Two High-Order Difference Schemes for Solving Time Distributed-Order Diffusion Equations
-
摘要: 基于复化Simpson公式和复化两点Gauss-Legendre公式,构造了两个求解时间分布阶扩散方程的高阶有限差分格式.不同于以往文献中提出的时间一阶或二阶格式,这两种格式在时间方向都具有三阶精度,而在分布阶和空间方向可达到四阶精度.数值结果表明,两种算法都是稳定且收敛的,从而是有效的.两种格式的收敛速率也通过数值实验进行了验证,并且通过和文献中的算法对比可以得出其更为高效,Abstract: Based on the composite Simpson’s quadrature rule and the composite 2-point Gauss-Legendre quadrature rule, 2 high-order finite difference schemes were proposed for solving time distributed-order diffusion equations. Other than the existing methods whose convergence rates are only 1st-order or 2nd-order in the temporal domain, the proposed 2 schemes both have 3rd-order convergence rates in the temporal domain, and 4th-order rates in the spatial domain and the distributed order, respectively. Such high-order convergence rates were further verified with numerical examples. The results show that, both of the proposed 2 schemes are stable, and have higher accuracy and efficiency compared with existing algorithms.
-
[1] PODLUBNY I. Fractional Differential Equations: an Introduction to Fractional Derivatives, Fractional Differential Equations, to Methods of Their Solution and Some of Their Applications [M]. San Diego: Academic Press, 1999. [2] 张平奎, 杨绪君. 基于激励滑模控制的分数阶神经网络的修正投影同步研究[J]. 应用数学和力学, 2018,39(3): 343-354.(ZHANG Pingkui, YANG Xujun. Modified projective synchronization of a class of fractional-order neural networks based on active sliding mode control[J]. Applied Mathematics and Mechanics,2018,39(3): 343-354.(in Chinese)) [3] 杨旭, 梁英杰, 孙洪广, 等. 空间分数阶非Newton流体本构及圆管流动规律研究[J]. 应用数学和力学, 2018,39(11): 1213-1226.(YANG Xu, LIANG Yingjie, SUN Hongguang, et al. A study on the constitutive relation and the flow of spatial fractional non-Newtonian fluid in circular pipes[J]. Applied Mathematics and Mechanics,2018,39(11): 1213-1226.(in Chinese)) [4] CAPUTO M. Elasticità e Dissipazione [M]. Bologna: Zanichelli, 1969. [5] SINAI Y G. The limiting behavior of a one-dimensional random walk in a random medium[J]. Theory of Probability & Its Applications,1983,27(2): 256-268. [6] CHECHKIN A V, KLAFTER J, SOKOLOV I M. Fractional Fokker-Planck equation for ultraslow kinetics[J]. Europhysics Letters,2003,63(3): 326-332. [7] KOCHUBEI A N. Distributed order calculus and equations of ultraslow diffusion[J]. Journal of Mathematical Analysis and Applications,2008,340(1): 252-281. [8] JIAO Z, CHEN Y,PODLUBNY I. Distributed-Order Dynamic Systems: Stability, Simulation, Applications and Perspectives [M]. London: Springer, 2012. [9] CAPUTO M. Distributed order differential equations modelling dielectric induction and diffusion[J]. Fractional Calculus and Applied Analysis,2001,4(4): 421-442. [10] HARTLEY T T, LORENZO C F. Fractional system identification: an approach using continuous order-distributions: NASA/TM-1999-209640[R]. USA: NASA, 1999. [11] FORD N J, MORGADO M L, REBELO M. An implicit finite difference approximation for the solution of the diffusion equation with distributed order in time[J]. Electronic Transactions on Numerical Analysis,2015,44: 289-305. [12] GAO G H, SUN Z Z. Two unconditionally stable and convergent difference schemes with the extrapolation method for the one-dimensional distributed-order differential equations[J]. Numerical Methods for Partial Differential Equations,2016,32(2): 591-615. [13] GAO G H, SUN H W, SUN Z Z. Some high-order difference schemes for the distributed-order differential equations[J]. Journal of Computational Physics, 2015,298: 337-359. [14] HU J H, WANG J G, NIE Y F. Numerical algorithms for multidimensional time-fractional wave equation of distributed-order with a nonlinear source term[J]. Advances in Difference Equations,2018(1): 352. DOI: 10.1186/s13662-018-1817-2. [15] 郭晓斌, 尚德泉. 复化两点Gauss-Legendre公式及其误差分析[J]. 数学教学研究, 2010,29(4): 49-51.(GUO Xiaobin, SHANG Dequan. Composite two-point Gauss-Legendre formula and the error analysis[J]. Research of Mathematic Teaching-Learning,2010,29(4): 49-51.(in Chinese)) [16] TIAN W, ZHOU H, DENG W. A class of second order difference approximations for solving space fractional diffusion equations[J]. Mathematics of Computation,2015,84(294): 1703-1727. [17] SUN Z Z. The Method of Order Reduction and Its Application to the Numerical Solutions of Partial Differential Equations [M]. Beijing: Science Press, 2009. [18] ZHU Y, SUN Z Z. A high-order difference scheme for the space and time fractional Bloch-Torrey equation[J].Computational Methods in Applied Mathematics,2018,18(1): 147-164.
点击查看大图
计量
- 文章访问数: 929
- HTML全文浏览量: 122
- PDF下载量: 427
- 被引次数: 0