Convergence Analysis and Numerical Implementation of a Second Order Numerical Scheme for the Three-Dimensional Phase Field Crystal Equation

TL;DR

This paper provides a detailed convergence analysis and practical implementation of a second-order numerical scheme for the 3D phase field crystal equation, including multigrid solver efficiency and simulation results.

Contribution

It offers the first convergence proof for the scheme and details a multigrid method for solving the nonlinear system in 3D PFC simulations.

Findings

Convergence of the scheme is established with maximum norm estimates.

Multigrid solver demonstrates efficiency in solving the nonlinear system.

Numerical simulations include polycrystal growth and convergence tests.

Abstract

In this paper we analyze and implement a second-order-in-time numerical scheme for the three-dimensional phase field crystal (PFC) equation. The numerical scheme was proposed in [46], with the unique solvability and unconditional energy stability established. However, its convergence analysis remains open. We present a detailed convergence analysis in this article, in which the maximum norm estimate of the numerical solution over grid points plays an essential role. Moreover, we outline the detailed multigrid method to solve the highly nonlinear numerical scheme over a cubic domain, and various three-dimensional numerical results are presented, including the numerical convergence test, complexity test of the multigrid solver and the polycrystal growth simulation.