变分数阶Allen-Cahn方程的自适应指数时间差分方法
Adaptive Exponential Time Discretization Method for the Variable-Order Fractional Allen-Cahn Equation
摘要: 本文提出变分数阶Allen-Cahn方程的自适应指数时间差分方法，在空间方向上采用有限差分法和分步傅里叶谱方法进行离散，在时间方向上采用自适应控制与指数时间差分结合的方法，构建数值格式并通过数值实验验证在计算精度、效率和稳定性方面的优势。
Abstract: This paper proposes an adaptive exponential time differencing method for the variable-order fractional Allen-Cahn equation. Finite difference and split-step Fourier spectral methods are used for spatial discretization, while an adaptive control combined with the exponential time differencing is employed for time discretization. The numerical scheme is constructed and its advantages in computational accuracy, efficiency, and stability are verified through numerical experiments.
参考文献
|
[1]
|
Samko, S.G. and Ross, B. (1993) Integration and Differentiation to a Variable Fractional Order. Integral Transforms and Special Functions, 1, 277-300. [Google Scholar] [CrossRef]
|
|
[2]
|
Sun, H.G., Chen, W., Wei, H. and Chen, Y.Q. (2011) A Comparative Study of Constant-Order and Variable-Order Fractional Models in Characterizing Memory Property of Systems. The European Physical Journal Special Topics, 193, 185-192. [Google Scholar] [CrossRef]
|
|
[3]
|
Wang, G.T., Pei, K., Agarwal, R.P., Zhang, L. and Ahmad, B. (2018) Nonlocal Hadamard Fractional Boundary Value Problem with Hadamard Integral and Discrete Boundary Conditions on a Half-Line. Journal of Computational and Applied Mathematics, 343, 230-239. [Google Scholar] [CrossRef]
|
|
[4]
|
尹修草, 周均, 胡兵. 分数阶对流-弥散方程的有限差分方法[J]. 四川大学学报(自然科学版), 2013, 50(3): 409-413.
|
|
[5]
|
陶群群, 吴亚运. 变系数的分数阶非齐次线性微分方程[J]. 应用数学与计算数学学报, 2015, 29(2): 154-161.
|
|
[6]
|
宋光珍, 赵维加, 黄健飞. 时间分数阶扩散方程的一种数值解法[J]. 青岛大学学报(自然科学版), 2015, 28(3): 9-14.
|
|
[7]
|
曾宝思, 尹修草, 谢常平, 等. 带Robin知更鸟边界条件的分数阶对流-扩散方程的数值解法[J]. 四川大学学报(自然科学版), 2018, 55(1): 13-17.
|
|
[8]
|
李俊婵. 分数阶对流扩散方程的有限点方法研究[D]: [硕士学位论文]. 西安: 西安理工大学, 2019.
|
|
[9]
|
Cox, S.M. and Matthews, P.C. (2002) Exponential Time Differencing for Stiff Systems. Journal of Computational Physics, 176, 430-455. [Google Scholar] [CrossRef]
|