论文标题
通过完全隐式的Nesterov加速方案的相位场晶体和功能化的Cahn-Hilliard方程的基准计算
Benchmark computations of the phase field crystal and functionalized Cahn-Hilliard equations via fully implicit, Nesterov accelerated schemes
论文作者
论文摘要
我们在矩形结构域上引入了一个用于相位场晶体（PFC）的快速求解器和功能化的Cahn-Hilliard（FCH）方程，该方程具有周期性边界条件，该矩形域上具有预处理的Nesterov加速梯度下降（PAGD）方法。我们通过空间中的傅立叶搭配方法离散这些问题，并在时间上采用各种二阶方案。与预处理的梯度下降（PGD）方法相比，我们观察到该求解器的显着加速。使用PAGD求解器，完全隐式的，二阶的时间方案不仅可以解决PFC和FCH方程，而且在某些情况下，在考虑准确性问题的情况下，也比某些半显得方案更有效地做到这一点。对PFC和FCH方程进行了五种不同方案的基准计算，结果表明，对于FCH实验，完全隐式的方案（中值规则和BDF2）具有PAGD作为非线性时间行进的求解器，其在获得一定精确度的计算成本方面都比其IMEX效果更好。对于PFC而言，结果不像FCH实验那样结论，我们认为这是由于PFC中的非线性与FCH方程相比是温和的。我们还讨论了应用PAGD的一些实际问题。我们介绍了一个平均的牛顿预科人员和一种横流策略，作为选择良好的预处理参数的启发式方法。扫地策略的表现几乎与手动调谐参数一样出色。
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.
