TY - JOUR
T1 - Benchmark Computations of the Phase Field Crystal and Functionalized Cahn-Hilliard Equations via Fully Implicit, Nesterov Accelerated Schemes
AU - Park , Jea-Hyun
AU - Salgado , Abner J.
AU - Wise , Steven M.
JO - Communications in Computational Physics
VL - 2
SP - 367
EP - 398
PY - 2023
DA - 2023/03
SN - 33
DO - http://doi.org/ 10.4208/cicp.OA-2022-0117
UR - https://global-sci.org/intro/article_detail/cicp/21492.html
KW - Phase field crystal, functionalized Cahn-Hilliard, preconditioning, Nesterov acceleration, nonlinear solver.
AB - <p style="text-align: justify;">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’s 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 four 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.</p>