Close Search Advanced
Journals
Resources
About Us
Open Access

A High Order Adaptive Time-Stepping Strategy and Local Discontinuous Galerkin Method for the Modified Phase Field Crystal Equation

A High Order Adaptive Time-Stepping Strategy and Local Discontinuous Galerkin Method for the Modified Phase Field Crystal Equation
Preview
Add to basket

Year:    2018

Communications in Computational Physics, Vol. 24 (2018), Iss. 1 : pp. 123–151

Abstract

In this paper, we will develop a first order and a second order convex splitting, and a first order linear energy stable fully discrete local discontinuous Galerkin (LDG) methods for the modified phase field crystal (MPFC) equation. In which, the first order linear scheme is based on the invariant energy quadratization approach. The MPFC equation is a damped wave equation, and to preserve an energy stability, it is necessary to introduce a pseudo energy, which all increase the difficulty of constructing numerical methods comparing with the phase field crystal (PFC) equation. Due to the severe time step restriction of explicit time marching methods, we introduce the first order and second order semi-implicit schemes, which are proved to be unconditionally energy stable. In order to improve the temporal accuracy, the semi-implicit spectral deferred correction (SDC) method combining with the first order convex splitting scheme is employed. Numerical simulations of the MPFC equation always need long time to reach steady state, and then adaptive time-stepping method is necessary and of paramount importance. The schemes at the implicit time level are linear or nonlinear and we solve them by multigrid solver. Numerical experiments of the accuracy and long time simulations are presented demonstrating the capability and efficiency of the proposed methods, and the effectiveness of the adaptive time-stepping strategy.

Submit Article

You do not have full access to this article.

Already a Subscriber? Sign in as an individual or via your institution

Journal Article Details

Publisher Name:    Global Science Press

Language:    English

DOI:    https://doi.org/10.4208/cicp.OA-2017-0074

Communications in Computational Physics, Vol. 24 (2018), Iss. 1 : pp. 123–151

Published online:    2018-01

AMS Subject Headings:    Global Science Press

Copyright:    COPYRIGHT: © Global Science Press

Pages:    29

Keywords:    Adaptive time-stepping local discontinuous Galerkin method modified phase field crystal equation convex splitting pseudo energy unconditionally energy stable spectral deferred correction.

  1. Entropy dissipative higher order accurate positivity preserving time-implicit discretizations for nonlinear degenerate parabolic equations

    Yan, Fengna | Van der Vegt, J.J.W. | Xia, Yinhua | Xu, Yan

    Journal of Computational and Applied Mathematics, Vol. 441 (2024), Iss. P.115674

    https://doi.org/10.1016/j.cam.2023.115674 [Citations: 1]

  2. Error estimates for second-order SAV finite element method to phase field crystal model

    Wang, Liupeng | Huang, Yunqing

    Electronic Research Archive, Vol. 29 (2021), Iss. 1 P.1735

    https://doi.org/10.3934/era.2020089 [Citations: 6]

  3. Highly efficient, decoupled and unconditionally stable numerical schemes for a modified phase-field crystal model with a strong nonlinear vacancy potential

    Zhang, Xin | Wu, Jingwen | Tan, Zhijun

    Computers & Mathematics with Applications, Vol. 132 (2023), Iss. P.119

    https://doi.org/10.1016/j.camwa.2022.12.011 [Citations: 4]

  4. Efficient Numerical Methods for Computing the Stationary States of Phase Field Crystal Models

    Jiang, Kai | Si, Wei | Chen, Chang | Bao, Chenglong

    SIAM Journal on Scientific Computing, Vol. 42 (2020), Iss. 6 P.B1350

    https://doi.org/10.1137/20M1321176 [Citations: 8]

  5. Error estimates for the Scalar Auxiliary Variable (SAV) schemes to the modified phase field crystal equation

    Qi, Longzhao | Hou, Yanren

    Journal of Computational and Applied Mathematics, Vol. 417 (2023), Iss. P.114579

    https://doi.org/10.1016/j.cam.2022.114579 [Citations: 8]

  6. A new space-fractional modified phase field crystal equation and its numerical algorithm

    Bu, Linlin | Li, Rui | Mei, Liquan | Wang, Ying

    Applied Mathematics Letters, Vol. 158 (2024), Iss. P.109216

    https://doi.org/10.1016/j.aml.2024.109216 [Citations: 0]

  7. Optimal Error Estimates of the Local Discontinuous Galerkin Method and High-order Time Discretization Scheme for the Swift–Hohenberg Equation

    Zhou, Lingling | Guo, Ruihan

    Journal of Scientific Computing, Vol. 93 (2022), Iss. 2

    https://doi.org/10.1007/s10915-022-02014-3 [Citations: 2]

  8. A second-order and nonuniform time-stepping maximum-principle preserving scheme for time-fractional Allen-Cahn equations

    Liao, Hong-lin | Tang, Tao | Zhou, Tao

    Journal of Computational Physics, Vol. 414 (2020), Iss. P.109473

    https://doi.org/10.1016/j.jcp.2020.109473 [Citations: 101]

  9. Adaptive time step selection for spectral deferred correction

    Baumann, Thomas | Götschel, Sebastian | Lunet, Thibaut | Ruprecht, Daniel | Speck, Robert

    Numerical Algorithms, Vol. (2024), Iss.

    https://doi.org/10.1007/s11075-024-01964-z [Citations: 0]
  10. Geometric Partial Differential Equations - Part I

    The phase field method for geometric moving interfaces and their numerical approximations

    Du, Qiang | Feng, Xiaobing

    2020

    https://doi.org/10.1016/bs.hna.2019.05.001 [Citations: 33]

  11. Efficient unconditionally stable numerical schemes for a modified phase field crystal model with a strong nonlinear vacancy potential

    Pei, Shuaichao | Hou, Yanren | Yan, Wenjing

    Numerical Methods for Partial Differential Equations, Vol. 38 (2022), Iss. 1 P.65

    https://doi.org/10.1002/num.22828 [Citations: 6]

  12. Linear and unconditionally energy stable schemes for the modified phase field crystal equation

    Liang, Yihong | Jia, Hongen

    Computers & Mathematics with Applications, Vol. 153 (2024), Iss. P.197

    https://doi.org/10.1016/j.camwa.2023.11.008 [Citations: 2]

  13. Efficient linear and unconditionally energy stable schemes for the modified phase field crystal equation

    Li, Xiaoli | Shen, Jie

    Science China Mathematics, Vol. 65 (2022), Iss. 10 P.2201

    https://doi.org/10.1007/s11425-020-1867-8 [Citations: 10]

  14. Efficient numerical scheme for a dendritic solidification phase field model with melt convection

    Chen, Chuanjun | Yang, Xiaofeng

    Journal of Computational Physics, Vol. 388 (2019), Iss. P.41

    https://doi.org/10.1016/j.jcp.2019.03.017 [Citations: 98]

  15. A-stable spectral deferred correction method for nonlinear Allen-Cahn model

    Yao, Lin | Zhang, Xindong

    Alexandria Engineering Journal, Vol. 95 (2024), Iss. P.197

    https://doi.org/10.1016/j.aej.2024.03.091 [Citations: 0]

  16. L-stable spectral deferred correction methods and applications to phase field models

    Yao, Lin | Xia, Yinhua | Xu, Yan

    Applied Numerical Mathematics, Vol. 197 (2024), Iss. P.288

    https://doi.org/10.1016/j.apnum.2023.11.020 [Citations: 1]

  17. A linearly second-order, unconditionally energy stable scheme and its error estimates for the modified phase field crystal equation

    Pei, Shuaichao | Hou, Yanren | Li, Qi

    Computers & Mathematics with Applications, Vol. 103 (2021), Iss. P.104

    https://doi.org/10.1016/j.camwa.2021.10.029 [Citations: 11]

  18. Energy quadratization Runge–Kutta method for the modified phase field crystal equation

    Shin, Jaemin | Lee, Hyun Geun | Lee, June-Yub

    Modelling and Simulation in Materials Science and Engineering, Vol. 30 (2022), Iss. 2 P.024004

    https://doi.org/10.1088/1361-651X/ac466c [Citations: 5]