High-order energy stable schemes of incommensurate phase-field crystal model

TL;DR

This paper develops high-order energy stable numerical schemes for the incommensurate phase-field crystal model, enabling accurate simulation of aperiodic structures with multiple length scales.

Contribution

It introduces a second-order Crank-Nicolson scheme with energy stability and error estimates, and enhances accuracy using spectral deferred correction methods.

Findings

High-order schemes improve numerical accuracy.

Energy dissipation law is proven for the schemes.

Length scales significantly influence structure formation.

Abstract

This article focuses on the development of high-order energy stable schemes for the multi-length-scale incommensurate phase-field crystal model which is able to study the phase behavior of aperiodic structures. These high-order schemes based on the scalar auxiliary variable (SAV) and spectral deferred correction (SDC) approaches are suitable for the L 2 gradient flow equation, i.e., the Allen-Cahn dynamic equation. Concretely, we propose a second-order Crank-Nicolson (CN) scheme of the SAV system, prove the energy dissipation law, and give the error estimate in the almost periodic function sense. Moreover, we use the SDC method to improve the computational accuracy of the SAV/CN scheme. Numerical results demonstrate the advantages of high-order numerical methods in numerical computations and show the influence of length-scales on the formation of ordered structures.