An efficient method for computing stationary states of phase field crystal models

TL;DR

This paper introduces an adaptive accelerated proximal gradient method that efficiently computes stationary states in phase field crystal models, significantly speeding up convergence while ensuring energy dissipation.

Contribution

The paper presents a novel adaptive accelerated proximal gradient approach with proven energy dissipation and convergence for PFC models, connecting it to classical schemes.

Findings

Method accelerates convergence over semi-implicit schemes.

Successfully computes complex 3D and 2D crystal structures.

Reveals physical mechanisms in the Landau-Brazovskii model.

Abstract

Computing stationary states is an important topic for phase field crystal (PFC) models. Great efforts have been made for energy dissipation of the numerical schemes when using gradient flows. However, it is always time-consuming due to the requirement of small effective time steps. In this paper, we propose an adaptive accelerated proximal gradient method for finding the stationary states of PFC models. The energy dissipation is guaranteed and the convergence property is established for the discretized energy functional. Moreover, the connections between generalized proximal operator with classical (semi-)implicit and explicit schemes for gradient flow are given. Extensive numerical experiments, including two three dimensional periodic crystals in Landau-Brazovskii (LB) model and a two dimensional quasicrystal in Lifshitz-Petrich (LP) model, demonstrate that our approach has adaptive time steps which lead to significant acceleration over semi-implicit methods for computing complex structures. Furthermore, our result reveals a deep physical mechanism of the simple LB model via which the sigma phase is first discovered.