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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 and 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.

[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.

[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.

[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.

[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.

[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.

[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.

[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.

[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.

[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.

[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.

[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. 

[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.

[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.

[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.

[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.

[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.

[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.

[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.

[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.

[21]

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

[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.

[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.

[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.

[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.

[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.

[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.

[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.

[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.

[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.

[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.

[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.

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.

[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.

[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.

[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.

[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.

[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.

[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.

[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.

[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.

[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.

[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.

[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. 

[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.

[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.

[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.

[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.

[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.

[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.

[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.

[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.

[21]

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

[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.

[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.

[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.

[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.

[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.

[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.

[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.

[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.

[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.

[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.

[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.

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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