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求解一类时间分数阶偏微分方程的高阶格式
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Abstract:
本文研究了一类时间分数阶偏微分方程的数值解。首先，本文利用高阶加权本质无振荡(WENO)格式对空间变量进行离散化，使其在空间方向上达到高阶精度，因此得到一个只跟时间有关的常微分方程。接着在时间方向上应用指数和近似(SOE)时间分数阶Caputo导数，以减少内存和复杂度，达到快速计算的目的。其次，从理论上分析了WENO方法的高阶收敛性。最后通过数值实验验证了该方法的高阶精度。同时，应用该方法去数值求解含间断解的方程可以在非光滑区域保持本质无振荡，验证了该方法的有效性。
This paper investigates the numerical solution of a class of partial differential equations of time- fractional order. Firstly, the paper discretizes the spatial variables using the higher order weighted essential no oscillation (WENO) scheme to achieve high accuracy in the spatial direction, thus ob-taining an ordinary differential equation related only to time. Then, the exponential sum approxi-mation (SOE) to the time-fractional order Caputo derivative is applied in the time direction to re-duce memory and complexity for fast computation. Next, the higher-order convergence of the WENO method is analyzed theoretically. Finally, the higher-order accuracy of the method is verified by numerical experiments. Meanwhile, applying the method to solve the equations containing inter-rupted solutions numerically is found to maintain the essence of non-oscillation in the non- smooth region, which verifies the effectiveness of the method.
| [1] | Metzler, R. and Klafter, J. (2000) The Random Walk’s Guide to Anomalous Diffusion: A Fractional Dynamics Approach. Physics Reports, 339, 1-77. https://doi.org/10.1016/S0370-1573(00)00070-3 |
| [2] | 孙志忠, 高广花. 分数阶偏微分方程的有限差分方法[M]. 北京: 科学出版社, 2015. |
| [3] | Das, S. (2011) Functional Fractional Calculus. Springer, Berlin. https://doi.org/10.1007/978-3-642-20545-3 |
| [4] | Harten, A. and Osher, S. (1987) Uniformly High-Order Accurate Nonoscillatory Schemes. I. SIAM Journal on Numerical Analysis, 24, 279-309. https://doi.org/10.1137/0724022 |
| [5] | Liu, X.-D., Osher, S. and Chan, T. (1994) Weighted Essentially Non-Oscillatory Schemes. Journal of Computational Physics, 115, 200-212. https://doi.org/10.1006/jcph.1994.1187 |
| [6] | Jiang, G.-S. and Shu, C.-W. (1996) Efficient Implementation of Weighted ENO Schemes. Journal of Computational Physics, 126, 202-228. https://doi.org/10.1006/jcph.1996.0130 |
| [7] | Liu, Y.Y., Shu, C.W. and Zhang, M.P. (2011) High Order Finite Dif-ference WENO Schemes for Nonlinear Degenerate Parabolic Equations. SIAM Journal on Scientific Computing, 33, 939-965. https://doi.org/10.1137/100791002 |
| [8] | Liu, F., Anh, V. and Turner, I. (2004) Numerical Solution of the Space Fractional Fokker-Planck Equation. Journal of Computational and Applied Mathematics, 166, 209-219. https://doi.org/10.1016/j.cam.2003.09.028 |
| [9] | Meerschaert, M.M. and Tadjeran, C. (2004) Finite Difference Ap-proximations for Fractional Advection-Dispersion Flow Equations. Journal of Computational and Applied Mathematics, 172, 65-77.
https://doi.org/10.1016/j.cam.2004.01.033 |
| [10] | Deng, W. (2009) Finite Element Method for the Space and Time Fractional Fokker-Planck Equation. SIAM Journal on Numerical Analysis, 47, 204-226. https://doi.org/10.1137/080714130 |
| [11] | Li, X. and Xu, C. (2010) Existence and Uniqueness of the Weak Solution of the Space-Time Fractional Diffusion Equation and a Spectral Method Approximation. Communications in Computa-tional Physics, 8, 1016-1051.
https://doi.org/10.4208/cicp.020709.221209a |
| [12] | Deng, W.H., Du, S.D. and Wu, Y.J. (2013) High Order Finite Difference WENO Schemes for Fractional Differential Equations. Applied Mathematics Letters, 26, 362-366. https://doi.org/10.1016/j.aml.2012.10.005 |
| [13] | Zhang, Y., Deng, W.H. and Zhu, J. (2021) A New Sixth-Order Fi-nite Difference WENO Scheme for Fractional Differential Equations. Journal of Scientific Computing, 87, Article No. 73. https://doi.org/10.1007/s10915-021-01486-z |
| [14] | Fan, C., Zhang, X.X. and Qiu, J.X. (2022) Positivity-Preserving High Order Finite Difference WENO Schemes for Compressible Navier-Stokes Equations. Journal of Computational Physics, 467, Article ID: 111446.
https://doi.org/10.1016/j.jcp.2022.111446 |
| [15] | Du, J., Shu, C.-W. and Zhong, X.H. (2022) An Improved Simple WENO Limiter for Discontinuous Galerkin Methods Solving Hyperbolic Systems on Unstructured Meshes. Journal of Computational Physics, 467, Article ID: 111424.
https://doi.org/10.1016/j.jcp.2022.111424 |
| [16] | Borges, R., Carmona, M., Costa, B. and Don, W.S. (2008) An Im-proved Weighted Essentially Non-Oscillatory Scheme for Hyperbolic Conservation Laws. Journal of Computational Physics, 227, 3191-3211.
https://doi.org/10.1016/j.jcp.2007.11.038 |
