American Institute of Mathematical Sciences

Advanced
March  2021, 26(3): 1447-1468. doi: 10.3934/dcdsb.2020168

Approximation methods for the distributed order calculus using the convolution quadrature

Baoli Yin 1, , Yang Liu 1,, , Hong Li 1, and  Zhimin Zhang 2,3,

1. 

School of Mathematical Sciences, Inner Mongolia University, Hohhot 010021, China
2. 

Beijing Computational Science Research Center, Beijing 100193, China
3. 

Department of Mathematics, Wayne State University Detroit, MI 48202, USA

* Corresponding author: mathliuyang@imu.edu.cn (Y. Liu)

Received  August 2019 Revised  March 2020 Published  May 2020

Fund Project: The second author is supported in part by the NSFC grant 11661058; The third author is supported in part by the NSFC grant 11761053, the NSF of Inner Mongolia 2017MS0107, and the program for Young Talents of Science and Technology in Universities of Inner Mongolia Autonomous Region NJYT-17-A07; The fourth author was supported in part by grants NSFC 11871092 and NSAF U1930402

In this article we generalize the convolution quadrature (CQ) method, which aims at approximating the fractional calculus, to the case for the distributed order calculus. Our method is a natural expansion that the approximation formulas, convergence results and correction technique reduce to the cases for the CQ method if the weight function $ \mu(\alpha) $ is defined by $ \delta(\alpha-\alpha_0) $. Further, we explore a new structure of the solution of an ODE with the distributed order fractional derivative, which differs from those of the solutions of traditional fractional ODEs, and propose a new correction technique for this new structure to restore the optimal convergence rate. Numerical tests with smooth and nonsmooth solutions confirm our theoretical results and the efficiency of our correction technique.

Keywords: Distributed order calculus, correction technique, convolution quadrature, optimal convergence, regularity.
Mathematics Subject Classification: Primary: 26A33, 65D25; Secondary: 65D30.
Citation: Baoli Yin, Yang Liu, Hong Li, Zhimin Zhang. Approximation methods for the distributed order calculus using the convolution quadrature. Discrete & Continuous Dynamical Systems - B, 2021, 26 (3) : 1447-1468. doi: 10.3934/dcdsb.2020168
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show all references

References:
[1]

M. Abbaszadeh and M. Dehghan, An improved meshless method for solving two-dimensional distributed order time-fractional diffusion-wave equation with error estimate, Numer. Algor., 75 (2017), 173-211.  doi: 10.1007/s11075-016-0201-0.  Google Scholar

[2]

D. Baffet and J. S. Hesthaven, A kernel compression scheme for fractional differential equations, SIAM J. Numer. Anal., 55 (2017), 496-520.  doi: 10.1137/15M1043960.  Google Scholar

[3]

M. Caputo, Mean fractional-order-derivatives differential equations and filters, Ann. Univ. Ferrara Sez. Ⅶ (N.S.), 41 (1995), 73-84.   Google Scholar

[4]

A. V. Chechkin, R. Gorenflo, I. M. Sokolov and V. Y. Gonchar, Distributed order time fractional diffusion equation, Fract. Calc. Appl. Anal., 6 (2003), 259-279.   Google Scholar

[5]

M. H. Chen and W. H. Deng, Fourth order difference approximations for space Riemann-Liouville derivatives based on weighted and shifted Lubich difference operators, Commu. Comput. Phys., 16 (2014), 516-540.  doi: 10.4208/cicp.120713.280214a.  Google Scholar

[6]

K. Diethelm and N. J. Ford, Numerical analysis for distributed-order differential equations, J. Comput. Appl. Math., 225 (2009), 96-104.  doi: 10.1016/j.cam.2008.07.018.  Google Scholar

[7]

H. F. Ding and C. P. Li, High-order numerical algorithms for Riesz derivatives via constructing new generating functions, J. Sci. Comput., 71 (2017), 759-784.  doi: 10.1007/s10915-016-0317-3.  Google Scholar

[8]

Y. W. Du, Y. Liu, H. Li, Z. C. Fang and S. He, Local discontinuous Galerkin method for a nonlinear time-fractional fourth-order partial differential equation, J. Comput. Phys., 344 (2017), 108-126.  doi: 10.1016/j.jcp.2017.04.078.  Google Scholar

[9]

W. Feller, An Introduction to Probability Theory and its Applications. Vol. II, Second edition, John Wiley & Sons, Inc., New York-London-Sydney, 1971.  Google Scholar

[10]

L. B. Feng, F. W. Liu and I. Turner, Finite difference/finite element method for a novel 2D multi-term time-fractional mixed sub-diffusion and diffusion-wave equation on convex domains, Commu. Nonlinear Sci. Numer. Simulat., 70 (2019), 354-371.  doi: 10.1016/j.cnsns.2018.10.016.  Google Scholar

[11]

G.-H. Gao, H.-W. Sun and Z.-Z. Sun, Some high-order difference schemes for the distributed-order differential equations, J. Comput. Phys., 298 (2015), 337-359.  doi: 10.1016/j.jcp.2015.05.047.  Google Scholar

[12]

P. Gatto and J. S. Hesthaven, Numerical approximation of the fractional Laplacian via $hp$-finite elements, with an application to image denoising, J. Sci. Comput., 65 (2015), 249-270.  doi: 10.1007/s10915-014-9959-1.  Google Scholar

[13]

S. M. Guo, L. Q. Mei, Z. Q. Zhang and Y. T. Jiang, Finite difference/spectral-Galerkin method for a two-dimensional distributed-order time-space fractional reaction-diffusion equation, Appl. Math. Letters, 85 (2018), 157-163.  doi: 10.1016/j.aml.2018.06.005.  Google Scholar

[14]

J. H. Jia and H. Wang, A fast finite difference method for distributed-order space-fractional partial differential equations on convex domains, Comput. Math. Appl., 75 (2018), 2031-2043.  doi: 10.1016/j.camwa.2017.09.003.  Google Scholar

[15]

B. T. Jin, B. Y. Li and Z. Zhou, Correction of high-order BDF convolution quadrature for fractional evolution equations, SIAM J. Sci. Comput., 39 (2017), A3129–A3152. doi: 10.1137/17M1118816.  Google Scholar

[16]

B. T. Jin, R. Lazarov, D. Sheen and Z. Zhou, Error estimates for approximations of distributed order time fractional diffusion with nonsmooth data, Fract. Calc. Appl. Anal., 19 (2016), 69-93.  doi: 10.1515/fca-2016-0005.  Google Scholar

[17]

A. N. Kochubei, Distributed order calculus and equations of ultraslow diffusion, J. Math. Anal. Appl., 340 (2008), 252-281.  doi: 10.1016/j.jmaa.2007.08.024.  Google Scholar

[18]

J. C. Li, Y. Q. Huang and Y. P. Lin, Developing finite element methods for Maxwell's equations in a Cole-Cole dispersive medium, SIAM J. Sci. Comput., 33 (2011), 3153-3174.  doi: 10.1137/110827624.  Google Scholar

[19] C. P. Li and F. H. Zeng, Numerical Methods for Fractional Calculus, Chapman & Hall/CRC Numerical Analysis and Scientific Computing, CRC Press, Boca Raton, FL, 2015.   Google Scholar
[20]

D. F. Li, J. W. Zhang and Z. M. Zhang, Unconditionally optimal error estimates of a linearized Galerkin method for nonlinear time fractional reaction-subdiffusion equations, J. Sci. Comput., 76 (2018), 848-866.  doi: 10.1007/s10915-018-0642-9.  Google Scholar

[21]

B. J. Li, H. Luo and X. P. Xie, Analysis of a time-stepping scheme for time fractional diffusion problems with nonsmooth data, SIAM J. Numer. Anal., 57 (2019), 779-798.  doi: 10.1137/18M118414X.  Google Scholar

[22]

Z. Y. Li, Y. Luchko and M. Yamamoto, Asymptotic estimates of solutions to initial-boundary-value problems for distributed order time-fractional diffusion equations, Fract. Calc. Appl. Anal., 17 (2014), 1114-1136.  doi: 10.2478/s13540-014-0217-x.  Google Scholar

[23]

H.-L. Liao, W. McLean and J. W. Zhang, A discrete Grönwall inequality with applications to numerical schemes for subdiffusion problems, SIAM J. Numer. Anal., 57 (2019), 218-237.  doi: 10.1137/16M1175742.  Google Scholar

[24]

Y. Liu, Y.-W. Du, H. Li and J.-F. Wang, A two-grid finite element approximation for a nonlinear time-fractional Cable equation, Nonlinear Dyn., 85 (2016), 2535-2548.  doi: 10.1007/s11071-016-2843-9.  Google Scholar

[25]

Y. Liu, Y. W. Du, H. Li, F. W. Liu and Y. J. Wang, Some second-order $\theta$ schemes combined with finite element method for nonlinear fractional Cable equation, Numer. Algor., 80 (2019), 533-555.  doi: 10.1007/s11075-018-0496-0.  Google Scholar

[26]

Y. Liu, B. Yin, H. Li and Z. Zhang, The unified theory of shifted convolution quadrature for fractional calculus, arXiv: 1908.01136v3. Google Scholar

[27]

C. Lubich, Discretized fractional calculus, SIAM J. Math. Anal., 17 (1986), 704-719.  doi: 10.1137/0517050.  Google Scholar

[28]

C. Lubich, Convolution quadrature and discretized operational calculus. I, Numer. Math., 52 (1988), 129-145.  doi: 10.1007/BF01398686.  Google Scholar

[29]

Y. Luchko, Boundary value problems for the generalized time-fractional diffusion equation of distributed order, Fract. Calc. Appl. Anal., 12 (2009), 409-422.   Google Scholar

[30]

S. Mashayekhi and M. Razzaghi, Numerical solution of distributed order fractional differential equations by hybrid functions, J. Comput. Phys., 315 (2016), 169-181.  doi: 10.1016/j.jcp.2016.01.041.  Google Scholar

[31]

W. McLean and K. Mustapha, Time-stepping error bounds for fractional diffusion problems with non-smooth initial data, J. Comput. Phys., 293 (2015), 201-217.  doi: 10.1016/j.jcp.2014.08.050.  Google Scholar

[32]

M. M. Meerschaert and H. P. Scheffler, Stochastic model for ultraslow diffusion, Stoch. Proc. Appl., 116 (2006), 1215-1235.  doi: 10.1016/j.spa.2006.01.006.  Google Scholar

[33]

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

[34]

B. P. Moghaddam, J. A. Tenreiro Machado and M. L. Morgado, Numerical approach for a class of distributed order time fractional partial differential equations, Appl. Numer. Math., 136 (2019), 152-162.  doi: 10.1016/j.apnum.2018.09.019.  Google Scholar

[35] I. Podlubny, Fractional Differential Equations: An Introduction to Fractional Derivatives, Fractional Differential Equations, to Methods of Their Solution and Some of Their Applications, Mathematics in Science and Engineering, 198. Academic Press, Inc., San Diego, CA, 1999.   Google Scholar
[36]

A. Quarteroni, R. Sacco and F. Saleri, Numerical Mathematics, Springer-Verlag Italia, Milan, 1998.  Google Scholar

[37]

M. H. Ran and C. J. Zhang, New compact difference scheme for solving the fourth-order time fractional sub-diffusion equation of the distributed order, Appl. Numer. Math., 129 (2018), 58-70.  doi: 10.1016/j.apnum.2018.03.005.  Google Scholar

[38]

J. C. Ren and Z.-Z. Sun, Efficient and stable numerical methods for multi-term time fractional sub-diffusion equations, E. Asian J. Appl. Math., 4 (2014), 242-266.  doi: 10.4208/eajam.181113.280514a.  Google Scholar

[39]

Y. H. Shi, F. Liu, Y. M. Zhao, F. L. Wang and I. Turner, An unstructured mesh finite element method for solving the multi-term time fractional and Riesz space distributed-order wave equation on an irregular convex domain, Appl. Math. Model., 73 (2019), 615-636.  doi: 10.1016/j.apm.2019.04.023.  Google Scholar

[40]

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

[41]

P. D. Wang and C. M. Huang, An implicit midpoint difference scheme for the fractional Ginzburg-Landau equation, J. Comput. Phys., 312 (2016), 31-49.  doi: 10.1016/j.jcp.2016.02.018.  Google Scholar

[42]

Y. B. Yan, M. Khan and N. J. Ford, An analysis of the modified L1 scheme for time-fractional partial differential equations with nonsmooth data, SIAM J. Numer. Anal., 56 (2018), 210-227.  doi: 10.1137/16M1094257.  Google Scholar

[43]

B. L. Yin, Y. Liu, H. Li and S. He, Fast algorithm based on TT-M FE system for space fractional Allen-Cahn equations with smooth and non-smooth solutions, J. Comput. Phys., 379 (2019), 351-372.  doi: 10.1016/j.jcp.2018.12.004.  Google Scholar

[44]

B. Yin, Y. Liu, H. Li and Z. Zhang, Finite element methods based on two families of second-order numerical formulas for the fractional Cable model with smooth solutions, arXiv: 1911.08166v1. Google Scholar

[45]

B. L. Yin, Y. Liu and H. Li, A class of shifted high-order numerical methods for the fractional mobile/immobile transport equations, Appl. Math. Comput., 368 (2020), 124799, 20 pp. doi: 10.1016/j.amc.2019.124799.  Google Scholar

[46]

B. Yin, Y. Liu, H. Li and Z. Zhang, Two families of novel second-order fractional numerical formulas and their applications to fractional differential equations, preprint, arXiv: 1906.01242v1. Google Scholar

[47]

F. H. Zeng, Z. Q. Zhang and G. E. Karniadakis, Second-order numerical methods for multi-term fractional differential equations: Smooth and non-smooth solutions, Comput. Methods Appl. Mech. Eng., 327 (2017), 478-502.  doi: 10.1016/j.cma.2017.08.029.  Google Scholar

[48]

H. Zhang, F. W. Liu, X. Y. Jiang, F. H. Zeng and I. Turner, A Crank-Nicolson ADI Galerkin-Legendre spectral method for the two-dimensional Riesz space distributed-order advection-diffusion equation, Comput. Math. Appl., 76 (2018), 2460-2476.  doi: 10.1016/j.camwa.2018.08.042.  Google Scholar

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Figure 1.  Figures of $ f_n(x) $ and the numerical solution
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Figure 2.  Error plot at each node with correction terms, $ h = \frac{1}{40} $
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Figure 3.  Error plot at each node without correction terms, $ h = \frac{1}{40} $
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Table 2.  Comparison of rates for methods with and without correction terms, $ h = \frac{1}{2000} $
Methods $ \tau $ $ E_U $ Rate CPU(s) $ E_C $ Rate CPU(s)
D-Euler
($ \ell $=0.1)		 1/10 2.66E-01 – 0.441 5.14E-02 – 0.435
1/20 2.90E-01 – 0.426 2.55E-02 1.01 0.461
1/40 3.05E-01 – 0.480 1.28E-02 1.00 0.496
1/80 3.10E-01 – 0.611 6.40E-03 1.00 0.611
D-BDF2
($ \ell $=0.5)		 1/10 5.54E-02 – 0.442 4.20E-03 – 0.435
1/20 4.97E-02 0.16 0.443 1.11E-03 1.92 0.447
1/40 4.23E-02 0.23 0.461 2.85E-04 1.96 0.481
1/80 3.43E-02 0.30 0.501 7.15E-05 1.99 0.810
D-BT-$ \theta $
($ \theta $=0.45, $ \ell $=0.3)		 1/10 1.55E-01 – 0.450 1.87E-03 – 0.428
1/20 1.50E-01 0.05 0.457 4.70E-04 2.00 0.461
1/40 1.38E-01 0.11 0.476 1.17E-04 2.00 0.498
1/80 1.23E-01 0.17 0.589 2.88E-05 2.02 0.605
D-BN-$ \theta $
($ \theta $=1, $ \ell $=0.8)		 1/10 1.59E-02 – 0.448 6.76E-03 – 0.444
1/20 1.27E-02 0.33 0.467 1.82E-03 1.90 0.414
1/40 9.47E-03 0.42 0.478 4.70E-04 1.95 0.491
1/80 6.69E-03 0.50 0.619 1.19E-04 1.98 0.570
D-BDF3
($ \ell $=0.9)		 1/10 1.48E-02 – 0.437 3.98E-04 – 0.428
1/20 1.03E-02 0.53 0.437 5.08E-05 2.97 0.443
1/40 6.74E-03 0.60 0.494 5.76E-06 3.14 0.479
1/80 4.21E-03 0.68 0.582 5.20E-07 3.47 0.585
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Methods $ \tau $ $ E_U $ Rate CPU(s) $ E_C $ Rate CPU(s)
D-Euler
($ \ell $=0.1)		 1/10 2.66E-01 – 0.441 5.14E-02 – 0.435
1/20 2.90E-01 – 0.426 2.55E-02 1.01 0.461
1/40 3.05E-01 – 0.480 1.28E-02 1.00 0.496
1/80 3.10E-01 – 0.611 6.40E-03 1.00 0.611
D-BDF2
($ \ell $=0.5)		 1/10 5.54E-02 – 0.442 4.20E-03 – 0.435
1/20 4.97E-02 0.16 0.443 1.11E-03 1.92 0.447
1/40 4.23E-02 0.23 0.461 2.85E-04 1.96 0.481
1/80 3.43E-02 0.30 0.501 7.15E-05 1.99 0.810
D-BT-$ \theta $
($ \theta $=0.45, $ \ell $=0.3)		 1/10 1.55E-01 – 0.450 1.87E-03 – 0.428
1/20 1.50E-01 0.05 0.457 4.70E-04 2.00 0.461
1/40 1.38E-01 0.11 0.476 1.17E-04 2.00 0.498
1/80 1.23E-01 0.17 0.589 2.88E-05 2.02 0.605
D-BN-$ \theta $
($ \theta $=1, $ \ell $=0.8)		 1/10 1.59E-02 – 0.448 6.76E-03 – 0.444
1/20 1.27E-02 0.33 0.467 1.82E-03 1.90 0.414
1/40 9.47E-03 0.42 0.478 4.70E-04 1.95 0.491
1/80 6.69E-03 0.50 0.619 1.19E-04 1.98 0.570
D-BDF3
($ \ell $=0.9)		 1/10 1.48E-02 – 0.437 3.98E-04 – 0.428
1/20 1.03E-02 0.53 0.437 5.08E-05 2.97 0.443
1/40 6.74E-03 0.60 0.494 5.76E-06 3.14 0.479
1/80 4.21E-03 0.68 0.582 5.20E-07 3.47 0.585
Table 3.  Temporal convergence rates for methods with and without correction terms
Methods $ h $ $ E_U $ Rate CPU(s) $ E_C $ Rate CPU(s)
D-Euler 1/20 1.82E-02 – 1.516 5.18E-03 – 1.579
1/40 9.85E-03 0.89 3.526 2.64E-03 0.97 3.266
1/80 5.23E-03 0.91 7.992 1.33E-03 0.99 8.236
1/160 2.74E-03 0.93 33.652 6.65E-04 1.00 34.729
D-BDF2 1/20 5.32E-02 – 1.525 2.79E-04 – 1.507
1/40 3.28E-02 0.70 3.285 7.09E-05 1.97 3.203
1/80 1.96E-02 0.75 7.897 1.79E-05 1.99 8.142
1/160 1.19E-02 0.72 34.870 4.49E-06 1.99 32.841
D-BT-$ \theta $
($ \theta $=0.2)		 1/20 5.94E-02 – 1.468 1.95E-04 – 1.523
1/40 3.68E-02 0.69 3.254 4.97E-05 1.97 3.308
1/80 2.20E-02 0.74 7.876 1.25E-05 1.99 8.160
1/160 1.28E-02 0.78 34.141 3.14E-06 1.99 34.439
D-BN-$ \theta $
($ \theta $=0.5)		 1/20 5.44E-02 – 1.509 2.46E-04 – 1.517
1/40 3.35E-02 0.70 3.281 6.24E-05 1.98 3.479
1/80 1.99E-02 0.75 8.090 1.57E-05 1.99 7.998
1/160 1.20E-02 0.73 35.025 3.95E-06 1.99 34.751
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Methods $ h $ $ E_U $ Rate CPU(s) $ E_C $ Rate CPU(s)
D-Euler 1/20 1.82E-02 – 1.516 5.18E-03 – 1.579
1/40 9.85E-03 0.89 3.526 2.64E-03 0.97 3.266
1/80 5.23E-03 0.91 7.992 1.33E-03 0.99 8.236
1/160 2.74E-03 0.93 33.652 6.65E-04 1.00 34.729
D-BDF2 1/20 5.32E-02 – 1.525 2.79E-04 – 1.507
1/40 3.28E-02 0.70 3.285 7.09E-05 1.97 3.203
1/80 1.96E-02 0.75 7.897 1.79E-05 1.99 8.142
1/160 1.19E-02 0.72 34.870 4.49E-06 1.99 32.841
D-BT-$ \theta $
($ \theta $=0.2)		 1/20 5.94E-02 – 1.468 1.95E-04 – 1.523
1/40 3.68E-02 0.69 3.254 4.97E-05 1.97 3.308
1/80 2.20E-02 0.74 7.876 1.25E-05 1.99 8.160
1/160 1.28E-02 0.78 34.141 3.14E-06 1.99 34.439
D-BN-$ \theta $
($ \theta $=0.5)		 1/20 5.44E-02 – 1.509 2.46E-04 – 1.517
1/40 3.35E-02 0.70 3.281 6.24E-05 1.98 3.479
1/80 1.99E-02 0.75 8.090 1.57E-05 1.99 7.998
1/160 1.20E-02 0.73 35.025 3.95E-06 1.99 34.751
Table 1.  Temporal convergence rates for smooth solutions, $ h = \frac{1}{2000} $
Methods $ \tau $ $ E_T $ Rate CPU(s) $ E_S $ Rate CPU(s)
D-Euler 1/10 9.03E-02 – 0.401 9.10E-02 – 0.441
1/20 4.59E-02 0.98 0.513 4.60E-02 0.98 0.453
1/40 2.31E-02 0.99 0.554 2.31E-02 0.99 0.492
1/80 1.16E-02 1.00 0.688 1.16E-02 1.00 0.617
D-BDF2 1/10 1.48E-02 – 0.441 1.49E-02 – 0.426
1/20 3.98E-03 1.89 0.453 3.98E-03 1.90 0.435
1/40 1.03E-03 1.95 0.474 1.03E-03 1.95 0.468
1/80 2.62E-04 1.98 0.579 2.62E-04 1.98 0.607
D-BT-$ \theta $
($ \theta $=0.45)		 1/10 5.36E-03 – 0.405 5.41E-03 – 0.452
1/20 1.36E-03 1.97 0.454 1.37E-03 1.98 0.449
1/40 3.44E-04 1.99 0.516 3.44E-04 1.99 0.469
1/80 8.61E-05 2.00 0.604 8.61E-05 2.00 0.598
D-BN-$ \theta $
($ \theta $=1)		 1/10 2.63E-02 – 0.413 2.65E-02 – 0.438
1/20 7.17E-03 1.88 0.453 7.18E-03 1.88 0.437
1/40 1.87E-03 1.94 0.490 1.87E-03 1.94 0.494
1/80 4.76E-04 1.97 0.623 4.76E-04 1.97 0.583
D-BDF3 1/10 2.00E-03 – 0.473 2.03E-03 – 0.407
1/20 2.65E-04 2.92 0.475 2.66E-04 2.93 0.440
1/40 3.39E-05 2.97 0.496 3.39E-05 2.97 0.496
1/80 4.20E-06 3.01 0.611 4.20E-06 3.01 0.614
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Methods $ \tau $ $ E_T $ Rate CPU(s) $ E_S $ Rate CPU(s)
D-Euler 1/10 9.03E-02 – 0.401 9.10E-02 – 0.441
1/20 4.59E-02 0.98 0.513 4.60E-02 0.98 0.453
1/40 2.31E-02 0.99 0.554 2.31E-02 0.99 0.492
1/80 1.16E-02 1.00 0.688 1.16E-02 1.00 0.617
D-BDF2 1/10 1.48E-02 – 0.441 1.49E-02 – 0.426
1/20 3.98E-03 1.89 0.453 3.98E-03 1.90 0.435
1/40 1.03E-03 1.95 0.474 1.03E-03 1.95 0.468
1/80 2.62E-04 1.98 0.579 2.62E-04 1.98 0.607
D-BT-$ \theta $
($ \theta $=0.45)		 1/10 5.36E-03 – 0.405 5.41E-03 – 0.452
1/20 1.36E-03 1.97 0.454 1.37E-03 1.98 0.449
1/40 3.44E-04 1.99 0.516 3.44E-04 1.99 0.469
1/80 8.61E-05 2.00 0.604 8.61E-05 2.00 0.598
D-BN-$ \theta $
($ \theta $=1)		 1/10 2.63E-02 – 0.413 2.65E-02 – 0.438
1/20 7.17E-03 1.88 0.453 7.18E-03 1.88 0.437
1/40 1.87E-03 1.94 0.490 1.87E-03 1.94 0.494
1/80 4.76E-04 1.97 0.623 4.76E-04 1.97 0.583
D-BDF3 1/10 2.00E-03 – 0.473 2.03E-03 – 0.407
1/20 2.65E-04 2.92 0.475 2.66E-04 2.93 0.440
1/40 3.39E-05 2.97 0.496 3.39E-05 2.97 0.496
1/80 4.20E-06 3.01 0.611 4.20E-06 3.01 0.614
Table 4.  Temporal convergence rates for methods with and without correction terms
Methods $ h $ $ E_U $ CPU(s) $ E_C $ CPU(s)
D-Euler 1/20 1.8231582E-02 1.010 1.40274270E-02 1.054
1/40 9.8461370E-03 2.195 2.16534296E-03 2.278
1/80 5.2308982E-03 5.646 1.18484214E-03 5.556
1/160 2.7366136E-03 22.973 6.24937949E-04 23.143
D-BDF2 1/20 5.3244469E-02 0.982 1.40274270E-02 0.998
1/40 3.2848868E-02 2.202 2.16534296E-03 2.233
1/80 1.9587693E-02 5.452 2.69407811E-04 5.554
1/160 1.1890696E-02 23.290 4.62329665E-06 23.297
D-BT-$ \theta $
($ \theta $=0.2)		 1/20 5.9360151E-02 0.971 1.40274270E-02 0.998
1/40 3.6795394E-02 2.226 2.16534295E-03 2.187
1/80 2.1969727E-02 5.443 2.69407811E-04 5.579
1/160 1.2752778E-02 23.064 4.62329717E-06 23.388
D-BN-$ \theta $
($ \theta $=0.5)		 1/20 5.4412292E-02 0.959 1.40274270E-02 1.013
1/40 3.3527144E-02 2.210 2.16534296E-03 2.199
1/80 1.9922404E-02 5.520 2.69407811E-04 5.530
1/160 1.2015071E-02 23.077 4.62329657E-06 23.571
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Methods $ h $ $ E_U $ CPU(s) $ E_C $ CPU(s)
D-Euler 1/20 1.8231582E-02 1.010 1.40274270E-02 1.054
1/40 9.8461370E-03 2.195 2.16534296E-03 2.278
1/80 5.2308982E-03 5.646 1.18484214E-03 5.556
1/160 2.7366136E-03 22.973 6.24937949E-04 23.143
D-BDF2 1/20 5.3244469E-02 0.982 1.40274270E-02 0.998
1/40 3.2848868E-02 2.202 2.16534296E-03 2.233
1/80 1.9587693E-02 5.452 2.69407811E-04 5.554
1/160 1.1890696E-02 23.290 4.62329665E-06 23.297
D-BT-$ \theta $
($ \theta $=0.2)		 1/20 5.9360151E-02 0.971 1.40274270E-02 0.998
1/40 3.6795394E-02 2.226 2.16534295E-03 2.187
1/80 2.1969727E-02 5.443 2.69407811E-04 5.579
1/160 1.2752778E-02 23.064 4.62329717E-06 23.388
D-BN-$ \theta $
($ \theta $=0.5)		 1/20 5.4412292E-02 0.959 1.40274270E-02 1.013
1/40 3.3527144E-02 2.210 2.16534296E-03 2.199
1/80 1.9922404E-02 5.520 2.69407811E-04 5.530
1/160 1.2015071E-02 23.077 4.62329657E-06 23.571
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