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February  2022, 27(2): 903-920. doi: 10.3934/dcdsb.2021073

A fast high order method for time fractional diffusion equation with non-smooth data

Haili Qiao and  Aijie Cheng ,

School of Mathematics, Shandong University, Jinan, Shandong 250100, China

* Corresponding author: Aijie Cheng

Received  May 2020 Revised  December 2020 Published  February 2022 Early access  March 2021

Fund Project: This work was supported in part by the National Natural Science Foundation of China under Grants 91630207, 11971272

In this paper, we consider the time fractional diffusion equation with Caputo fractional derivative. Due to the singularity of the solution at the initial moment, it is difficult to achieve an ideal convergence order on uniform meshes. Therefore, in order to improve the convergence order, we discrete the Caputo time fractional derivative by a new $ L1-2 $ format on graded meshes, while the spatial derivative term is approximated by the classical central difference scheme on uniform meshes. We analyze the approximation about the time fractional derivative, and obtain the time truncation error, but the stability analysis remains an open problem. On the other hand, considering that the computational cost is extremely large, we present a reduced-order finite difference extrapolation algorithm for the time-fraction diffusion equation by means of proper orthogonal decomposition (POD) technique, which effectively reduces the computational cost. Finally, several numerical examples are given to verify the convergence of the scheme and the effectiveness of the reduced order extrapolation algorithm.

Keywords: Time fractional differential equation, finite difference method, graded meshes, L1-2 formula, POD.
Mathematics Subject Classification: Primary: 26A33, 65M06; Secondary: 65M15.
Citation: Haili Qiao, Aijie Cheng. A fast high order method for time fractional diffusion equation with non-smooth data. Discrete & Continuous Dynamical Systems - B, 2022, 27 (2) : 903-920. doi: 10.3934/dcdsb.2021073
References:
[1]

A. A. Alikhanov, A new difference scheme for the time fractional diffusion equation, J. Comput. Phys., 280 (2015), 424-438.  doi: 10.1016/j.jcp.2014.09.031.  Google Scholar

[2]

H. Brunner, L. Ling and M. Yamamoto, Numerical simulations of 2D fractional subdiffusion problems, J. Comput. Phys., 229 (2010), 6613-6622.  doi: 10.1016/j.jcp.2010.05.015.  Google Scholar

[3]

H. Chen and M. Stynes, A high order method on graded meshes for a time-fractional diffusion problem, Finite Difference Methods, 15–27, Lecture Notes in Comput. Sci., 11386, Springer, Cham, 2019. doi: 10.1007/978-3-030-11539-5_2.  Google Scholar

[4]

H. Chen and M. Stynes, Error analysis of a second-order method on fitted meshes for a time-fractional diffusion problem, J. Sci. Comput., 79 (2019), 624-647.  doi: 10.1007/s10915-018-0863-y.  Google Scholar

[5]

N. J. Ford, M. L. Morgado and M. Rebelo, Nonpolynomial collocation approximation of solutions to fractional differential equations, Fract. Calc. Appl. Anal., 16 (2013), 874-891.  doi: 10.2478/s13540-013-0054-3.  Google Scholar

[6]

N. J. Ford and Y. Yan, An approach to construct higher order time discretisation schemes for time fractional partial differential equations with nonsmooth data, Fract. Calc. Appl. Anal., 20 (2017), 1076-1105.  doi: 10.1515/fca-2017-0058.  Google Scholar

[7]

G.-H. Gao and Z.-Z. Sun, A compact finite difference scheme for the fractional sub-diffusion equations, J. Comput. Phys., 230 (2011), 586-595.  doi: 10.1016/j.jcp.2010.10.007.  Google Scholar

[8]

G.-H. Gao, Z.-Z. Sun and H.-W. Zhang, A new fractional numerical differentiation formula to approximate the Caputo fractional derivative and its applications, J. Comput. Phys., 259 (2014), 33-50.  doi: 10.1016/j.jcp.2013.11.017.  Google Scholar

[9]

M. Giona, S. Cerbelli and H. E. Roman, Fractional diffusion equation and relaxation in complex viscoelastic materials, Physica A: Statistical Mechanics and its Applications, 191 (1992), 449-453.  doi: 10.1016/0378-4371(92)90566-9.  Google Scholar

[10]

R. Gorenflo, F. Mainardi, D. Moretti and P. Paradisi, Time fractional diffusion: A discrete random walk approach, Nonlinear Dynam., 29 (2002), 129-143.  doi: 10.1023/A:1016547232119.  Google Scholar

[11]

Y. Lin and C. Xu, Finite difference/spectral approximations for the time-fractional diffusion equation, J. Comput. Phys., 225 (2007), 1533-1552.  doi: 10.1016/j.jcp.2007.02.001.  Google Scholar

[12]

F. Liu, S. Shen, V. Anh and I. Turner, Analysis of a discrete non-Markovian random walk approximation for the time fractional diffusion equation, ANZIAM J., 46 (2004), 488-504.   Google Scholar

[13]

Z. Luo, J. Chen, J. Zhu, R. Wang and I. M. Navon, An optimizing reduced order FDS for the tropical Pacific Ocean reduced gravity model, Internat. J. Numer. Methods Fluids, 55 (2007), 143-161.  doi: 10.1002/fld.1452.  Google Scholar

[14]

Z. Luo, H. Li, P. Sun and J. Gao, A reduced-order finite difference extrapolation algorithm based on POD technique for the non-stationary navier–stokes equations, Appl. Math. Model., 37 (2013), 5464-5473.  doi: 10.1016/j.apm.2012.10.051.  Google Scholar

[15]

Z. Luo, F. Teng and H. Xia, A reduced-order extrapolated Crank–Nicolson finite spectral element method based on POD for the 2D non-stationary Boussinesq equations, J. Math. Anal. Appl., 471 (2019), 564-583.  doi: 10.1016/j.jmaa.2018.10.092.  Google Scholar

[16]

C. Lv and C. Xu, Error analysis of a high order method for time-fractional diffusion equations, SIAM J. Sci. Comput., 38 (2016), A2699–A2724. doi: 10.1137/15M102664X.  Google Scholar

[17]

Z. Mao and J. Shen, Efficient spectral–Galerkin methods for fractional partial differential equations with variable coefficients, J. Comput. Phys., 307 (2016), 243-261.  doi: 10.1016/j.jcp.2015.11.047.  Google Scholar

[18]

R. Metzler and J. Klafter, The random walk's guide to anomalous diffusion: A fractional dynamics approach, Phys. Rep., 339 (2000), 1-77.  doi: 10.1016/S0370-1573(00)00070-3.  Google Scholar

[19]

R. R. Nigmatullin, The realization of the generalized transfer equation in a medium with fractal geometry, Physica Status Solidi (B), 133 (1986), 425-430.  doi: 10.1002/pssb.2221330150.  Google Scholar

[20]

K. B. Oldham and J. Spanier, The Fractional Calculus, Mathematics in Science and Engineering, Vol. 111. Academic Press [A subsidiary of Harcourt Brace Jovanovich, Publishers], New York-London, 1974.  Google Scholar

[21]

I. Podlubny, Fractional Differential Equations, in Mathematics in Science and Engineering, 1999.  Google Scholar

[22]

K. Sakamoto and M. Yamamoto, Initial value/boundary value problems for fractional diffusion-wave equations and applications to some inverse problems, J. Math. Anal. Appl., 382 (2011), 426-447.  doi: 10.1016/j.jmaa.2011.04.058.  Google Scholar

[23]

M. Stynes, Too much regularity may force too much uniqueness, Fract. Calc. Appl. Anal., 19 (2016), 1554-1562.  doi: 10.1515/fca-2016-0080.  Google Scholar

[24]

M. Stynes, E. O'Riordan and J. L. 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.  doi: 10.1137/16M1082329.  Google Scholar

[25]

Y. Xing and Y. Yan, A higher order numerical method for time fractional partial differential equations with nonsmooth data, J. Comput. Phys., 357 (2018), 305-323.  doi: 10.1016/j.jcp.2017.12.035.  Google Scholar

[26]

B. Xu and X. Zhang, A reduced fourth-order compact difference scheme based on a proper orthogonal decomposition technique for parabolic equations, Bound. Value Probl., 2019 (2019), 130. doi: 10.1186/s13661-019-1243-8.  Google Scholar

[27]

Y. Yang, Y. Yan and N. J. Ford, Some time stepping methods for fractional diffusion problems with nonsmooth data, Comput. Methods Appl. Math., 18 (2018), 129-146.  doi: 10.1515/cmam-2017-0037.  Google Scholar

[28]

F. Zeng, Z. Zhang and G. E. Karniadakis, A generalized spectral collocation method with tunable accuracy for variable-order fractional differential equations, SIAM J. Sci. Comput., 37 (2015), A2710–A2732. doi: 10.1137/141001299.  Google Scholar

[29]

Y.-N. Zhang and Z.-Z. Sun, Alternating direction implicit schemes for the two-dimensional fractional sub-diffusion equation, J. Comput. Phys., 230 (2011), 8713-8728.  doi: 10.1016/j.jcp.2011.08.020.  Google Scholar

[30]

Z. Zhang, F. Zeng and G. E. Karniadakis, Optimal error estimates of spectral Petrov–Galerkin and collocation methods for initial value problems of fractional differential equations, SIAM J. Numer. Anal., 53 (2015), 2074-2096.  doi: 10.1137/140988218.  Google Scholar

[31]

H. Zhu and C. Xu, A fast high order method for the time-fractional diffusion equation, SIAM J. Numer. Anal., 57 (2019), 2829-2849.  doi: 10.1137/18M1231225.  Google Scholar

[32]

P. Zhuang and F. Liu, Implicit difference approximation for the time fractional diffusion equation, J. Appl. Math. Comput., 22 (2006), 87-99.  doi: 10.1007/BF02832039.  Google Scholar

show all references

References:
[1]

A. A. Alikhanov, A new difference scheme for the time fractional diffusion equation, J. Comput. Phys., 280 (2015), 424-438.  doi: 10.1016/j.jcp.2014.09.031.  Google Scholar

[2]

H. Brunner, L. Ling and M. Yamamoto, Numerical simulations of 2D fractional subdiffusion problems, J. Comput. Phys., 229 (2010), 6613-6622.  doi: 10.1016/j.jcp.2010.05.015.  Google Scholar

[3]

H. Chen and M. Stynes, A high order method on graded meshes for a time-fractional diffusion problem, Finite Difference Methods, 15–27, Lecture Notes in Comput. Sci., 11386, Springer, Cham, 2019. doi: 10.1007/978-3-030-11539-5_2.  Google Scholar

[4]

H. Chen and M. Stynes, Error analysis of a second-order method on fitted meshes for a time-fractional diffusion problem, J. Sci. Comput., 79 (2019), 624-647.  doi: 10.1007/s10915-018-0863-y.  Google Scholar

[5]

N. J. Ford, M. L. Morgado and M. Rebelo, Nonpolynomial collocation approximation of solutions to fractional differential equations, Fract. Calc. Appl. Anal., 16 (2013), 874-891.  doi: 10.2478/s13540-013-0054-3.  Google Scholar

[6]

N. J. Ford and Y. Yan, An approach to construct higher order time discretisation schemes for time fractional partial differential equations with nonsmooth data, Fract. Calc. Appl. Anal., 20 (2017), 1076-1105.  doi: 10.1515/fca-2017-0058.  Google Scholar

[7]

G.-H. Gao and Z.-Z. Sun, A compact finite difference scheme for the fractional sub-diffusion equations, J. Comput. Phys., 230 (2011), 586-595.  doi: 10.1016/j.jcp.2010.10.007.  Google Scholar

[8]

G.-H. Gao, Z.-Z. Sun and H.-W. Zhang, A new fractional numerical differentiation formula to approximate the Caputo fractional derivative and its applications, J. Comput. Phys., 259 (2014), 33-50.  doi: 10.1016/j.jcp.2013.11.017.  Google Scholar

[9]

M. Giona, S. Cerbelli and H. E. Roman, Fractional diffusion equation and relaxation in complex viscoelastic materials, Physica A: Statistical Mechanics and its Applications, 191 (1992), 449-453.  doi: 10.1016/0378-4371(92)90566-9.  Google Scholar

[10]

R. Gorenflo, F. Mainardi, D. Moretti and P. Paradisi, Time fractional diffusion: A discrete random walk approach, Nonlinear Dynam., 29 (2002), 129-143.  doi: 10.1023/A:1016547232119.  Google Scholar

[11]

Y. Lin and C. Xu, Finite difference/spectral approximations for the time-fractional diffusion equation, J. Comput. Phys., 225 (2007), 1533-1552.  doi: 10.1016/j.jcp.2007.02.001.  Google Scholar

[12]

F. Liu, S. Shen, V. Anh and I. Turner, Analysis of a discrete non-Markovian random walk approximation for the time fractional diffusion equation, ANZIAM J., 46 (2004), 488-504.   Google Scholar

[13]

Z. Luo, J. Chen, J. Zhu, R. Wang and I. M. Navon, An optimizing reduced order FDS for the tropical Pacific Ocean reduced gravity model, Internat. J. Numer. Methods Fluids, 55 (2007), 143-161.  doi: 10.1002/fld.1452.  Google Scholar

[14]

Z. Luo, H. Li, P. Sun and J. Gao, A reduced-order finite difference extrapolation algorithm based on POD technique for the non-stationary navier–stokes equations, Appl. Math. Model., 37 (2013), 5464-5473.  doi: 10.1016/j.apm.2012.10.051.  Google Scholar

[15]

Z. Luo, F. Teng and H. Xia, A reduced-order extrapolated Crank–Nicolson finite spectral element method based on POD for the 2D non-stationary Boussinesq equations, J. Math. Anal. Appl., 471 (2019), 564-583.  doi: 10.1016/j.jmaa.2018.10.092.  Google Scholar

[16]

C. Lv and C. Xu, Error analysis of a high order method for time-fractional diffusion equations, SIAM J. Sci. Comput., 38 (2016), A2699–A2724. doi: 10.1137/15M102664X.  Google Scholar

[17]

Z. Mao and J. Shen, Efficient spectral–Galerkin methods for fractional partial differential equations with variable coefficients, J. Comput. Phys., 307 (2016), 243-261.  doi: 10.1016/j.jcp.2015.11.047.  Google Scholar

[18]

R. Metzler and J. Klafter, The random walk's guide to anomalous diffusion: A fractional dynamics approach, Phys. Rep., 339 (2000), 1-77.  doi: 10.1016/S0370-1573(00)00070-3.  Google Scholar

[19]

R. R. Nigmatullin, The realization of the generalized transfer equation in a medium with fractal geometry, Physica Status Solidi (B), 133 (1986), 425-430.  doi: 10.1002/pssb.2221330150.  Google Scholar

[20]

K. B. Oldham and J. Spanier, The Fractional Calculus, Mathematics in Science and Engineering, Vol. 111. Academic Press [A subsidiary of Harcourt Brace Jovanovich, Publishers], New York-London, 1974.  Google Scholar

[21]

I. Podlubny, Fractional Differential Equations, in Mathematics in Science and Engineering, 1999.  Google Scholar

[22]

K. Sakamoto and M. Yamamoto, Initial value/boundary value problems for fractional diffusion-wave equations and applications to some inverse problems, J. Math. Anal. Appl., 382 (2011), 426-447.  doi: 10.1016/j.jmaa.2011.04.058.  Google Scholar

[23]

M. Stynes, Too much regularity may force too much uniqueness, Fract. Calc. Appl. Anal., 19 (2016), 1554-1562.  doi: 10.1515/fca-2016-0080.  Google Scholar

[24]

M. Stynes, E. O'Riordan and J. L. 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.  doi: 10.1137/16M1082329.  Google Scholar

[25]

Y. Xing and Y. Yan, A higher order numerical method for time fractional partial differential equations with nonsmooth data, J. Comput. Phys., 357 (2018), 305-323.  doi: 10.1016/j.jcp.2017.12.035.  Google Scholar

[26]

B. Xu and X. Zhang, A reduced fourth-order compact difference scheme based on a proper orthogonal decomposition technique for parabolic equations, Bound. Value Probl., 2019 (2019), 130. doi: 10.1186/s13661-019-1243-8.  Google Scholar

[27]

Y. Yang, Y. Yan and N. J. Ford, Some time stepping methods for fractional diffusion problems with nonsmooth data, Comput. Methods Appl. Math., 18 (2018), 129-146.  doi: 10.1515/cmam-2017-0037.  Google Scholar

[28]

F. Zeng, Z. Zhang and G. E. Karniadakis, A generalized spectral collocation method with tunable accuracy for variable-order fractional differential equations, SIAM J. Sci. Comput., 37 (2015), A2710–A2732. doi: 10.1137/141001299.  Google Scholar

[29]

Y.-N. Zhang and Z.-Z. Sun, Alternating direction implicit schemes for the two-dimensional fractional sub-diffusion equation, J. Comput. Phys., 230 (2011), 8713-8728.  doi: 10.1016/j.jcp.2011.08.020.  Google Scholar

[30]

Z. Zhang, F. Zeng and G. E. Karniadakis, Optimal error estimates of spectral Petrov–Galerkin and collocation methods for initial value problems of fractional differential equations, SIAM J. Numer. Anal., 53 (2015), 2074-2096.  doi: 10.1137/140988218.  Google Scholar

[31]

H. Zhu and C. Xu, A fast high order method for the time-fractional diffusion equation, SIAM J. Numer. Anal., 57 (2019), 2829-2849.  doi: 10.1137/18M1231225.  Google Scholar

[32]

P. Zhuang and F. Liu, Implicit difference approximation for the time fractional diffusion equation, J. Appl. Math. Comput., 22 (2006), 87-99.  doi: 10.1007/BF02832039.  Google Scholar

Figure 1.  The absolute error of solutions obtained by the two formats with $ M = 500 $, $ N = 500 $, $ \alpha = 0.6 $, $ r = \frac{3-\alpha}{\alpha} $ for Example 1
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Figure 2.  The absolute error of solutions obtained by the two formats with $ M = 500 $, $ N = 500 $, $ \alpha = 0.6 $, $ r = \frac{3-\alpha}{\alpha} $ for Example 2
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Figure 3.  The absolute error of solutions obtained by FD and RFD $ M = 200 $, $ N = 200 $, $ \alpha = 0.6 $, $ r = \frac{3-\alpha}{\alpha} $ for Example 3
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Table 1.  The CPU time consumed by FD and RFD with different mesh sizes with $ M = 10000 $, $ \alpha = 0.6 $, $ r = \frac{3-\alpha}{\alpha} $ for Example 1
$ N $ $ 2^5 $ $ 2^6 $ $ 2^7 $ $ 2^8 $ $ 2^9 $ $ 2^{10} $
FD (CPU) 235.9671 460.4681 922.5431 1821.1869 3661.3747 7446.4113
RFD (CPU) 0.0624 0.1092 0.2496 0.9360 3.2448 13.5721
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$ N $ $ 2^5 $ $ 2^6 $ $ 2^7 $ $ 2^8 $ $ 2^9 $ $ 2^{10} $
FD (CPU) 235.9671 460.4681 922.5431 1821.1869 3661.3747 7446.4113
RFD (CPU) 0.0624 0.1092 0.2496 0.9360 3.2448 13.5721
Table 2.  $ M = 10000 $, $ r = 1 $, the $ L^{\infty} $ error and convergence rates for Example 1 on graded meshes
$ N $ $ \alpha=0.4 $ $ \alpha=0.6 $ $ \alpha=0.8 $
$ Max\_err $ rate $ Max\_err $ rate $ Max\_err $ rate
$ 2^{5} $ 4.4813e-2 2.1049e-2 6.0226e-3
$ 2^{6} $ 3.5203e-2 0.3482 1.5005e-2 0.4883 4.2132e-3 0.5155
$ 2^{7} $ 2.7395e-2 0.3618 1.0381e-2 0.5316 2.6654e-3 0.6606
$ 2^{8} $ 2.1178e-2 0.3713 7.0548e-3 0.5572 1.6099e-3 0.7273
$ 2^{9} $ 1.6293e-2 0.3783 4.7427e-3 0.5729 9.5001e-4 0.7610
$ 2^{10} $ 1.2489e-2 0.3835 3.1668e-3 0.5827 5.5376e-4 0.7787
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$ N $ $ \alpha=0.4 $ $ \alpha=0.6 $ $ \alpha=0.8 $
$ Max\_err $ rate $ Max\_err $ rate $ Max\_err $ rate
$ 2^{5} $ 4.4813e-2 2.1049e-2 6.0226e-3
$ 2^{6} $ 3.5203e-2 0.3482 1.5005e-2 0.4883 4.2132e-3 0.5155
$ 2^{7} $ 2.7395e-2 0.3618 1.0381e-2 0.5316 2.6654e-3 0.6606
$ 2^{8} $ 2.1178e-2 0.3713 7.0548e-3 0.5572 1.6099e-3 0.7273
$ 2^{9} $ 1.6293e-2 0.3783 4.7427e-3 0.5729 9.5001e-4 0.7610
$ 2^{10} $ 1.2489e-2 0.3835 3.1668e-3 0.5827 5.5376e-4 0.7787
Table 3.  $ M = 10000 $, $ r = 1/\alpha $, the $ L^{\infty} $ error and convergence rates for Example 1 on graded meshes
$ N $ $ \alpha=0.4 $ $ \alpha=0.6 $ $ \alpha=0.8 $
$ Max\_err $ rate $ Max\_err $ rate $ Max\_err $ rate
$ 2^{5} $ 1.1780e-2 6.1861e-3 3.7802e-3
$ 2^{6} $ 6.0418e-3 0.9633 3.1668e-3 0.9660 2.0794e-3 0.8623
$ 2^{7} $ 3.0601e-3 0.9814 1.6014e-3 0.9837 1.0853e-3 0.9380
$ 2^{8} $ 1.5400e-3 0.9906 8.0511e-4 0.9921 5.5376e-4 0.9708
$ 2^{9} $ 7.7251e-4 0.9953 4.0364e-4 0.9961 2.7944e-4 0.9867
$ 2^{10} $ 3.8687e-4 0.9977 2.0209e-4 0.9981 1.4039e-4 0.9931
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$ N $ $ \alpha=0.4 $ $ \alpha=0.6 $ $ \alpha=0.8 $
$ Max\_err $ rate $ Max\_err $ rate $ Max\_err $ rate
$ 2^{5} $ 1.1780e-2 6.1861e-3 3.7802e-3
$ 2^{6} $ 6.0418e-3 0.9633 3.1668e-3 0.9660 2.0794e-3 0.8623
$ 2^{7} $ 3.0601e-3 0.9814 1.6014e-3 0.9837 1.0853e-3 0.9380
$ 2^{8} $ 1.5400e-3 0.9906 8.0511e-4 0.9921 5.5376e-4 0.9708
$ 2^{9} $ 7.7251e-4 0.9953 4.0364e-4 0.9961 2.7944e-4 0.9867
$ 2^{10} $ 3.8687e-4 0.9977 2.0209e-4 0.9981 1.4039e-4 0.9931
Table 4.  $ M = 10000 $, $ r = 2/\alpha $, the $ L^{\infty} $ error and convergence rates for Example 1 on graded meshes
$ N $ $ \alpha=0.4 $ $ \alpha=0.6 $ $ \alpha=0.8 $
$ Max\_err $ rate $ Max\_err $ rate $ Max\_err $ rate
$ 2^{5} $ 9.3428e-3 2.5627e-3 9.3566e-4
$ 2^{6} $ 2.3654e-3 1.9818 6.5273e-4 1.9731 2.5124e-4 1.8969
$ 2^{7} $ 5.9358e-4 1.9946 1.6398e-4 1.9929 6.4435e-5 1.9632
$ 2^{8} $ 1.4888e-4 1.9953 4.1050e-5 1.9981 1.6243e-5 1.9880
$ 2^{9} $ 3.7600e-5 1.9854 1.0266e-5 1.9995 4.0700e-6 1.9967
$ 2^{10} $ 9.7725e-6 1.9439 2.5667e-6 1.9999 1.0181e-6 1.9992
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$ N $ $ \alpha=0.4 $ $ \alpha=0.6 $ $ \alpha=0.8 $
$ Max\_err $ rate $ Max\_err $ rate $ Max\_err $ rate
$ 2^{5} $ 9.3428e-3 2.5627e-3 9.3566e-4
$ 2^{6} $ 2.3654e-3 1.9818 6.5273e-4 1.9731 2.5124e-4 1.8969
$ 2^{7} $ 5.9358e-4 1.9946 1.6398e-4 1.9929 6.4435e-5 1.9632
$ 2^{8} $ 1.4888e-4 1.9953 4.1050e-5 1.9981 1.6243e-5 1.9880
$ 2^{9} $ 3.7600e-5 1.9854 1.0266e-5 1.9995 4.0700e-6 1.9967
$ 2^{10} $ 9.7725e-6 1.9439 2.5667e-6 1.9999 1.0181e-6 1.9992
Table 5.  $ M = 10000 $, $ r = (3-\alpha)/\alpha $, the $ L^{\infty} $ error and convergence rates for Example 1 on graded meshes
$ N $ $ \alpha=0.4 $ $ \alpha=0.6 $ $ \alpha=0.8 $
$ Max\_err $ rate $ Max\_err $ rate $ Max\_err $ rate
$ 2^{5} $ 8.1920e-3 1.9021e-3 8.5782e-4
$ 2^{6} $ 1.3564e-3 2.5944 3.6421e-4 2.3848 2.0153e-4 2.0897
$ 2^{7} $ 2.2375e-4 2.5999 6.9143e-5 2.3971 4.5568e-5 2.1449
$ 2^{8} $ 3.6784e-5 2.6048 1.3105e-5 2.3995 1.0109e-5 2.1724
$ 2^{9} $ 5.9417e-6 2.6301 2.4831e-6 2.3999 2.2188e-6 2.1877
$ 2^{10} $ 8.5465e-7 2.7975 4.7049e-7 2.3999 4.8440e-7 2.1955
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$ N $ $ \alpha=0.4 $ $ \alpha=0.6 $ $ \alpha=0.8 $
$ Max\_err $ rate $ Max\_err $ rate $ Max\_err $ rate
$ 2^{5} $ 8.1920e-3 1.9021e-3 8.5782e-4
$ 2^{6} $ 1.3564e-3 2.5944 3.6421e-4 2.3848 2.0153e-4 2.0897
$ 2^{7} $ 2.2375e-4 2.5999 6.9143e-5 2.3971 4.5568e-5 2.1449
$ 2^{8} $ 3.6784e-5 2.6048 1.3105e-5 2.3995 1.0109e-5 2.1724
$ 2^{9} $ 5.9417e-6 2.6301 2.4831e-6 2.3999 2.2188e-6 2.1877
$ 2^{10} $ 8.5465e-7 2.7975 4.7049e-7 2.3999 4.8440e-7 2.1955
Table 6.  The CPU time consumed by FD and RFD with different mesh sizes with $ M = 10000 $, $ \alpha = 0.6 $, $ r = \frac{3-\alpha}{\alpha} $ for Example2
$ N $ $ 2^5 $ $ 2^6 $ $ 2^7 $ $ 2^8 $ $ 2^9 $ $ 2^{10} $
FD (CPU) 216.2954 435.1960 882.8253 1751.2672 3460.3362 7030.6999
RFD(CPU) 0.0156 0.0624 0.1872 0.7488 2.8080 14.0713
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$ N $ $ 2^5 $ $ 2^6 $ $ 2^7 $ $ 2^8 $ $ 2^9 $ $ 2^{10} $
FD (CPU) 216.2954 435.1960 882.8253 1751.2672 3460.3362 7030.6999
RFD(CPU) 0.0156 0.0624 0.1872 0.7488 2.8080 14.0713
Table 7.  Take $ M = 10000 $, $ r = (3-\alpha)/\alpha $ the $ L^{\infty} $ error and convergence rates for Example 2 on graded meshes
$ N $ $ \alpha=0.4 $ $ \alpha=0.6 $ $ \alpha=0.8 $
$ Max\_err $ rate $ Max\_err $ rate $ Max\_err $ rate
$ 2^{5} $ 9.2288e-3 2.1343e-3 9.6882e-4
$ 2^{6} $ 1.5290e-3 2.5936 4.0781e-4 2.3878 2.2141e-4 2.1296
$ 2^{7} $ 2.5250e-4 2.5982 7.7390e-5 2.3977 4.9476e-5 2.1619
$ 2^{8} $ 4.1777e-5 2.5955 1.4667e-5 2.3996 1.0907e-5 2.1815
$ 2^{9} $ 7.0163e-6 2.5739 2.7790e-6 2.3999 2.3867e-6 2.1922
$ 2^{10} $ 1.2826e-6 2.4516 5.2652e-7 2.4000 5.2036e-7 2.1974
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$ N $ $ \alpha=0.4 $ $ \alpha=0.6 $ $ \alpha=0.8 $
$ Max\_err $ rate $ Max\_err $ rate $ Max\_err $ rate
$ 2^{5} $ 9.2288e-3 2.1343e-3 9.6882e-4
$ 2^{6} $ 1.5290e-3 2.5936 4.0781e-4 2.3878 2.2141e-4 2.1296
$ 2^{7} $ 2.5250e-4 2.5982 7.7390e-5 2.3977 4.9476e-5 2.1619
$ 2^{8} $ 4.1777e-5 2.5955 1.4667e-5 2.3996 1.0907e-5 2.1815
$ 2^{9} $ 7.0163e-6 2.5739 2.7790e-6 2.3999 2.3867e-6 2.1922
$ 2^{10} $ 1.2826e-6 2.4516 5.2652e-7 2.4000 5.2036e-7 2.1974
Table 8.  $ L1 $ format, $ M = 10000 $, $ r = (2-\alpha)/\alpha $ the $ L^{\infty} $ error and convergence rates for Example 2 on graded meshes
$ N $ $ \alpha=0.4 $ $ \alpha=0.6 $ $ \alpha=0.8 $
$ Max\_err $ rate $ Max\_err $ rate $ Max\_err $ rate
$ 2^{5} $ 2.0114e-3 3.4273e-3 5.1455e-3
$ 2^{6} $ 7.1710e-4 1.4880 1.3922e-3 1.2997 2.3995e-3 1.1006
$ 2^{7} $ 2.4927e-4 1.5245 5.5229e-4 1.3339 1.1033e-3 1.1209
$ 2^{8} $ 8.5555e-5 1.5428 2.1578e-4 1.3558 5.0211e-4 1.1358
$ 2^{9} $ 2.9144e-5 1.5537 8.3468e-5 1.3703 2.2670e-4 1.1472
$ 2^{10} $ 9.8709e-6 1.5619 3.2075e-5 1.3798 1.0172e-4 1.1562
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$ N $ $ \alpha=0.4 $ $ \alpha=0.6 $ $ \alpha=0.8 $
$ Max\_err $ rate $ Max\_err $ rate $ Max\_err $ rate
$ 2^{5} $ 2.0114e-3 3.4273e-3 5.1455e-3
$ 2^{6} $ 7.1710e-4 1.4880 1.3922e-3 1.2997 2.3995e-3 1.1006
$ 2^{7} $ 2.4927e-4 1.5245 5.5229e-4 1.3339 1.1033e-3 1.1209
$ 2^{8} $ 8.5555e-5 1.5428 2.1578e-4 1.3558 5.0211e-4 1.1358
$ 2^{9} $ 2.9144e-5 1.5537 8.3468e-5 1.3703 2.2670e-4 1.1472
$ 2^{10} $ 9.8709e-6 1.5619 3.2075e-5 1.3798 1.0172e-4 1.1562
Table 9.  $ L2-1_{\sigma} $ format, $ M = 10000 $, $ r = 2/\alpha $ the $ L^{\infty} $ error and convergence rates for Example 2 on graded meshes
$ N $ $ \alpha=0.4 $ $ \alpha=0.6 $ $ \alpha=0.8 $
$ Max\_err $ rate $ Max\_err $ rate $ Max\_err $ rate
$ 2^{5} $ 2.3712e-4 1.9954e-4 1.2946e-4
$ 2^{6} $ 6.1882e-5 1.9380 5.2396e-5 1.9291 3.5332e-5 1.8735
$ 2^{7} $ 1.5706e-5 1.9782 1.3317e-5 1.9762 9.2562e-6 1.9325
$ 2^{8} $ 3.9412e-6 1.9946 3.3471e-6 1.9923 2.3634e-6 1.9695
$ 2^{9} $ 9.8622e-7 1.9986 8.3790e-7 1.9981 5.9543e-7 1.9889
$ 2^{10} $ 2.4661e-7 1.9997 2.0955e-7 1.9995 1.4920e-7 1.9967
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$ N $ $ \alpha=0.4 $ $ \alpha=0.6 $ $ \alpha=0.8 $
$ Max\_err $ rate $ Max\_err $ rate $ Max\_err $ rate
$ 2^{5} $ 2.3712e-4 1.9954e-4 1.2946e-4
$ 2^{6} $ 6.1882e-5 1.9380 5.2396e-5 1.9291 3.5332e-5 1.8735
$ 2^{7} $ 1.5706e-5 1.9782 1.3317e-5 1.9762 9.2562e-6 1.9325
$ 2^{8} $ 3.9412e-6 1.9946 3.3471e-6 1.9923 2.3634e-6 1.9695
$ 2^{9} $ 9.8622e-7 1.9986 8.3790e-7 1.9981 5.9543e-7 1.9889
$ 2^{10} $ 2.4661e-7 1.9997 2.0955e-7 1.9995 1.4920e-7 1.9967
Table 10.  The CPU time consumed by FD and RFD with different mesh sizes with $ M = 100 $, $ \alpha = 0.6 $, $ r = \frac{3-\alpha}{\alpha} $ for Example3
$ N $ $ 2^5 $ $ 2^6 $ $ 2^7 $ $ 2^8 $
FD (CPU) 2082.2857 4202.6201 8426.9712 16835.3939
RFD(CPU) 0.0468 0.0624 0.1872 0.8112
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$ N $ $ 2^5 $ $ 2^6 $ $ 2^7 $ $ 2^8 $
FD (CPU) 2082.2857 4202.6201 8426.9712 16835.3939
RFD(CPU) 0.0468 0.0624 0.1872 0.8112
Table 11.  $ M = 500 $, $ r = (3-\alpha)/\alpha $, the $ L^{\infty} $ error and convergence rates for Example 3 on graded meshes
$ N $ $ \alpha=0.4 $ $ \alpha=0.6 $ $ \alpha=0.8 $
$ Max\_err $ rate $ Max\_err $ rate $ Max\_err $ rate
$ 2^{5} $ 9.1847e-3 2.1075e-3 9.0565e-4
$ 2^{6} $ 1.5275e-3 2.5880 4.0685e-4 2.3730 2.1443e-4 2.0785
$ 2^{7} $ 2.5217e-4 2.5987 7.7358e-5 2.3949 4.8728e-5 2.1377
$ 2^{8} $ 4.1476e-5 2.6040 1.4667e-5 2.3990 1.0858e-5 2.1660
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$ N $ $ \alpha=0.4 $ $ \alpha=0.6 $ $ \alpha=0.8 $
$ Max\_err $ rate $ Max\_err $ rate $ Max\_err $ rate
$ 2^{5} $ 9.1847e-3 2.1075e-3 9.0565e-4
$ 2^{6} $ 1.5275e-3 2.5880 4.0685e-4 2.3730 2.1443e-4 2.0785
$ 2^{7} $ 2.5217e-4 2.5987 7.7358e-5 2.3949 4.8728e-5 2.1377
$ 2^{8} $ 4.1476e-5 2.6040 1.4667e-5 2.3990 1.0858e-5 2.1660
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