# 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 combined with SAV and SDC methods, proving energy stability and improving computational accuracy for the model.

## Key findings

- High-order schemes are effective for simulating incommensurate phase structures.
- The proposed methods demonstrate energy dissipation and improved accuracy.
- Numerical results reveal the impact of length-scales on 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.