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

Kai Jiang; Wei Si; Kai Jiang; Wei Si

doi:10.3934/era.2020059

Electronic Research Archive

2020, Volume 28, Issue 2: 1077-1093. doi: 10.3934/era.2020059

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

Kai Jiang ^,,
Wei Si

School of Mathematics and Computational Science, Hunan Key Laboratory for Computation and Simulation in Science and Engineering, Xiangtan University, Hunan, Xiangtan 411105, China

Received: 01 April 2020 Revised: 01 May 2020
Primary: 35K55, 35K35; Secondary: 65M70

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.
- Incommensurate phase-field crystal model,
- high-order energy stable schemes,
- spectral deferred correction method,
- projection method,
- quasicrystals
Citation: Kai Jiang, Wei Si. High-order energy stable schemes of incommensurate phase-field crystal model[J]. Electronic Research Archive, 2020, 28(2): 1077-1093. doi: 10.3934/era.2020059

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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.

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