• 论文 • 上一篇    下一篇

分布阶扩散—波动方程的有限元解的误差估计

高兴华1, 李宏2, 刘洋2   

  1. 1. 内蒙古师范大学数学科学学院, 呼和浩特 010021;
    2. 内蒙古大学数学科学学院, 呼和浩特 010021
  • 收稿日期:2020-06-16 出版日期:2021-11-14 发布日期:2021-11-12
  • 基金资助:
    内蒙古自然科学基金(2021MS01018,2020MS01003),内蒙古草原英才和内蒙古自治区高等学校青年科技英才支持计划(NJYT-17-A07)资助.

高兴华, 李宏, 刘洋. 分布阶扩散—波动方程的有限元解的误差估计[J]. 计算数学, 2021, 43(4): 493-505.

Gao Xinghua, Li Hong, Liu Yang. ERROR ESTIMATION OF FINITE ELEMENT SOLUTION FOR A DISTRIBUTED-ORDER DIFFUSION-WAVE EQUATION[J]. Mathematica Numerica Sinica, 2021, 43(4): 493-505.

ERROR ESTIMATION OF FINITE ELEMENT SOLUTION FOR A DISTRIBUTED-ORDER DIFFUSION-WAVE EQUATION

Gao Xinghua1, Li Hong2, Liu Yang2   

  1. 1. School of Mathematical Sciences, Inner Mongolia Normal University, Hohhot 010022, China;
    2. School of Mathematical Sciences, Inner Mongolia University, Hohhot 010021, China
  • Received:2020-06-16 Online:2021-11-14 Published:2021-11-12
本文考虑了分布阶时间分数阶扩散波动方程,其中时间分数阶导数是在Caputo意义上定义的,其阶次$\alpha,\beta$分别属于(0,1)和(1,2).文中提出了在计算上行之有效的数值方法来模拟分布阶时间分数阶扩散波动方程.在时间上,通过中点求积公式把分布阶项转换为多项的时间分数阶导数项,并且利用$L1$和$L2$公式来近似Caputo分数阶导数;空间上使用Galerkin有限元方法进行离散.给出了基于$H^1$范数的有限元解的稳定性和误差估计的详细证明,最后的数值算例结果说明了理论分析的正确性以及有效性.
In this paper, the distributed-order diffusion-wave equation is considered. The time fractional derivatives are defined in the Caputo sense, and their orders $\alpha$, $\beta$ belong to the intervals (0,1) and (1,2), respectively. Some computationally effective numerical methods are proposed for simulating the distributed-order time-fractional diffusion-wave equation. In the time direction, the mid-point quadrature rule is used to transform the distributed-order term into the multi-term time fractional terms, then $L1$ formula and $L2$ formula are chosen to approximate the Caputo fractional derivatives. Further, the spatial direction is discretized by the Galerkin finite element method. The stability and error estimation of the fully discrete scheme based on the $H^1$ norm are proved. Finally, a numerical example is given to illustrate the correctness and effectiveness of the theoretical analysis.

MR(2010)主题分类: 

()
[1] Liu Y, Du Y, Li H, He S, Gao W. Finite difference/finite element method for a nonlinear timefractional fourth-order reaction-diffusion problem[J]. Comput. Math. Appl., 2015, 70:573-591.
[2] Shi D, Yang H. Superconvergence analysis of a new low order nonconforming MFEM for timefractional diffusion equation[J]. Appl. Numer. Math., 2018, 131:109-122.
[3] Deng W, Li C, Guo Q. Analysis of fractional differential equations with multi-orders[J]. Fractals, 2007, 15(2):173-182.
[4] Lin Y, Xu C. Finite difference/spectral approximations for the time-fractional diffusion equation[J]. J. Comput. Phys., 2007, 225:1533-1552.
[5] Yue X, Shu S, Xu X, Bu W, Pan K. Parallel-in-time multigrid for space-time finite element approximations of two-dimensional space-fractional diffusion equations[J]. Comput. Math. Appl., 2019, 78(11):3471-3484.
[6] Zheng M, Liu F, Liu Q, Burrage K, Simpson M J. Numerical solution of the time fractional reaction-diffusion equation with a moving boundary[J]. J. Comput. Phys., 2017, 338:493-510.
[7] Liu Y, Yu Z, Li H, Liu F, Wang J. Time two-mesh algorithm combined with finite element method for time fractional water wave model[J]. Int. J. Heat Mass Transfer, 2018, 120:1132-1145.
[8] Liu Y, Du Y, Li H, Wang J. A two-grid finite element approximation for a nonlinear time-fractional Cable equation[J]. Nonlinear Dyn., 2016, 85:2535-2548.
[9] Yin B, Liu Y, Li H, He S, Fast algorithm based on TT-M FE system for space fractional AllenCahn equations with smooth and non-smooth solution[J]. J. Comput. Phys., 2019, 379:351-372.
[10] Bu W, Tang Y, Wu Y, Yang J. Finite difference/finite element method for two dimensional space and time fractional Bloch-Torrey equations[J]. J. Comput. Phys., 2015, 293:264-279.
[11] Yang Z, Liu F, Nie Y, Turner I. An unstructured mesh finite difference/finite element method for the three-dimensional time-space fractional Bloch-Torrey equations on irregular domains[J]. J. Comput. Phys., 2020, 408:109284.
[12] Ding H, Li C. Numerical algorithms for the fractional diffusion-wave equation with reaction term[J]. Abstr. Appl. Anal., 2013, 2013:1-15.
[13] Luo Z, Wang H. A highly efficient reduced-order extrapolated finite difference algorithm for timespace tempered fractional diffusion-wave equation[J]. Appl. Math. Lett., 2020, 102:106090.
[14] Sun Z, Wu X. A fully discrete difference scheme for a diffusion-wave system[J]. Appl. Numer. Math., 2006, 56:193-209.
[15] Li L, Xu D, Luo M. Alternating direction implicit Galerkin finite element method for the twodimensional fractional diffusion-wave equation[J]. J. Comput. Phys., 2013, 255:471-485.
[16] Jin B T, Lazarov R, Zhou Z. Two fully discrete schemes for fractional diffusion and diffusion-wave equations with nonsmooth data[J]. SIAM J. Sci. Comput., 2016, 38(1):A146-A170.
[17] Liu Y, Fang Z, Li H, He S. A mixed finite element method for a time-fractional fourth-order partial differential equation[J]. Appl. Math. Comput., 2014, 243:703-717.
[18] Zeng F, Li C. A new Crank-Nicolson finite element method for the time-fractional subdiffusion equation[J]. Appl. Numer. Math., 2017, 121:82-95.
[19] Stojanovic M. Numerical method for solving diffusion-wave phenomena[J]. J. Comp. Appl. Math., 2011, 235:3121-3137.
[20] Liu F, Meerschaert M M, Mcgough R J, Zhuang P, Liu Q. Numerical methods for solving the multi-term time-fractional wave-diffusion equations[J]. Fract. Calc. Appl. Anal., 2013, 16(1):9-25.
[21] Caputo M. Distributed order differential equations modeling dielectric induction and diffusion[J]. Fract. Calc. Appl. Anal., 2001, 4:421-442.
[22] Mainardi F, Pagnini G, Gorenflo R. Some aspects of fractional diffusion equations of single and distributed order[J]. Appl. Math. Comput., 2007, 187(1):295-305.
[23] Chechkin A V, Gorenflo R, Sokolov I M. Retarding subdiffusion and accelerating superdiffusion governed by distributed-order fractional diffusion equations[J]. Phys. Rev. E., 2002, 66(4):046129.
[24] Gao X, Liu F, Li H, Liu Y, Turner I, Yin B. A novel finite element method for the distributed-order time fractional Cable equation in two dimensions[J]. Comput. Math. Appl., 2020, 80(5):923-939.
[25] Yin B, Liu Y, Li H, Zhang Z. Approximation methods for the distributed order calculus using the convolution quadrature[J]. Discrete Contin. Dyn. Syst.-Ser. B., 2020, doi:10.3934/dcdsb.2020168.
[26] Ford N J, Morgado M L. Distributed order equations as boundary value problems[J]. Comput. Math. Appl., 2012, 64(10):2973-2981.
[27] Sun Z, Wu X. A fully discrete difference scheme for a diffusion-wave system[J]. Appl. Numer. Math., 2006, 56(3):193-209.
[28] Liu F, Zhuang P, Liu Q. The applications and numerical methods of fractional differential equations[M]. Beijing:Science Press, 2015, 50-51.
[29] Feng L, Liu F, Turner I. Novel numerical analysis of multi-term time fractional viscoelastic nonNewtonian fluid models for simulating unsteady MHD Couette flow of a generalized Oldroyd-B fluid[J]. Fract. Calc. Appl. Anal., 2018, 21:1073-1103.
[1] 张铁,. Cahn-Hilliard方程的有限元分析[J]. 计算数学, 2006, 28(3): 281-292.
[2] 蔚喜军. 一维双曲守恒方程组的Taylor-Galerkin有限元方法[J]. 计算数学, 2001, 23(2): 199-208.
阅读次数
全文


摘要