# Parallel energy-stable phase field crystal simulations based on domain decomposition methods

## TL;DR

This paper introduces a parallel, energy-stable phase field crystal simulation algorithm that employs domain decomposition and adaptive time stepping, achieving high scalability and second-order accuracy in 2D and 3D cases.

## Contribution

It develops a novel parallel Newton–Krylov–Schwarz method with improved boundary conditions for efficient phase field crystal simulations.

## Key findings

- Second-order accuracy in space and time.
- Energy stability with large time steps.
- Scalability to over ten thousand cores.

## Abstract

In this paper, we present a parallel numerical algorithm for solving the phase field crystal equation. In the algorithm, a semi-implicit finite difference scheme is derived based on the discrete variational derivative method. Theoretical analysis is provided to show that the scheme is unconditionally energy stable and can achieve second-order accuracy in both space and time. An adaptive time step strategy is adopted such that the time step size can be flexibly controlled based on the dynamical evolution of the problem. At each time step, a nonlinear algebraic system is constructed from the discretization of the phase field crystal equation and solved by a domain decomposition based, parallel Newton--Krylov--Schwarz method with improved boundary conditions for subdomain problems. Numerical experiments with several two and three dimensional test cases show that the proposed algorithm is second-order accurate in both space and time, energy stable with large time steps, and highly scalable to over ten thousands processor cores on the Sunway TaihuLight supercomputer.