An Efficient Operator Splitting Method for Modified Allen-Cahn Equations With Mean Curvature Terms
edited-by
edited-by
(Recommended by Zhao Jingjun, Member of the Editorial Board of AMM)-
摘要: 含平均曲率项的修正Allen-Cahn方程可有效模拟与曲率相关的物理过程, 但非线性项和梯度模的存在导致难以构造高效求解格式. 该文提出了一种求解此模型的高效数值格式. 基于Strang算子分裂方法, 在时间方向上将原方程分解为三个子方程: 非线性方程解析求解；平均曲率方程采用二阶Runge-Kutta方法结合中心差分建立全离散显格式；热传导方程采用Crank-Nicolson格式离散求解, 并构造交替方向隐式(ADI)快速求解策略. 理论分析表明，构造格式具有二阶收敛精度. 最后, 通过数值算例验证格式收敛阶和有效性.
-
关键词:
- 修正Allen-Cahn方程 /
- 算子分裂格式 /
- ADI /
- 收敛性分析
Abstract: The modified Allen-Cahn equation with the mean curvature source term can effectively model curvature-driven physical processes. However, the nonlinear term and gradient magnitude make it difficult to design efficient numerical schemes. An efficient numerical scheme was proposed for solving the equation. Based on the Strang operator splitting method, the original equation was discretized in time into three subproblems: the nonlinear equation was solved analytically; the mean curvature equation was discretized using the second-order Runge-Kutta method combined with central differences to establish a fully discrete explicit scheme, while the heat equation was discretized with the Crank-Nicolson method and solved via the alternating direction implicit (ADI) fast scheme. Theoretical analysis shows that, the proposed scheme achieves second order convergence accuracy. Finally, numerical experiments were presented to validate the convergence rate and effectiveness of the scheme.-
Key words:
- modified Allen-Cahn equation /
- operator splitting scheme /
- ADI /
- convergence analysis
edited-byedited-by1) (本刊编委赵景军推荐) -
图 2 λ=0时，不同时刻数值解和对应等高线图
Figure 2. The numerical solution results for λ=0 and the corresponding contour plots
图 3 λ=1时，不同时刻数值解和对应等高线图
Figure 3. The numerical solution results for λ=1 and the corresponding contour plots
图 4 λ=1.5时，不同时刻数值解和对应等高线图
Figure 4. The numerical solution results for λ=1.5 and the corresponding contour plots
图 5 τ=0.01, 0.1, 1时，T=100的数值解与等高线图
Figure 5. Numerical solutions and contour plots at T=100 for τ=0.01, 0.1, 1
表 1 空间误差和收敛阶
Table 1. Spatial errors and convergence orders
N E∞ Rrate E2 Rrate 40 5.02E-5 - 4.59E-5 - 80 1.32E-5 1.93 1.28E-5 1.84 160 3.33E-6 1.98 3.29E-6 1.97 320 8.36E-7 2.00 8.22E-7 2.00 
下载: 导出CSV
表 2 时间方向L2范数误差、收敛阶以及CPU计算时间
Table 2. Temporal L2 norm errors, convergence orders and CPU computation times
K operator splitting scheme backward Euler implict method implict method E2 Rrate tCPU/s E2 Rrate tCPU/s E2 Rrate tCPU/s 200 3.63E-4 - 2.35 2.25E-3 - 258.93 2.01E-4 - 359.82 400 9.35E-5 1.96 4.66 1.59E-3 0.50 429.28 5.06E-5 1.99 667.97 800 2.36E-5 1.99 9.16 9.56E-4 0.73 1 482.47 1.27E-5 1.99 1 170.62 1 600 5.91E-6 2.00 18.64 5.26E-4 0.86 2 738.52 3.17E-6 2.00 2 182.16
下载: 导出CSV
-
[1] Allen S M, Cahn J W. 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] Yue P, Feng J J, Liu C, et al. Diffuse-interface simulations of drop coalescence and retraction in viscoelastic fluids[J]. Journal of Non-Newtonian Fluid Mechanics, 2005, 129 (3): 163-176. doi: 10.1016/j.jnnfm.2005.07.002 [3] Kim J. Phase-field models for multi-component fluid flows[J]. Communications in Computational Physics, 2012, 12 (3): 613-661. doi: 10.4208/cicp.301110.040811a [4] Kobayashi R. Modeling and numerical simulations of dendritic crystal growth[J]. Physica D: Nonlinear Phenomena, 1993, 63 (3/4): 410-423. [5] Li Y, Lee H G, Kim J. A fast, robust, and accurate operator splitting method for phase-field simulations of crystal growth[J]. Journal of Crystal Growth, 2011, 321 (1): 176-182. doi: 10.1016/j.jcrysgro.2011.02.042 [6] Wheeler A A, Boettinger W J, McFadden G B. Phase-field model for isothermal phase transitions in binary alloys[J]. Physical Review A, 1992, 45 (10): 7424-7439. doi: 10.1103/PhysRevA.45.7424 [7] Elliott C M, Stinner B. Computation of two-phase biomembranes with phase dependent material parameters using surface finite elements[J]. Communications in Computational Physics, 2013, 13 (2): 325-360. doi: 10.4208/cicp.170611.130112a [8] Cohen D S, Murray J D. A generalized diffusion model for growth and dispersal in a population[J]. Journal of Mathematical Biology, 1981, 12 (2): 237-249. doi: 10.1007/BF00276132 [9] Wang J, Li Y, Choi Y, et al. Fast and accurate smoothing method using a modified Allen-Cahn equation[J]. Computer-Aided Design, 2020, 120: 102804. doi: 10.1016/j.cad.2019.102804 [10] Hazewinkel M, Kaashoek J F, Leynse B. Pattern formation for a one dimensional evolution equation based on thom's river basin model[M]//Kilmister C W. Disequilibrium and Self-Organisation. Dordrecht: Springer Netherlands, 1986: 23-46. [11] Wang J, Lee C, Lee H G, et al. Phase-field modeling and numerical simulation for ice melting[J]. Numerical Mathematics: Theory, Methods and Applications, 2021, 14 (2): 540-558. doi: 10.4208/nmtma.OA-2020-0023 [12] Xu J, Li Y, Wu S, et al. On the stability and accuracy of partially and fully implicit schemes for phase field modeling[J]. Computer Methods in Applied Mechanics and Engineering, 2019, 345: 826-853. doi: 10.1016/j.cma.2018.09.017 [13] Feng X, Tang T, Yang J. Stabilized Crank-Nicolson/Adams-Bashforth schemes for phase field models[J]. East Asian Journal on Applied Mathematics, 2013, 3 (1): 59-80. doi: 10.4208/eajam.200113.220213a [14] He Y, Liu Y, Tang T. On large time-stepping methods for the Cahn-Hilliard equation[J]. Applied Numerical Mathematics, 2007, 57 (5/6/7): 616-628. [15] Shen J, Yang X. Numerical approximations of Allen-Cahn and Cahn-Hilliard equations[J]. Discrete and Continuous Dynamical Systems, 2010, 28 (4): 1669-1691. doi: 10.3934/dcds.2010.28.1669 [16] Hochbruck M, Ostermann A. Explicit exponential Runge-Kutta methods for semilinear parabolic problems[J]. SIAM Journal on Numerical Analysis, 2005, 43 (3): 1069-1090. doi: 10.1137/040611434 [17] Hochbruck M, Ostermann A. Exponential integrators[J]. Acta Numerica, 2010, 19: 209-286. doi: 10.1017/S0962492910000048 [18] Tang T, Yang J. Implicit-explicit scheme for the Allen-Cahn equation preserves the maximum principle[J]. Journal of Computational Mathematics, 2025, 34 (5): 451-461. [19] Shen J, Tang T, Yang J. On the maximum principle preserving schemes for the generalized Allen-Cahn equation[J]. Communications in Mathematical Sciences, 2016, 14 (6): 1517-1534. doi: 10.4310/CMS.2016.v14.n6.a3 [20] Shen J, Xu J, Yang J. A new class of efficient and robust energy stable schemes for gradient flows[J]. SIAM Review, 2019, 61 (3): 474-506. doi: 10.1137/17M1150153 [21] Strang G. On the construction and comparison of difference schemes[J]. SIAM Journal on Numerical Analysis, 1968, 5 (3): 506-517. doi: 10.1137/0705041 [22] Zhai S, Wang D, Weng Z, et al. Error analysis and numerical simulations of strang splitting method for space fractional nonlinear Schrödinger equation[J]. Journal of Scientific Computing, 2019, 81 (2): 965-989. doi: 10.1007/s10915-019-01050-w [23] Zhai S, Weng Z, Yang Y. A high order operator splitting method based on spectral deferred correction for the nonlocal viscous Cahn-Hilliard equation[J]. Journal of Computational Physics, 2021, 446: 110636. doi: 10.1016/j.jcp.2021.110636 [24] Yang J, Kang S, Kwak S, et al. The Allen-Cahn equation with a space-dependent mobility and a source term for general motion by mean curvature[J]. Journal of Computational Science, 2024, 77: 102252. doi: 10.1016/j.jocs.2024.102252 [25] Cao Z, Weng Z, Zhai S. An effective operator splitting scheme for general motion by mean curvature using a modified Allen-Cahn equation[J]. Applied Mathematics Letters, 2025, 163: 109472. doi: 10.1016/j.aml.2025.109472 [26] Zhai S, Weng Z, Feng X, et al. Stability and error estimate of the operator splitting method for the phase field crystal equation[J]. Journal of Scientific Computing, 2021, 86: 8. doi: 10.1007/s10915-020-01386-8 [27] 崔晨, 吴哲, 翟术英. 非局部守恒Allen-Cahn方程的高效算子分裂格式[J]. 华侨大学学报(自然科学版), 2022, 43 (5): 698-704.Cui Chen, Wu Zhe, Zhai Shuying. Effective operator splitting scheme for conservative nonlocal Allen-Cahn equation[J]. Journal of Huaqiao University (Natural Science), 2022, 43 (5): 698-704. (in Chinese) [28] Qiao Y Y, Zhai S Y, Feng X L. An operator splitting method for image inpainting based on the Allen-Cahn equation[J]. Chinese Journal of Engineering Mathematics, 2018, 35 (6): 722-732. [29] Weng Z, Zhai S, Dai W, et al. Stability and error estimates of Strang splitting method for the nonlocal ternary conservative Allen-Cahn model[J]. Journal of Computational and Applied Mathematics, 2024, 441: 115668. doi: 10.1016/j.cam.2023.115668 [30] Zhai S, Weng Z, Mo Y, et al. Energy dissipation and maximum bound principle preserving scheme for solving a nonlocal ternary Allen-Cahn model[J]. Computers & Mathematics With Applications, 2024, 155: 150-164. [31] 吴龙渊, 汪精英, 翟术英. 求解二维Allen-Cahn方程的两种ADI格式[J]. 华侨大学学报(自然科学版), 2019, 40 (3): 412-420.Wu Longyuan, Wang Jingying, Zhai Shuying. Two ADI schemes for solving two-dimensional Alleb-Cahn equations[J]. Journal of Huaqiao University (Natural Science), 2019, 40 (3): 412-420. (in Chinese) [32] D'Yakonov Y G. Difference schemes with a separable operator for general second order parabolic equations with variable coefficients[J]. USSR Computational Mathematics and Mathematical Physics, 1964, 4 (2): 92-110. doi: 10.1016/0041-5553(64)90108-9 [33] Du Q, Ju L, Li X, et al. Stabilized linear semi-implicit schemes for the nonlocal Cahn-Hilliard equation[J]. Journal of Computational Physics, 2018, 363: 39-54. doi: 10.1016/j.jcp.2018.02.023 -
图(5) / 表(2)
计量
- 文章访问数: 30
- HTML全文浏览量: 10
- PDF下载量: 10
- 被引次数: 0
渝公网安备50010802005915号