A transformed $ L1 $ Legendre-Galerkin spectral method for time fractional Fokker-Planck equations

Diandian Huang; Xin Huang; Tingting Qin; Yongtao Zhou; Diandian Huang; Xin Huang; Tingting Qin; Yongtao Zhou

doi:10.3934/nhm.2023034

Networks and Heterogeneous Media

2023, Volume 18, Issue 2: 799-812. doi: 10.3934/nhm.2023034

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A transformed $ L1 $ Legendre-Galerkin spectral method for time fractional Fokker-Planck equations

1.
School of Mathematics and Statistics, Huazhong University of Science and Technology, Wuhan 430074, China
2.
Hubei Key Laboratory of Engineering Modeling and Scientific Computing, Huazhong University of Science and Technology, Wuhan 430074, China
3.
Applied and Computational Mathematics Division, Beijing Computational Science Research Center, Beijing 100193, China
4.
School of Science, Qingdao University of Technology, Qingdao 266033, China

Received: 19 December 2022 Revised: 12 February 2023 Accepted: 23 February 2023 Published: 06 March 2023

The numerical solutions of time $ \alpha $-order $ (\alpha \in (0, 1)) $ Caputo fractional Fokker-Planck equations is considered. The constructed method is consist of the transformed $ L1 $ ($ TL1 $) scheme in the temporal direction and the Legendre-Galerkin spectral method in the spatial direction. It has been shown that the $ TL1 $ Legendre-Galerkin spectral method in $ L^2 $-norm is exponential order convergent in space and ($ 2-\alpha $)-th order convergent in time. Several numerical examples are given to verify the obtained theoretical results.
- time fractional Fokker-Planck equations,
- error estimate,
- Legendre-Galerkin spectral method,
- transformed $ L1 $ scheme,
- change of variable
Citation: Diandian Huang, Xin Huang, Tingting Qin, Yongtao Zhou. A transformed $ L1 $ Legendre-Galerkin spectral method for time fractional Fokker-Planck equations[J]. Networks and Heterogeneous Media, 2023, 18(2): 799-812. doi: 10.3934/nhm.2023034

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Abstract

The numerical solutions of time $ \alpha $-order $ (\alpha \in (0, 1)) $ Caputo fractional Fokker-Planck equations is considered. The constructed method is consist of the transformed $ L1 $ ($ TL1 $) scheme in the temporal direction and the Legendre-Galerkin spectral method in the spatial direction. It has been shown that the $ TL1 $ Legendre-Galerkin spectral method in $ L^2 $-norm is exponential order convergent in space and ($ 2-\alpha $)-th order convergent in time. Several numerical examples are given to verify the obtained theoretical results.

References

[1]	R. Metzler, T. Nonnenmacher, Space-and time-fractional diffusion and wave equations, fractional Fokker-Planck equations, and physical motivation, Chem. Phys., 284 (2002), 67–90. https://doi.org/10.1016/S0301-0104(02)00537-2 doi: 10.1016/S0301-0104(02)00537-2
[2]	E. Barkai, R. Metzler, J. Klafter, From continuous time random walks to the fractional Fokker-Planck equation, Phys. Rev. E., 61 (2000), 132–138. https://doi.org/10.1103/PhysRevE.61.132 doi: 10.1103/PhysRevE.61.132
[3]	E. Barkai, Fractional Fokker-Planck equation, solution, and application, Phys. Rev. E., 63 (2001), 046118. https://doi.org/10.1103/PhysRevE.63.046118 doi: 10.1103/PhysRevE.63.046118
[4]	G. Liu, M. Liu, Smoothed Particle Hydrodynamics: a Meshfree Particle Method, World Scientific, 2003.
[5]	H. Fu, G. Wu, G. Yang, L. Huang, Continuous time random walk to a general fractional Fokker-Planck equation on fractal media, Eur. Phys. J. Spec. Top., 230 (2021), 3927–3933. https://doi.org/10.1140/epjs/s11734-021-00323-6 doi: 10.1140/epjs/s11734-021-00323-6
[6]	I. Goychuk, E. Heinsalu, M. Patriarca, G. Schmid, P. Hänggi, Current and universal scaling in anomalous transport, Phys. Rev. E, 73 (2006), 020101. https://doi.org/10.1103/PhysRevE.73.020101 doi: 10.1103/PhysRevE.73.020101
[7]	M. Stynes, Too much regularity may force too much uniqueness, Fract. Calc. Appl. Anal., 19 (2016), 1554–1562. https://doi.org/10.1515/fca-2016-0080 doi: 10.1515/fca-2016-0080
[8]	W. Deng, Numerical algorithm for the time fractional Fokker-Planck equation, J. Comput. Phys., 227 (2007), 1510–1522. https://doi.org/10.1016/j.jcp.2007.09.015 doi: 10.1016/j.jcp.2007.09.015
[9]	S. Vong, Z. Wang, A high order compact finite difference scheme for time fractional Fokker-Planck equations, Appl. Math. Lett., 43 (2015), 38–43. https://doi.org/10.1016/j.aml.2014.11.007 doi: 10.1016/j.aml.2014.11.007
[10]	A. Mahdy, Numerical solutions for solving model time-fractional Fokker-Planck equation, Numer. Methods Partial Differ. Equ., 37 (2021), 1120–1135. https://doi.org/10.1002/num.22570 doi: 10.1002/num.22570
[11]	X. Yang, H. Zhang, Q. Zhang, G. Yuan, Z. Sheng, The finite volume scheme preserving maximum principle for two-dimensional time-fractional Fokker-Planck equations on distorted meshes, Appl. Math. Lett., 97 (2019), 99–106. https://doi.org/10.1016/j.aml.2019.05.030 doi: 10.1016/j.aml.2019.05.030
[12]	F. Zeng, C. Li, F. Liu, I. Turner, The use of finite difference/element approaches for solving the time-fractional subdiffusion equation, SIAM J. Sci. Comput., 35 (2013), A2976–A3000. https://doi.org/10.1137/130910865 doi: 10.1137/130910865
[13]	A. Mohebbi, M. Abbaszadeh, M. Dehghan, A high-order and unconditionally stable scheme for the modified anomalous fractional sub-diffusion equation with a nonlinear source term, J. Comput. Phys., 240 (2013), 36–48. https://doi.org/10.1016/j.jcp.2012.11.052 doi: 10.1016/j.jcp.2012.11.052
[14]	Y. Jiang, A new analysis of stability and convergence for finite difference schemes solving the time fractional Fokker-Planck equation, Appl. Math. Model., 39 (2015), 1163–1171. https://doi.org/10.1016/j.apm.2014.07.029 doi: 10.1016/j.apm.2014.07.029
[15]	C. Zhang, Y. Zhou, A preconditioned implicit difference scheme for semilinear two-dimensional time-space fractional Fokker-Planck equations, Numer. Linear Algebra Appl., 28 (2021), e2357. https://doi.org/10.1002/nla.2357 doi: 10.1002/nla.2357
[16]	W. Deng, Finite element method for the space and time fractional Fokker-Planck equation, SIAM J. Numer. Anal., 47 (2008), 204–226. https://doi.org/10.1137/080714130 doi: 10.1137/080714130
[17]	S. Chen, F. Liu, P. Zhuang, V. Anh, Finite difference approximations for the fractional Fokker-Planck equation, Appl. Math. Model., 33 (2009), 256–273. https://doi.org/10.1016/j.apm.2007.11.005 doi: 10.1016/j.apm.2007.11.005
[18]	B. Jin, B. Li, Z. Zhou, Numerical analysis of nonlinear subdiffusion equations, SIAM J. Numer. Anal., 56 (2018), 1–23. https://doi.org/10.1137/16M1089320 doi: 10.1137/16M1089320
[19]	C. Lubich, Convolution quadrature and discretized operational calculus. I, Numer. Math., 52 (1988), 129–145. https://doi.org/10.1007/BF01398686 doi: 10.1007/BF01398686
[20]	N. Kopteva, Error analysis of the $L1$ method on graded and uniform meshes for a fractional-derivative problem in two and three dimensions, Math. Comput., 88 (2019), 2135–2155. https://doi.org/10.1090/mcom/3410 doi: 10.1090/mcom/3410
[21]	D. Li, C. Zhang, Long time numerical behaviors of fractional pantograph equations, Math. Comput. Simul., 172 (2020), 244–257. https://doi.org/10.1016/j.matcom.2019.12.004 doi: 10.1016/j.matcom.2019.12.004
[22]	A. Alikhanov, A new difference scheme for the time fractional diffusion equation, J. Comput. Phys., 280 (2015), 424–438. https://doi.org/10.1016/j.jcp.2014.09.031 doi: 10.1016/j.jcp.2014.09.031
[23]	Z. Sun, X. Wu, A fully discrete difference scheme for a diffusion-wave system, Appl. Numer. Math., 56 (2006), 193–209. https://doi.org/10.1016/j.apnum.2005.03.003 doi: 10.1016/j.apnum.2005.03.003
[24]	H. Liao, W. Mclean, J. Zhang, A discrete Grönwall inequality with application to numerical schemes for subdiffusion problems, SIAM J. Numer. Anal., 57 (2019), 218–237. https://doi.org/10.1137/16M1175742 doi: 10.1137/16M1175742
[25]	B. Zhou, X. Chen, D. Li, Nonuniform Alikhanov linearized Galerkin finite element methods for nonlinear time-fractional parabolic equations, J. Sci. Comput., 85 (2020), 39. https://doi.org/10.1007/s10915-020-01350-6 doi: 10.1007/s10915-020-01350-6
[26]	M. Stynes, E. O'Riordan, J. Gracia, Error analysis of a finite difference method on graded meshes for a time-fractional diffusion equation, SIAM J. Numer. Anal., 55 (2017), 1057–1079. https://doi.org/10.1137/16M1082329 doi: 10.1137/16M1082329
[27]	J. Shen, Z. Sun, R. Du, Fast finite difference schemes for time-fractional diffusion equations with a weak singularity at initial time, East Asian J. Appl. Math., 8 (2018), 834–858. https://doi.org/10.4208/eajam.010418.020718 doi: 10.4208/eajam.010418.020718
[28]	J. Shen, Z. Sun, W. Cao, A finite difference scheme on graded meshes for time-fractional nonlinear Korteweg-de Vries equation, Appl. Math. Comput., 361 (2019), 752–765. https://doi.org/10.1016/j.amc.2019.06.023 doi: 10.1016/j.amc.2019.06.023
[29]	D. Li, C. Wu, Z. Zhang, Linearized Galerkin FEMs for nonlinear time fractional parabolic problems with non-smooth solutions in time direction, J. Sci. Comput., 80 (2019), 403–419. https://doi.org/10.1007/s10915-019-00943-0 doi: 10.1007/s10915-019-00943-0
[30]	Y. Zhou, M. Stynes, Optimal convergence rates in time-fractional discretisations: the ${\rm L}1$, $\overline{{\rm L}1}$ and Alikhanov schemes, East Asian J. Appl. Math., 12 (2022), 503–520. https://doi.org/10.4208/eajam.290621.220921 doi: 10.4208/eajam.290621.220921
[31]	J. Shen, F. Zeng, M. Stynes, Second-order error analysis of the averaged L1 scheme L1 for time-fractional initial-value and subdiffusion problems, Sci. China Math., 2023. https://doi.org/10.1007/s11425-022-2078-4
[32]	B. Ji, H. Liao, Y. Gong, L. Zhang, Adaptive second-order Crank-Nicolson time-stepping schemes for time-fractional molecular beam epitaxial growth models, SIAM J. Sci. Comput., 42 (2020) B738–B760. https://doi.org/10.1137/19M1259675
[33]	S. Jiang, J. Zhang, Q. Zhang, Z. Zhang, Fast evaluation of the Caputo fractional derivative and its applications to fractional diffusion equations, Commu. Comput. Phys., 21 (2017), 650–678. https://doi.org/10.4208/cicp.OA-2016-0136 doi: 10.4208/cicp.OA-2016-0136
[34]	Y. Yan, Z. Sun, J. Zhang, Fast evaluation of the Caputo fractional derivative and its applications to fractional diffusion equations: a second-order scheme, Commun. Comput. Phys., 22 (2017) 1028–1048. https://doi.org/10.4208/cicp.OA-2017-0019
[35]	Y. Zhao, X. Gu, A. Ostermann, A preconditioning technique for an all-at-once system from Volterra subdiffusion equations with graded time steps, J. Sci. Comput., 88 (2021), 11. https://doi.org/10.1007/s10915-021-01527-7 doi: 10.1007/s10915-021-01527-7
[36]	X. Gu, S. Wu, A parallel-in-time iterative algorithm for Volterra partial integro-differential problems with weakly singular kernel, J. Comput. Phys., 417 (2020), 109576. https://doi.org/10.1016/j.jcp.2020.109576 doi: 10.1016/j.jcp.2020.109576
[37]	D. Li, W. Sun, C. Wu, A novel numerical approach to time-fractional parabolic equations with nonsmooth solutions, Numer. Math. Theory Methods Appl., 14 (2021), 355–376. https://doi.org/10.4208/nmtma.oa-2020-0129 doi: 10.4208/nmtma.oa-2020-0129
[38]	H. Qin, D. Li, Z. Zhang, A novel scheme to capture the initial dramatic evolutions of nonlinear subdiffusion equations, J. Sci. Comput., 89 (2021), 1–20. https://doi.org/10.1007/s10915-021-01672-z doi: 10.1007/s10915-021-01672-z
[39]	K. Mustapha, D. Schotzau, Well-posedness of hp-version discontinuous Galerkin methods for fractional diffusion wave equations, IMA. J. Numer. Anal., 34 (2014), 1426–1446. https://doi.org/10.1093/imanum/drt048 doi: 10.1093/imanum/drt048
[40]	K. Mustapha, W. Mclean, Superconvergence of a discontinuous Galerkin method for fractional diffusion and wave equations, SIAM. J. Numer. Anal., 51 (2013), 491–515. https://doi.org/10.1137/120880719 doi: 10.1137/120880719
[41]	D. Li, C. Zhang, Superconvergence of a discontinuous Galerkin method for first-order linear delay differential equations, J. Comput. Math., 29 (2011), 574–588. https://doi.org/10.4208/jcm.1107-m3433 doi: 10.4208/jcm.1107-m3433
[42]	M. Cui, A high-order compact exponential scheme for the fractional convection-diffusion equation, J. Comput. Appl. Math., 255 (2014), 404–416. https://doi.org/10.1016/j.cam.2013.06.001 doi: 10.1016/j.cam.2013.06.001
[43]	A. Mohebbi, M. Abbaszadeh, Compact finite difference scheme for the solution of time fractional advection-dispersion equation, Numer. Algorithms, 63 (2013), 431–452. https://doi.org/10.1007/s11075-012-9631-5 doi: 10.1007/s11075-012-9631-5
[44]	M. She, L. Li, R. Tang, D. Li, A novel numerical scheme for a time fractional Black-Scholes equation, J. Appl. Math. Comput., 66 (2021) 853–870. https://doi.org/10.1007/s12190-020-01467-9
[45]	J. Shen, T. Tang, L. Wang, Spectral Methods: Algorithms, Analysis and Applications, Berlin: Springer, 2011.
[46]	W. Cao, F. Zeng, Z. Zhang, G. E. Karniadakis, Implicit-explicit difference schemes for nonlinear fractional differential equations with nonsmooth solutions, SIAM J. Sci. Comput., 38 (2016), A3070–A3093. https://doi.org/10.1137/16M1070323 doi: 10.1137/16M1070323
[47]	K. Mustapha, FEM for time-fractional diffusion equations, novel optimal error analyses, Math. Comput., 87 (2018), 2259–2272. https://doi.org/10.1090/mcom/3304 doi: 10.1090/mcom/3304
[48]	C. Canuto, M. Hussaini, A. Quarteront, Spectral Methods in Fluid Dynamics, New York: Springer-Verlag, 1987.
[49]	W. Yuan, D. Li, C. Zhang, Linearized transformed $L1$ Galerkin FEMs with unconditional convergence for nonlinear time fractional Schödinger equations, Numer. Math. Theory Methods Appl., 2023. https://doi:10.4208/nmtma.OA-2022-0087 doi: 10.4208/nmtma.OA-2022-0087

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