An Efficient and Accurate Method for the Conservative Swift–Hohenberg Equation and Its Numerical Implementation

Hyun Geun Lee

doi:10.3390/math8091502

Abstract

The conservative Swift–Hohenberg equation was introduced to reformulate the phase-field crystal model. A challenge in solving the conservative Swift–Hohenberg equation numerically is how to treat the nonlinear term to preserve mass conservation without compromising efficiency and accuracy. To resolve this problem, we present a linear, high-order, and mass conservative method by placing the linear and nonlinear terms in the implicit and explicit parts, respectively, and employing the implicit-explicit Runge–Kutta method. We show analytically that the method inherits the mass conservation. Numerical experiments are presented demonstrating the efficiency and accuracy of the proposed method. In particular, long time simulation for pattern formation in 2D is carried out, where the phase diagram can be observed clearly. The MATLAB code for numerical implementation of the proposed method is provided in Appendix.

Highlights

The phase-field crystal (PFC) model describes the microstructure of two-phase systems on atomic length and diffusive time scales and has been used to study grain growth, dendritic and eutectic solidification, and epitaxial growth [1,2]
We show analytically that the method inherits the mass conservation
We developed linear, first, second, and third-order, and mass conservative methods for the conservative SH equation by placing the linear and nonlinear terms in the implicit and explicit parts, respectively, and employing the implicit-explicit RK method

Summary

Introduction

The phase-field crystal (PFC) model describes the microstructure of two-phase systems on atomic length and diffusive time scales and has been used to study grain growth, dendritic and eutectic solidification, and epitaxial growth [1,2]. Conservative SH equations were introduced to reformulate the PFC model [4,5]. Ω ( Φ ( φ ( x,t ))) dx and proposed mass conservative first- and second-order operator splitting methods. The aim of this paper is to present an efficient and accurate method that preserves mass conservation for solving the conservative SH Equation (3). Our method is linear, high-order accurate in time, and mass conservative. We show analytically that the method inherits the mass conservation. High-order (up to third-order), and mass conservative method for solving. For the method S1, Equation (7) can be transformed into the discrete Fourier space using (6): bn+1 =. The methods S2 and S3 inherit the mass conservation

Convergence Test

Efficiency of the Proposed Method

Phase Diagram in 2D

Comparison with Other Method

51–67. Phase field crystal model

Methods