析。在本文中,我们对该方案进行了推广,以解决一般的FCH质量守恒梯度流(FCH-MCGF)方程,并给出了关于独特的可溶性和能量稳定性的理论分析结果。通过重写双阱势函数以满足Lipschitz型条件,我们成功地证明了该方案在理论上是唯一可解的且能量稳定的。理论上也给出了稳定参数的范围。另外,提出了另一种类似的能量稳定方案,该方案在理论上稍微拓宽了稳定参数的范围,并且具有与先前方案几乎相同的精度。给出了详细的数值程序和稳定参数的选择。最后,基于这些方案对FCH-MCGF方程进行了几个数值实验。特别地,在2D和3D的相分离模拟方案中考虑了自适应时间步长,因为根据我们的理论结果可以使用任何时间步长。数值结果表明,这些方案具有能量稳定性,并且可以在计算中使用较大的时间步长。此外,通过对稳定的半隐式方案,凸分裂方案和完全隐式方案之间的稳定性和准确性进行全面比较,我们得出结论,稳定的半隐式方案的性能是最好的,并且凸分裂方案的性能比完全隐式方案更好。

更新日期:2021-03-12
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