Benchmark Computations of the Phase Field Crystal and Functionalized Cahn-Hilliard Equations via Fully Implicit, Nesterov Accelerated Schemes (Communications in Computational Physics, 2023)
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 methodin 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 schemesare 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 aretaken into account. Benchmark computations of four different schemes for the PFCand FCH equations are conducted and the results indicate that, for the FCH experiments, the fully implicit schemes (midpoint rule and BDF2 equipped with the PAGDas a nonlinear time marching solver) perform better than their IMEX versions in termsof computational cost needed to achieve a certain precision. For the PFC, the resultsare not as conclusive as in the FCH experiments, which, we believe, is due to the factthat the nonlinearity in the PFC is milder nature compared to the FCH equation. Wealso discuss some practical matters in applying the PAGD. We introduce an averagedNewton preconditioner and a sweeping-friction strategy as heuristic ways to choose goodpreconditioner parameters. The sweeping-friction strategy exhibits almost as gooda performance as the case of the best manually tuned parameters.
Keywords: Phase field crystal; functionalized Cahn-Hilliard; preconditioning; Nesterov acceleration; nonlinear solver.