The SAV Scheme Based on the Barycentric Interpolation Collocation Method for the Allen-Cahn Equation

HUANG Rong; DENG Yangfang; WENG Zhifeng

doi:10.21656/1000-0887.430149

Volume 44 Issue 5

May 2023

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HUANG Rong, DENG Yangfang, WENG Zhifeng. The SAV Scheme Based on the Barycentric Interpolation Collocation Method for the Allen-Cahn Equation[J]. Applied Mathematics and Mechanics, 2023, 44(5): 573-582. doi: 10.21656/1000-0887.430149

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The SAV Scheme Based on the Barycentric Interpolation Collocation Method for the Allen-Cahn Equation

doi: 10.21656/1000-0887.430149

School of Mathematical Sciences, Huaqiao University, Quanzhou, Fujian 362021, P.R.China

Received Date: 2022-04-27
Rev Recd Date: 2022-06-11
Publish Date: 2023-05-01

Abstract

Abstract

The scalar auxiliary variable (SAV) approach combined with the barycentric interpolation collocation method was proposed to solve the 2D Allen-Cahn equation. Two unconditional energy-stable SAV schemes were constructed based on the Crank-Nicolson scheme and the 2nd-order backward difference scheme for discretization in time, respectively, and the barycentric Lagrange interpolation collocation method for discretization in space. Moreover, the approximation properties of the barycentric Lagrange interpolation were presented. Numerical experiments show that the time-convergence rates of the 2 types of SAV schemes are of the 2nd order and both schemes satisfy the energy decay law. Compared with the finite difference method in space, the barycentric Lagrange interpolation collocation scheme features exponential convergence.
- Allen-Cahn equation,
- barycentric Lagrange interpolation collocation method,
- scalar auxiliary variable scheme,
- energy stability

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References

[1]	ALLEN S, CAHN J. A microscopic theory for antiphase boundary motion and its application to antiphase domain coarsening[J]. Acta Metallurgica, 1979, 27(6): 1085-1095. doi: 10.1016/0001-6160(79)90196-2
[2]	BENNES M, CHALUPECKY V, MIKULA K. Geometrical image segmentation by the Allen-Cahn equation[J]. Applied Numerical Mathematics, 2004, 51(2-3): 187-205. doi: 10.1016/j.apnum.2004.05.001
[3]	KOBAYASHI R. Physicai modeling and numerical simulations of dendritic crystal growth[J]. Physica D: Nonlinear Phenomena, 1993, 63(3/4): 410-423. doi: 10.1016/0167-2789(93)90120-p
[4]	FENG X B, PROHL A. Numerical analysis of the Allen-Cahn equation and approximation for mean curvature flows[J]. Numerische Mathematik, 2003, 94(1): 33-65. doi: 10.1007/s00211-002-0413-1
[5]	DU Q, YANG J. Asymptotically compatible Fourier spectral approximations of nonlocal Allen-Cahn equations[J]. SIAM Journal on Numerical Analysis, 2016, 54(3): 1899-1919. doi: 10.1137/15M1039857
[6]	ZHAI S, FENG X, HE Y. Numerical simulation of the three dimensional Allen-Cahn equation by the high-order compact ADI method[J]. Computer Physics Communications, 2014, 185(10): 2449-2455. doi: 10.1016/j.cpc.2014.05.017
[7]	WENG Z, TANG L. Analysis of the operator splitting scheme for the Allen-Cahn equation[J]. Numerical Heat Transfer (Part B): Fundamentals, 2016, 70(5): 472-483. doi: 10.1080/10407790.2016.1215714
[8]	LI C, HUANG Y, YI N. An unconditionally energy stable second order finite element method for solving the Allen-Cahn equation[J]. Journal of Computational and Applied Mathematics, 2019, 353: 38-48. doi: 10.1016/j.cam.2018.12.024
[9]	LIAO H, TANG T, ZHOU T. On energy stable, maximum-principle preserving, second order BDF scheme with variable steps for the Allen-Cahn equation[J]. SIAM Journal on Numerical Analysis, 2020, 58(4): 2294-2314. doi: 10.1137/19M1289157
[10]	LI H, SONG Z, HU J. Numerical analysis of a second-order IPDGFE method for the Allen-Cahn equation and the curvature-driven geometric flow[J]. Computers & Mathematics With Applications, 2021, 86: 49-62.
[11]	LI J, JU L, CAI Y, et al. Unconditionally maximum bound principle preserving linear schemes for the conservative Allen-Cahn equation with nonlocal constraint[J]. Journal of Scientific Computing, 2021, 87(3): 1-32. doi: 10.1007/s10915-021-01512-0
[12]	汪精英, 翟术英. 分数阶Cahn-Hilliard方程的高效数值算法[J]. 应用数学和力学, 2021, 42(8): 832-840. doi: 10.21656/1000-0887.420008 WANG Jingying, ZHAI Shuying. Efficient numerical algorithm for the fractional Cahn-Hilliard equation[J]. Applied Mathematics and Mechanics, 2021, 42(8): 832-840. (in Chinese) doi: 10.21656/1000-0887.420008
[13]	曾维鸿, 傅卓佳, 汤卓超. 水槽动力特性数值模拟的新型局部无网格配点法[J]. 应用数学和力学, 2022, 43(4): 392-400. doi: 10.21656/1000-0887.420246 ZENG Weihong, FU Zhuojia, TANG Zhuochao. A novel localized meshless collocation method for numerical simulation of flume dynamic characteristics[J]. Applied Mathematics and Mechanics, 2022, 43(4): 392-400. (in Chinese) doi: 10.21656/1000-0887.420246
[14]	吴迪, 李小林. 时间分数阶扩散波方程的无单元Galerkin法分析[J]. 应用数学和力学, 2022, 43(2): 215-223. doi: 10.21656/1000-0887.420172 WU Di, LI Xiaolin. An element-free Galerkin method for time-fractional diffusion-wave equations[J]. Applied Mathematics and Mechanics, 2022, 43(2): 215-223. (in Chinese) doi: 10.21656/1000-0887.420172
[15]	王兆清, 徐子康. 基于平面问题的位移压力混合配点法[J]. 计算物理, 2018, 35(1): 77-86. https://www.cnki.com.cn/Article/CJFDTOTAL-JSWL201801010.htm WANG Zhaoqing, XU Zikang. Displacement pressure mixed collocation method based on plane problem[J]. Computational physics, 2018, 35(1): 77-86. (in Chinese) https://www.cnki.com.cn/Article/CJFDTOTAL-JSWL201801010.htm
[16]	HU Y, PENG A, CHEN L, et al. Analysis of the barycentric interpolation collocation scheme for the Burgers equation[J]. Science Asia, 2021, 47(6): 758.
[17]	DAREHMIRAKI M, REZAZADEH A, AHMADIAN A, et al. An interpolation method for the optimal control problem governed by the elliptic convection-diffusion equation[J]. Numerical Methods for Partial Differential Equations, 2022, 38(2): 137-159. doi: 10.1002/num.22625/abstract
[18]	DENG Y, WENG Z. Barycentric interpolation collocation method based on Crank-Nicolson scheme for the Allen-Cahn equation[J]. AIMS Mathematics, 2021, 6(4): 3857-3873. http://www.researchgate.net/publication/348874624_Barycentric_interpolation_collocation_method_based_on_Crank-Nicolson_scheme_for_the_Allen-Cahn_equation
[19]	LIU H, HUANG J, ZHANG W, et al. Meshfree approach for solving multi-dimensional systems of Fredholm integral equations via barycentric Lagrange interpolation[J]. Applied Mathematics and Computation, 2019, 346: 295-304. http://www.onacademic.com/detail/journal_1000040906451410_249d.html
[20]	YI S, YAO L. A steady barycentric Lagrange interpolation method for the 2D higher-order time fractional telegraph equation with nonlocal boundary condition with error analysis[J]. Numerical Methods for Partial Differential Equations, 2019, 35(5): 1694-1716. http://www.sciencedirect.com/science/article/pii/S096007791930267X
[21]	SHEN J, XU J, YANG J. The scalar auxiliary variable (SAV) approach for gradient flows[J]. Journal of Computational Physics, 2018, 353: 407-416. http://www.math.purdue.edu/~shen/pub/SXY17.pdf
[22]	HUANG F, SHEN J, YANG Z. A highly efficient and accurate new scalar auxiliary variable approach for gradient flows[J]. SIAM Journal on Scientific Computing, 2020, 42(4): A2514-A2536.
[23]	CHENG Q, LIU C, SHEN J. Generalized SAV approaches for gradient systems[J]. Journal of Computational and Applied Mathematics, 2021, 394: 113532. http://www.sciencedirect.com/science/article/pii/S0377042721001515
[24]	LI X, SHEN J. Stability and error estimates of the SAV Fourier-spectral method for the phase field crystal equation[J]. Advances in Computational Mathematics, 2020, 46(3): 48. http://www.xueshufan.com/publication/3031773824
[25]	SHEN J, XU J. Convergence and error analysis for the scalar auxiliary variable (SAV) schemes to gradient flows[J]. SIAM Journal on Numerical Analysis, 2018, 56(5): 2895-2912. http://www.nstl.gov.cn/paper_detail.html?id=b274cd7358273bcdbfd1fa987c6fdfd3
[26]	KLEIN G, BERRUT J. Linear rational finite differences from derivatives of barycentric rational interpolants[J]. SIAM Journal on Numerical Analysis, 2012, 50(2): 643-656. http://d.wanfangdata.com.cn/periodical/88c1707a69313eb241ecc25eea2bfcd3