Close Search Advanced
Journals
Resources
About Us
Open Access

High Order Local Discontinuous Galerkin Methods for the Allen-Cahn Equation: Analysis and Simulation

High Order Local Discontinuous Galerkin Methods for the Allen-Cahn Equation: Analysis and Simulation
Preview
Add to basket

Year:    2016

Author:    Ruihan Guo, Liangyue Ji, Yan Xu

Journal of Computational Mathematics, Vol. 34 (2016), Iss. 2 : pp. 135–158

Abstract

In this paper, we present a local discontinuous Galerkin (LDG) method for the Allen-Cahn equation. We prove the energy stability, analyze the optimal convergence rate of $k + 1$ in $L^2$ norm and present the $(2k + 1)$-th order negative-norm estimate of the semi-discrete LDG method for the Allen-Cahn equation with smooth solution. To relax the severe time step restriction of explicit time marching methods, we construct a first order semi-implicit scheme based on the convex splitting principle of the discrete Allen-Cahn energy and prove the corresponding unconditional energy stability. To achieve high order temporal accuracy, we employ the semi-implicit spectral deferred correction (SDC) method. Combining with the unconditionally stable convex splitting scheme, the SDC method can be high order accurate and stable in our numerical tests. To enhance the efficiency of the proposed methods, the multigrid solver is adapted to solve the resulting nonlinear algebraic systems. Numerical studies are presented to confirm that we can achieve optimal accuracy of $\mathcal{O}(h^{k+1})$ in $L^2$ norm and improve the LDG solution from $\mathcal{O}(h^{k+1})$ to $\mathcal{O}(h^{2k+1})$ with the accuracy enhancement post-processing technique.

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/jcm.1510-m2014-0002

Journal of Computational Mathematics, Vol. 34 (2016), Iss. 2 : pp. 135–158

Published online:    2016-01

AMS Subject Headings:   

Copyright:    COPYRIGHT: © Global Science Press

Pages:    24

Keywords:    Local discontinuous Galerkin method Allen-Cahn equation Energy stability Convex splitting Spectral deferred correction A priori error estimate Negative norm error estimate.

Author Details

Ruihan Guo

Liangyue Ji

Yan Xu

  1. 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]

  2. Local Discontinuous Galerkin Scheme for Space Fractional Allen–Cahn Equation

    Li, Can | Liu, Shuming

    Communications on Applied Mathematics and Computation, Vol. 2 (2020), Iss. 1 P.73

    https://doi.org/10.1007/s42967-019-00034-9 [Citations: 7]

  3. Error Estimates of Energy Stable Numerical Schemes for Allen–Cahn Equations with Nonlocal Constraints

    Sun, Shouwen | Jing, Xiaobo | Wang, Qi

    Journal of Scientific Computing, Vol. 79 (2019), Iss. 1 P.593

    https://doi.org/10.1007/s10915-018-0867-7 [Citations: 7]

  4. Stability and Error Estimates of High Order BDF-LDG Discretizations for the Allen–Cahn Equation

    Yan, Fengna | Cheng, Ziqiang

    Computational Mathematics and Mathematical Physics, Vol. 63 (2023), Iss. 12 P.2551

    https://doi.org/10.1134/S0965542523120229 [Citations: 0]

  5. Comparison of different time discretization schemes for solving the Allen–Cahn equation

    Ayub, Sana | Rauf, Abdul | Affan, Hira | Shah, Abdullah

    International Journal of Nonlinear Sciences and Numerical Simulation, Vol. 23 (2022), Iss. 3-4 P.603

    https://doi.org/10.1515/ijnsns-2019-0283 [Citations: 2]

  6. Solving Allen-Cahn equations with periodic and nonperiodic boundary conditions using mimetic finite-difference operators

    Orizaga, Saulo | González-Parra, Gilberto | Forman, Logan | Villegas-Villanueva, Jesus

    Applied Mathematics and Computation, Vol. 484 (2025), Iss. P.128993

    https://doi.org/10.1016/j.amc.2024.128993 [Citations: 0]

  7. Stability analysis and error estimates of local discontinuous Galerkin methods with semi-implicit spectral deferred correction time-marching for the Allen–Cahn equation

    Yan, Fengna | Xu, Yan

    Journal of Computational and Applied Mathematics, Vol. 376 (2020), Iss. P.112857

    https://doi.org/10.1016/j.cam.2020.112857 [Citations: 11]

  8. Convergence Analysis of the Fully Discrete Hybridizable Discontinuous Galerkin Method for the Allen–Cahn Equation Based on the Invariant Energy Quadratization Approach

    Wang, Jiangxing | Pan, Kejia | Yang, Xiaofeng

    Journal of Scientific Computing, Vol. 91 (2022), Iss. 2

    https://doi.org/10.1007/s10915-022-01822-x [Citations: 6]

  9. Interior Penalty Discontinuous Galerkin Based Isogeometric Analysis for Allen-Cahn Equations on Surfaces

    Zhang, Futao | Xu, Yan | Chen, Falai | Guo, Ruihan

    Communications in Computational Physics, Vol. 18 (2015), Iss. 5 P.1380

    https://doi.org/10.4208/cicp.010914.180315a [Citations: 5]
  10. Model Reduction of Parametrized Systems

    Energy Stable Model Order Reduction for the Allen-Cahn Equation

    Uzunca, Murat | Karasözen, Bülent

    2017

    https://doi.org/10.1007/978-3-319-58786-8_25 [Citations: 4]

  11. Adaptive discontinuous Galerkin finite element methods for the Allen-Cahn equation on polygonal meshes

    Li, Rui | Gao, Yali | Chen, Zhangxin

    Numerical Algorithms, Vol. 95 (2024), Iss. 4 P.1981

    https://doi.org/10.1007/s11075-023-01635-5 [Citations: 1]

  12. Local discontinuous Galerkin methods with implicit–explicit BDF time marching for Newell–Whitehead–Segel equations

    Wang, Haijin | Shi, Xiaobin | Shao, Rumeng | Zhu, Hongqiang | Chen, Yanping

    International Journal of Computer Mathematics, Vol. (2024), Iss. P.1

    https://doi.org/10.1080/00207160.2024.2423658 [Citations: 0]
  13. 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: 34]

  14. Local Discontinuous Galerkin Method Coupled with Nonuniform Time Discretizations for Solving the Time-Fractional Allen-Cahn Equation

    Wang, Zhen | Sun, Luhan | Cao, Jianxiong

    Fractal and Fractional, Vol. 6 (2022), Iss. 7 P.349

    https://doi.org/10.3390/fractalfract6070349 [Citations: 0]

  15. Numerical approximation of SAV finite difference method for the Allen–Cahn equation

    Chen, Hang | Huang, Langyang | Zhuang, Qingqu | Weng, Zhifeng

    International Journal of Modeling, Simulation, and Scientific Computing, Vol. 14 (2023), Iss. 05

    https://doi.org/10.1142/S1793962324500016 [Citations: 1]

  16. Convergence analysis of an accurate and efficient method for nonlinear Maxwell's equations

    Wang, Jiangxing

    Discrete & Continuous Dynamical Systems - B, Vol. 26 (2021), Iss. 5 P.2429

    https://doi.org/10.3934/dcdsb.2020185 [Citations: 0]

  17. Arbitrarily High Order and Fully Discrete Extrapolated RK–SAV/DG Schemes for Phase-field Gradient Flows

    Tang, Tao | Wu, Xu | Yang, Jiang

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

    https://doi.org/10.1007/s10915-022-01995-5 [Citations: 6]

  18. 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]

  19. Numerical investigation into the dependence of the Allen–Cahn equation on the free energy

    Kim, Yunho | Lee, Dongsun

    Advances in Computational Mathematics, Vol. 48 (2022), Iss. 3

    https://doi.org/10.1007/s10444-022-09955-1 [Citations: 5]

  20. Numerical simulation of Allen–Cahn equation with nonperiodic boundary conditions by the local discontinuous Galerkin method

    Chand, Abhilash | Saha Ray, S.

    International Journal of Modern Physics B, Vol. 37 (2023), Iss. 02

    https://doi.org/10.1142/S0217979223500194 [Citations: 5]

  21. The Allen-Cahn equation with a time Caputo-Hadamard derivative: Mathematical and Numerical Analysis

    Wang, Zhen | Sun, Luhan

    Communications in Analysis and Mechanics, Vol. 15 (2023), Iss. 4 P.611

    https://doi.org/10.3934/cam.2023031 [Citations: 5]

  22. A two‐grid finite element method for the Allen‐Cahn equation with the logarithmic potential

    Wang, Danxia | Li, Yanan | Jia, Hongen

    Numerical Methods for Partial Differential Equations, Vol. 39 (2023), Iss. 2 P.1251

    https://doi.org/10.1002/num.22932 [Citations: 6]

  23. A structure-preserving explicit numerical scheme for the Allen–Cahn equation with a logarithmic potential

    Ham, Seokjun | Choi, Jaeyong | Kwak, Soobin | Kim, Junseok

    Journal of Mathematical Analysis and Applications, Vol. 538 (2024), Iss. 1 P.128425

    https://doi.org/10.1016/j.jmaa.2024.128425 [Citations: 0]

  24. Energy Stable Discontinuous Galerkin Finite Element Method for the Allen–Cahn Equation

    Karasözen, Bülent | Uzunca, Murat | Sariaydin-Fi̇li̇beli̇oğlu, Ayşe | Yücel, Hamdullah

    International Journal of Computational Methods, Vol. 15 (2018), Iss. 03 P.1850013

    https://doi.org/10.1142/S0219876218500135 [Citations: 17]

  25. Arbitrarily high order structure-preserving algorithms for the Allen-Cahn model with a nonlocal constraint

    Hong, Qi | Gong, Yuezheng | Zhao, Jia | Wang, Qi

    Applied Numerical Mathematics, Vol. 170 (2021), Iss. P.321

    https://doi.org/10.1016/j.apnum.2021.08.002 [Citations: 8]

  26. 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]