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Unconditionally energy stable second-order numerical schemes for the Functionalized Cahn–Hilliard gradient flow equation based on the SAV approach
Computers & Mathematics with Applications ( IF 2.9 ) Pub Date : 2020-12-26 , DOI: 10.1016/j.camwa.2020.12.003
Chenhui Zhang , Jie Ouyang

In this paper, we devise and analyse three highly efficient second-order accurate (in time) schemes for solving the Functionalized Cahn–Hilliard (FCH) gradient flow equation where an asymmetric double-well potential function is considered. Based on the Scalar Auxiliary Variable (SAV) approach, we construct these schemes by splitting the FCH free energy in a novel and ingenious way. Utilizing the Crank–Nicolson formula, we firstly construct two semi-discrete second-order numerical schemes, which we denote by CN-SAV and CN-SAV-A, respectively. To be more specific, the CN-SAV scheme is constructed based on the fixed time step, while the CN-SAV-A scheme is a variable time step scheme. The BDF2-SAV scheme is another second-order scheme in which the fixed time step should be used. It is designed by applying the second-order backward difference (BDF2) formula. All the constructed schemes are proved to be unconditionally energy stable and uniquely solvable in theory. To the best of our knowledge, the CN-SAV-A scheme is the first unconditionally energy stable, second-order scheme with variable time steps for the FCH gradient flow equation. In addition, an effective adaptive time selection strategy introduced in Christlieb et al., (2014) is slightly modified and then adopted to select the time step for the CN-SAV-A scheme. Finally, several numerical experiments based on the Fourier pseudo-spectral method are carried out in two and three dimensions, respectively, to confirm the numerical accuracy and efficiency of the constructed schemes.



中文翻译:

基于SAV方法的功能化Cahn-Hilliard梯度流方程的无条件能量稳定二阶数值格式

在本文中,我们设计并分析了三种高效的二阶精确(及时)方案,用于解决考虑非对称双阱势函数的功能化Cahn-Hilliard(FCH)梯度流方程。基于标量辅助变量(SAV)方法,我们通过新颖新颖的方式拆分FCH自由能来构造这些方案。利用Crank-Nicolson公式,我们首先构造了两个半离散的二阶数值方案,分别用CN-SAV和CN-SAV-A表示。更具体地,基于固定时间步长构造CN-SAV方案,而CN-SAV-A方案是可变时间步长方案。BDF2-SAV方案是另一种应使用固定时间步长的二阶方案。通过应用二阶后向差分(BDF2)公式进行设计。事实证明,所有构建的方案都是无条件能量稳定的,并且在理论上可以唯一求解。据我们所知,CN-SAV-A方案是FCH梯度流方程的第一个无条件能量稳定,具有可变时间步长的二阶方案。另外,对Christlieb等人(2014)中引入的有效自适应时间选择策略进行了稍微修改,然后采用它来选择CN-SAV-A方案的时间步长。最后,分别在二维和三维上进行了基于傅立叶伪谱方法的几个数值实验,以验证所构造方案的数值精度和效率。事实证明,所有构建的方案都是无条件能量稳定的,并且在理论上可以唯一求解。据我们所知,CN-SAV-A方案是FCH梯度流方程的第一个无条件能量稳定,具有可变时间步长的二阶方案。另外,对Christlieb等人(2014)中引入的有效自适应时间选择策略进行了稍微修改,然后采用它来选择CN-SAV-A方案的时间步长。最后,分别在二维和三维上进行了基于傅立叶伪谱方法的几个数值实验,以验证所构造方案的数值精度和效率。事实证明,所有构建的方案都是无条件能量稳定的,并且在理论上可以唯一求解。据我们所知,CN-SAV-A方案是FCH梯度流方程的第一个无条件能量稳定,具有可变时间步长的二阶方案。另外,对Christlieb等人(2014)中引入的有效自适应时间选择策略进行了稍微修改,然后采用它来选择CN-SAV-A方案的时间步长。最后,分别在二维和三维上进行了基于傅立叶伪谱方法的几个数值实验,以验证所构造方案的数值精度和效率。FCH梯度流方程具有可变时间步长的二阶方案。另外,对Christlieb等人(2014)中引入的有效自适应时间选择策略进行了稍微修改,然后采用它来选择CN-SAV-A方案的时间步长。最后,分别在二维和三维上进行了基于傅立叶伪谱方法的几个数值实验,以验证所构造方案的数值精度和效率。FCH梯度流方程具有可变时间步长的二阶方案。另外,对Christlieb等人(2014)中引入的有效自适应时间选择策略进行了稍微修改,然后采用它来选择CN-SAV-A方案的时间步长。最后,分别在二维和三维上进行了基于傅立叶伪谱方法的几个数值实验,以验证所构造方案的数值精度和效率。

更新日期:2020-12-26
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