  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.
