Benchmark computations of the phase field crystal and functionalized Cahn-Hilliard equations via fully implicit, Nesterov accelerated schemes
Benchmark computations of the phase field crystal and functionalized Cahn-Hilliard equations via fully implicit, Nesterov accelerated schemes
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DOI：
10.4208/cicp.oa-2022-0117
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发表时间：
2022-04
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通讯作者：
Jea-Hyun Park;A. Salgado;S. Wise
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作者：
Jea-Hyun Park;A. Salgado;S. Wise
我们介绍了一个快速求解器的相位场晶体（PFC）和功能化的Cahn-Hilliard（FCH）方程的矩形域上的周期性边界条件，具有预处理Nesterov加速梯度下降（PAGD）方法。我们离散这些问题的傅立叶配置法在空间上，并采用各种二阶格式的时间。我们观察到一个显着的加速与此求解器相比，预条件梯度下降（PGD）方法。使用PAGD求解器，全隐式二阶时间格式不仅可以求解PFC和FCH方程，而且在某些考虑精度问题的情况下比某些半隐式格式更有效。对PFC和FCH方程的五种不同格式进行了基准计算，结果表明，对于FCH实验，全隐式格式（中点规则和BDF 2配备PAGD作为非线性时间推进求解器）在达到一定精度所需的计算成本方面优于其IMEX版本。对于PFC，结果不像FCH实验那样具有决定性，我们认为，这是由于PFC中的非线性比FCH方程更温和。本文还讨论了应用PAGD的一些实际问题。我们引入了平均牛顿预条件和清扫摩擦策略作为启发式方法来选择好的预条件参数。清扫摩擦策略表现出几乎一样好的性能的情况下，最好的手动调整参数。
We introduce a fast solver for the phase field crystal (PFC) and functionalized Cahn-Hilliard (FCH) equations with periodic boundary conditions on a rectangular domain that features the preconditioned Nesterov accelerated gradient descent (PAGD) method. We discretize these problems with a Fourier collocation method in space, and employ various second-order schemes in time. We observe a significant speedup with this solver when compared to the preconditioned gradient descent (PGD) method. With the PAGD solver, fully implicit, second-order-in-time schemes are not only feasible to solve the PFC and FCH equations, but also do so more efficiently than some semi-implicit schemes in some cases where accuracy issues are taken into account. Benchmark computations of five different schemes for the PFC and FCH equations are conducted and the results indicate that, for the FCH experiments, the fully implicit schemes (midpoint rule and BDF2 equipped with the PAGD as a nonlinear time marching solver) perform better than their IMEX versions in terms of computational cost needed to achieve a certain precision. For the PFC, the results are not as conclusive as in the FCH experiments, which, we believe, is due to the fact that the nonlinearity in the PFC is milder nature compared to the FCH equation. We also discuss some practical matters in applying the PAGD. We introduce an averaged Newton preconditioner and a sweeping-friction strategy as heuristic ways to choose good preconditioner parameters. The sweeping-friction strategy exhibits almost as good a performance as the case of the best manually tuned parameters.