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Two finite difference methods based on an H2N2 interpolation for two-dimensional time fractional mixed diffusion and diffusion-wave equations
A fast high order method for time fractional diffusion equation with non-smooth data
School of Mathematics, Shandong University, Jinan, Shandong 250100, China |
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.
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. |



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 |
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 |
rate | rate | rate | ||||||
4.4813e-2 | 2.1049e-2 | 6.0226e-3 | ||||||
3.5203e-2 | 0.3482 | 1.5005e-2 | 0.4883 | 4.2132e-3 | 0.5155 | |||
2.7395e-2 | 0.3618 | 1.0381e-2 | 0.5316 | 2.6654e-3 | 0.6606 | |||
2.1178e-2 | 0.3713 | 7.0548e-3 | 0.5572 | 1.6099e-3 | 0.7273 | |||
1.6293e-2 | 0.3783 | 4.7427e-3 | 0.5729 | 9.5001e-4 | 0.7610 | |||
1.2489e-2 | 0.3835 | 3.1668e-3 | 0.5827 | 5.5376e-4 | 0.7787 |
rate | rate | rate | ||||||
4.4813e-2 | 2.1049e-2 | 6.0226e-3 | ||||||
3.5203e-2 | 0.3482 | 1.5005e-2 | 0.4883 | 4.2132e-3 | 0.5155 | |||
2.7395e-2 | 0.3618 | 1.0381e-2 | 0.5316 | 2.6654e-3 | 0.6606 | |||
2.1178e-2 | 0.3713 | 7.0548e-3 | 0.5572 | 1.6099e-3 | 0.7273 | |||
1.6293e-2 | 0.3783 | 4.7427e-3 | 0.5729 | 9.5001e-4 | 0.7610 | |||
1.2489e-2 | 0.3835 | 3.1668e-3 | 0.5827 | 5.5376e-4 | 0.7787 |
rate | rate | rate | ||||||
1.1780e-2 | 6.1861e-3 | 3.7802e-3 | ||||||
6.0418e-3 | 0.9633 | 3.1668e-3 | 0.9660 | 2.0794e-3 | 0.8623 | |||
3.0601e-3 | 0.9814 | 1.6014e-3 | 0.9837 | 1.0853e-3 | 0.9380 | |||
1.5400e-3 | 0.9906 | 8.0511e-4 | 0.9921 | 5.5376e-4 | 0.9708 | |||
7.7251e-4 | 0.9953 | 4.0364e-4 | 0.9961 | 2.7944e-4 | 0.9867 | |||
3.8687e-4 | 0.9977 | 2.0209e-4 | 0.9981 | 1.4039e-4 | 0.9931 |
rate | rate | rate | ||||||
1.1780e-2 | 6.1861e-3 | 3.7802e-3 | ||||||
6.0418e-3 | 0.9633 | 3.1668e-3 | 0.9660 | 2.0794e-3 | 0.8623 | |||
3.0601e-3 | 0.9814 | 1.6014e-3 | 0.9837 | 1.0853e-3 | 0.9380 | |||
1.5400e-3 | 0.9906 | 8.0511e-4 | 0.9921 | 5.5376e-4 | 0.9708 | |||
7.7251e-4 | 0.9953 | 4.0364e-4 | 0.9961 | 2.7944e-4 | 0.9867 | |||
3.8687e-4 | 0.9977 | 2.0209e-4 | 0.9981 | 1.4039e-4 | 0.9931 |
rate | rate | rate | ||||||
9.3428e-3 | 2.5627e-3 | 9.3566e-4 | ||||||
2.3654e-3 | 1.9818 | 6.5273e-4 | 1.9731 | 2.5124e-4 | 1.8969 | |||
5.9358e-4 | 1.9946 | 1.6398e-4 | 1.9929 | 6.4435e-5 | 1.9632 | |||
1.4888e-4 | 1.9953 | 4.1050e-5 | 1.9981 | 1.6243e-5 | 1.9880 | |||
3.7600e-5 | 1.9854 | 1.0266e-5 | 1.9995 | 4.0700e-6 | 1.9967 | |||
9.7725e-6 | 1.9439 | 2.5667e-6 | 1.9999 | 1.0181e-6 | 1.9992 |
rate | rate | rate | ||||||
9.3428e-3 | 2.5627e-3 | 9.3566e-4 | ||||||
2.3654e-3 | 1.9818 | 6.5273e-4 | 1.9731 | 2.5124e-4 | 1.8969 | |||
5.9358e-4 | 1.9946 | 1.6398e-4 | 1.9929 | 6.4435e-5 | 1.9632 | |||
1.4888e-4 | 1.9953 | 4.1050e-5 | 1.9981 | 1.6243e-5 | 1.9880 | |||
3.7600e-5 | 1.9854 | 1.0266e-5 | 1.9995 | 4.0700e-6 | 1.9967 | |||
9.7725e-6 | 1.9439 | 2.5667e-6 | 1.9999 | 1.0181e-6 | 1.9992 |
rate | rate | rate | ||||||
8.1920e-3 | 1.9021e-3 | 8.5782e-4 | ||||||
1.3564e-3 | 2.5944 | 3.6421e-4 | 2.3848 | 2.0153e-4 | 2.0897 | |||
2.2375e-4 | 2.5999 | 6.9143e-5 | 2.3971 | 4.5568e-5 | 2.1449 | |||
3.6784e-5 | 2.6048 | 1.3105e-5 | 2.3995 | 1.0109e-5 | 2.1724 | |||
5.9417e-6 | 2.6301 | 2.4831e-6 | 2.3999 | 2.2188e-6 | 2.1877 | |||
8.5465e-7 | 2.7975 | 4.7049e-7 | 2.3999 | 4.8440e-7 | 2.1955 |
rate | rate | rate | ||||||
8.1920e-3 | 1.9021e-3 | 8.5782e-4 | ||||||
1.3564e-3 | 2.5944 | 3.6421e-4 | 2.3848 | 2.0153e-4 | 2.0897 | |||
2.2375e-4 | 2.5999 | 6.9143e-5 | 2.3971 | 4.5568e-5 | 2.1449 | |||
3.6784e-5 | 2.6048 | 1.3105e-5 | 2.3995 | 1.0109e-5 | 2.1724 | |||
5.9417e-6 | 2.6301 | 2.4831e-6 | 2.3999 | 2.2188e-6 | 2.1877 | |||
8.5465e-7 | 2.7975 | 4.7049e-7 | 2.3999 | 4.8440e-7 | 2.1955 |
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 |
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 |
rate | rate | rate | ||||||
9.2288e-3 | 2.1343e-3 | 9.6882e-4 | ||||||
1.5290e-3 | 2.5936 | 4.0781e-4 | 2.3878 | 2.2141e-4 | 2.1296 | |||
2.5250e-4 | 2.5982 | 7.7390e-5 | 2.3977 | 4.9476e-5 | 2.1619 | |||
4.1777e-5 | 2.5955 | 1.4667e-5 | 2.3996 | 1.0907e-5 | 2.1815 | |||
7.0163e-6 | 2.5739 | 2.7790e-6 | 2.3999 | 2.3867e-6 | 2.1922 | |||
1.2826e-6 | 2.4516 | 5.2652e-7 | 2.4000 | 5.2036e-7 | 2.1974 |
rate | rate | rate | ||||||
9.2288e-3 | 2.1343e-3 | 9.6882e-4 | ||||||
1.5290e-3 | 2.5936 | 4.0781e-4 | 2.3878 | 2.2141e-4 | 2.1296 | |||
2.5250e-4 | 2.5982 | 7.7390e-5 | 2.3977 | 4.9476e-5 | 2.1619 | |||
4.1777e-5 | 2.5955 | 1.4667e-5 | 2.3996 | 1.0907e-5 | 2.1815 | |||
7.0163e-6 | 2.5739 | 2.7790e-6 | 2.3999 | 2.3867e-6 | 2.1922 | |||
1.2826e-6 | 2.4516 | 5.2652e-7 | 2.4000 | 5.2036e-7 | 2.1974 |
rate | rate | rate | ||||||
2.0114e-3 | 3.4273e-3 | 5.1455e-3 | ||||||
7.1710e-4 | 1.4880 | 1.3922e-3 | 1.2997 | 2.3995e-3 | 1.1006 | |||
2.4927e-4 | 1.5245 | 5.5229e-4 | 1.3339 | 1.1033e-3 | 1.1209 | |||
8.5555e-5 | 1.5428 | 2.1578e-4 | 1.3558 | 5.0211e-4 | 1.1358 | |||
2.9144e-5 | 1.5537 | 8.3468e-5 | 1.3703 | 2.2670e-4 | 1.1472 | |||
9.8709e-6 | 1.5619 | 3.2075e-5 | 1.3798 | 1.0172e-4 | 1.1562 |
rate | rate | rate | ||||||
2.0114e-3 | 3.4273e-3 | 5.1455e-3 | ||||||
7.1710e-4 | 1.4880 | 1.3922e-3 | 1.2997 | 2.3995e-3 | 1.1006 | |||
2.4927e-4 | 1.5245 | 5.5229e-4 | 1.3339 | 1.1033e-3 | 1.1209 | |||
8.5555e-5 | 1.5428 | 2.1578e-4 | 1.3558 | 5.0211e-4 | 1.1358 | |||
2.9144e-5 | 1.5537 | 8.3468e-5 | 1.3703 | 2.2670e-4 | 1.1472 | |||
9.8709e-6 | 1.5619 | 3.2075e-5 | 1.3798 | 1.0172e-4 | 1.1562 |
rate | rate | rate | ||||||
2.3712e-4 | 1.9954e-4 | 1.2946e-4 | ||||||
6.1882e-5 | 1.9380 | 5.2396e-5 | 1.9291 | 3.5332e-5 | 1.8735 | |||
1.5706e-5 | 1.9782 | 1.3317e-5 | 1.9762 | 9.2562e-6 | 1.9325 | |||
3.9412e-6 | 1.9946 | 3.3471e-6 | 1.9923 | 2.3634e-6 | 1.9695 | |||
9.8622e-7 | 1.9986 | 8.3790e-7 | 1.9981 | 5.9543e-7 | 1.9889 | |||
2.4661e-7 | 1.9997 | 2.0955e-7 | 1.9995 | 1.4920e-7 | 1.9967 |
rate | rate | rate | ||||||
2.3712e-4 | 1.9954e-4 | 1.2946e-4 | ||||||
6.1882e-5 | 1.9380 | 5.2396e-5 | 1.9291 | 3.5332e-5 | 1.8735 | |||
1.5706e-5 | 1.9782 | 1.3317e-5 | 1.9762 | 9.2562e-6 | 1.9325 | |||
3.9412e-6 | 1.9946 | 3.3471e-6 | 1.9923 | 2.3634e-6 | 1.9695 | |||
9.8622e-7 | 1.9986 | 8.3790e-7 | 1.9981 | 5.9543e-7 | 1.9889 | |||
2.4661e-7 | 1.9997 | 2.0955e-7 | 1.9995 | 1.4920e-7 | 1.9967 |
FD (CPU) | 2082.2857 | 4202.6201 | 8426.9712 | 16835.3939 | |
RFD(CPU) | 0.0468 | 0.0624 | 0.1872 | 0.8112 |
FD (CPU) | 2082.2857 | 4202.6201 | 8426.9712 | 16835.3939 | |
RFD(CPU) | 0.0468 | 0.0624 | 0.1872 | 0.8112 |
rate | rate | rate | ||||||
9.1847e-3 | 2.1075e-3 | 9.0565e-4 | ||||||
1.5275e-3 | 2.5880 | 4.0685e-4 | 2.3730 | 2.1443e-4 | 2.0785 | |||
2.5217e-4 | 2.5987 | 7.7358e-5 | 2.3949 | 4.8728e-5 | 2.1377 | |||
4.1476e-5 | 2.6040 | 1.4667e-5 | 2.3990 | 1.0858e-5 | 2.1660 |
rate | rate | rate | ||||||
9.1847e-3 | 2.1075e-3 | 9.0565e-4 | ||||||
1.5275e-3 | 2.5880 | 4.0685e-4 | 2.3730 | 2.1443e-4 | 2.0785 | |||
2.5217e-4 | 2.5987 | 7.7358e-5 | 2.3949 | 4.8728e-5 | 2.1377 | |||
4.1476e-5 | 2.6040 | 1.4667e-5 | 2.3990 | 1.0858e-5 | 2.1660 |
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