April  2022, 15(4): 773-795. doi: 10.3934/dcdss.2021044

Explicit multistep stochastic characteristic approximation methods for forward backward stochastic differential equations

1. 

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

2. 

College of Science, National University of Defense Technology, Changsha, Hunan 410073, China

* Corresponding author: wdzhao@sdu.edu.cn

Received  December 2020 Revised  January 2021 Published  April 2022 Early access  April 2021

Fund Project: This research is partially supported by the NSF of China (Nos. 12071261, 12001539, 11831010, 11871068), the Science Challenge Project (No. TZ2018001), the national key basic research program (No. 2018YFA0703903, No. 2018YFB0704304), the NSF of Hunan Province (No. 2020JJ5647), and China Postdoctoral Science Foundation (No. 2019TQ0073)

In this work, by combining with stochastic approximation methods, we proposed a new explicit multistep scheme for solving the forward backward stochastic differential equations. Compared with the one constructed by using derivative approximation method, the new one covers the approximation of the stochastic part and is more accurate and easier to realize. Several numerical tests are presented to show the stability and effectiveness of the proposed scheme.

Citation: Ying Liu, Yabing Sun, Weidong Zhao. Explicit multistep stochastic characteristic approximation methods for forward backward stochastic differential equations. Discrete and Continuous Dynamical Systems - S, 2022, 15 (4) : 773-795. doi: 10.3934/dcdss.2021044
References:
[1]

M. Abramowitz and I. Stegun, Handbook of Mathematical Functions, Dover, New York, 1972.

[2]

J. C. Butcher, Numerical Methods for Ordinary Differential Equations, John Wiley, Chichester, UK, Ltd, 2008. doi: 10.1002/9780470753767.

[3]

R. L. Burden and J. D. Faires, Numerical Analysis, 7th ed., Higher Education Press/Cengage Learning, Inc, 2001. Available from: https://www.scirp.org/reference/referencespapers.aspx?referenceid=696332.

[4]

R. BuckdahnJ. Li and S. Peng, Mean-field backward stochastic differential equations and related partial differential equations, Stochastic Process. Appl., 119 (2009), 3133-3154.  doi: 10.1016/j.spa.2009.05.002.

[5]

F. Delarue and S. Menozzi, An interpolated stochastic algorithm for quasi-linear pdes, Math. Comput., 77 (2008), 125-158.  doi: 10.1090/S0025-5718-07-02008-X.

[6]

A. FahimN. Touzi and X. Warin, A probabilistic numerical method for fully nonlinear parabolic PDEs, Ann. Appl. Probab., 21 (2011), 1322-1364.  doi: 10.1214/10-AAP723.

[7]

Y. FuW. Zhao and T. Zhou, Multistep schemes for forward backward stochastic differential equations with jumps, J. Sci. Comput., 69 (2016), 651-672.  doi: 10.1007/s10915-016-0212-y.

[8]

E. Gobet and C. Labart, Error expansion for the discretization of backward stochastic differential equations, Stochastic Process. Appl., 117 (2007), 803-829.  doi: 10.1016/j.spa.2006.10.007.

[9]

S. Hamadene and J. P. Lepeltier, Zero-sum stochastic differential games and backward equations, Systems Control Lett., 24 (1995), 259-263.  doi: 10.1016/0167-6911(94)00011-J.

[10]

Y. Hu and S. Peng, Solution of forward-backward stochastic differential equations, Probab. Theory Related Fields, 103 (1995), 273-283.  doi: 10.1007/BF01204218.

[11]

I. KharroubiN. Langrené and H. Pham, Discrete time approximation of fully nonlinear HJB equations via BSDEs with nonpositive jumps, Ann. Appl. Probab., 25 (2015), 2301-2338.  doi: 10.1214/14-AAP1049.

[12]

Y. LiuY. Sun and W. Zhao, A fully discrete explicit multistep scheme for solving coupled forward backward stochastic differential equations, Adv. Appl. Math. Mech., 12 (2020), 643-663.  doi: 10.4208/aamm.OA-2019-0079.

[13]

G. N. Milstein and M. V. Tretyakov, Discretization of forward-backward stochastic differential equations and related quasi-linear parabolic equations, SIAM J. Numer. Anal., 27 (2007), 24-44.  doi: 10.1093/imanum/drl019.

[14]

B. Øksendal, Stochastic Differential Equations, Springer-Verlag, Berlin, 2003.

[15]

S. Peng, Probabilistic interpretation for systems of quasilinear parabolic partial differential equations, Stochastics Stochastics Rep., 37 (1991), 61-74.  doi: 10.1080/17442509108833727.

[16]

S. Peng, A general stochastic maximum principle for optimal control problems, SIAM J. Control Optim., 28 (1990), 966-979.  doi: 10.1137/0328054.

[17]

S. Peng, A linear approximation algorithm using BSDE, Pacific Economic Review, 4 (1999), 285-292.  doi: 10.1111/1468-0106.00079.

[18]

E. Pardoux and S. Tang, Forward-backward stochastic differential equations and quasilinear parabolic PDEs, Probab. Theory Related Fields, 114 (1999), 123-150.  doi: 10.1007/s004409970001.

[19]

S. Peng and Z. Wu, Fully coupled forward-backward stochastic differential equations and applications to optimal control, SIAM J. Control Optim., 37 (1999), 825-843.  doi: 10.1137/S0363012996313549.

[20]

E. Pardoux and S. Peng, Adapted solution of a backward stochastic differential equation, Systems Control Lett., 14 (1990), 55-61.  doi: 10.1016/0167-6911(90)90082-6.

[21]

Y. Sun and W. Zhao, An explicit second-order numerical scheme for mean-field forward backward stochastic differential equations, Numer. Algorithms, 84 (2020), 253-283.  doi: 10.1007/s11075-019-00754-2.

[22]

Y. SunW. Zhao and T. Zhou, Explicit $\theta$-scheme for solving mean-field backward stochastic differential equations, SIAM J. Numer. Anal., 56 (2018), 2672-2697.  doi: 10.1137/17M1161944.

[23]

J. YangW. Zhao and T. Zhou, Explicit deferred correction methods for second-order forward backward stochastic differential equations, J. Sci. Comput., 79 (2019), 1409-1432.  doi: 10.1007/s10915-018-00896-w.

[24]

W. ZhaoL. Chen and S. Peng, A new kind of accurate numerical method for backward stochastic differential equations, SIAM J. Sci. Comput., 28 (2006), 1563-1581.  doi: 10.1137/05063341X.

[25]

W. ZhaoY. Fu and T. Zhou, New kinds of high-order multistep schemes for coupled forward backward stochastic differential equations, SIAM J. Sci. Comput., 36 (2014), 1731-1751.  doi: 10.1137/130941274.

[26]

W. ZhaoY. Li and Y. Fu, Second-order schemes for solving decoupled forward backward stochastic differential equations, Sci. China Math., 57 (2014), 665-686.  doi: 10.1007/s11425-013-4764-0.

[27]

W. ZhaoJ. Wang and S. Peng, Error estimates of the $\theta$-scheme for backward stochastic differential equations, Discrete Contin. Dyn. Syst. Ser. B, 12 (2009), 905-924.  doi: 10.3934/dcdsb.2009.12.905.

[28]

W. Zhao, W. Zhang and L. Ju, A numerical method and its error estimates for the decoupled forward-backward stochastic differential equations, Commun. Comput. Phys., 15 (2014), 618-646. doi: 10.4208/cicp.280113.190813a.

[29]

W. ZhaoT. Zhou and T. Kong, High order numerical schemes for second-order FBSDEs with applications to stochastic optimal control, Commun. Comput. Phys., 21 (2015), 808-834. 

show all references

References:
[1]

M. Abramowitz and I. Stegun, Handbook of Mathematical Functions, Dover, New York, 1972.

[2]

J. C. Butcher, Numerical Methods for Ordinary Differential Equations, John Wiley, Chichester, UK, Ltd, 2008. doi: 10.1002/9780470753767.

[3]

R. L. Burden and J. D. Faires, Numerical Analysis, 7th ed., Higher Education Press/Cengage Learning, Inc, 2001. Available from: https://www.scirp.org/reference/referencespapers.aspx?referenceid=696332.

[4]

R. BuckdahnJ. Li and S. Peng, Mean-field backward stochastic differential equations and related partial differential equations, Stochastic Process. Appl., 119 (2009), 3133-3154.  doi: 10.1016/j.spa.2009.05.002.

[5]

F. Delarue and S. Menozzi, An interpolated stochastic algorithm for quasi-linear pdes, Math. Comput., 77 (2008), 125-158.  doi: 10.1090/S0025-5718-07-02008-X.

[6]

A. FahimN. Touzi and X. Warin, A probabilistic numerical method for fully nonlinear parabolic PDEs, Ann. Appl. Probab., 21 (2011), 1322-1364.  doi: 10.1214/10-AAP723.

[7]

Y. FuW. Zhao and T. Zhou, Multistep schemes for forward backward stochastic differential equations with jumps, J. Sci. Comput., 69 (2016), 651-672.  doi: 10.1007/s10915-016-0212-y.

[8]

E. Gobet and C. Labart, Error expansion for the discretization of backward stochastic differential equations, Stochastic Process. Appl., 117 (2007), 803-829.  doi: 10.1016/j.spa.2006.10.007.

[9]

S. Hamadene and J. P. Lepeltier, Zero-sum stochastic differential games and backward equations, Systems Control Lett., 24 (1995), 259-263.  doi: 10.1016/0167-6911(94)00011-J.

[10]

Y. Hu and S. Peng, Solution of forward-backward stochastic differential equations, Probab. Theory Related Fields, 103 (1995), 273-283.  doi: 10.1007/BF01204218.

[11]

I. KharroubiN. Langrené and H. Pham, Discrete time approximation of fully nonlinear HJB equations via BSDEs with nonpositive jumps, Ann. Appl. Probab., 25 (2015), 2301-2338.  doi: 10.1214/14-AAP1049.

[12]

Y. LiuY. Sun and W. Zhao, A fully discrete explicit multistep scheme for solving coupled forward backward stochastic differential equations, Adv. Appl. Math. Mech., 12 (2020), 643-663.  doi: 10.4208/aamm.OA-2019-0079.

[13]

G. N. Milstein and M. V. Tretyakov, Discretization of forward-backward stochastic differential equations and related quasi-linear parabolic equations, SIAM J. Numer. Anal., 27 (2007), 24-44.  doi: 10.1093/imanum/drl019.

[14]

B. Øksendal, Stochastic Differential Equations, Springer-Verlag, Berlin, 2003.

[15]

S. Peng, Probabilistic interpretation for systems of quasilinear parabolic partial differential equations, Stochastics Stochastics Rep., 37 (1991), 61-74.  doi: 10.1080/17442509108833727.

[16]

S. Peng, A general stochastic maximum principle for optimal control problems, SIAM J. Control Optim., 28 (1990), 966-979.  doi: 10.1137/0328054.

[17]

S. Peng, A linear approximation algorithm using BSDE, Pacific Economic Review, 4 (1999), 285-292.  doi: 10.1111/1468-0106.00079.

[18]

E. Pardoux and S. Tang, Forward-backward stochastic differential equations and quasilinear parabolic PDEs, Probab. Theory Related Fields, 114 (1999), 123-150.  doi: 10.1007/s004409970001.

[19]

S. Peng and Z. Wu, Fully coupled forward-backward stochastic differential equations and applications to optimal control, SIAM J. Control Optim., 37 (1999), 825-843.  doi: 10.1137/S0363012996313549.

[20]

E. Pardoux and S. Peng, Adapted solution of a backward stochastic differential equation, Systems Control Lett., 14 (1990), 55-61.  doi: 10.1016/0167-6911(90)90082-6.

[21]

Y. Sun and W. Zhao, An explicit second-order numerical scheme for mean-field forward backward stochastic differential equations, Numer. Algorithms, 84 (2020), 253-283.  doi: 10.1007/s11075-019-00754-2.

[22]

Y. SunW. Zhao and T. Zhou, Explicit $\theta$-scheme for solving mean-field backward stochastic differential equations, SIAM J. Numer. Anal., 56 (2018), 2672-2697.  doi: 10.1137/17M1161944.

[23]

J. YangW. Zhao and T. Zhou, Explicit deferred correction methods for second-order forward backward stochastic differential equations, J. Sci. Comput., 79 (2019), 1409-1432.  doi: 10.1007/s10915-018-00896-w.

[24]

W. ZhaoL. Chen and S. Peng, A new kind of accurate numerical method for backward stochastic differential equations, SIAM J. Sci. Comput., 28 (2006), 1563-1581.  doi: 10.1137/05063341X.

[25]

W. ZhaoY. Fu and T. Zhou, New kinds of high-order multistep schemes for coupled forward backward stochastic differential equations, SIAM J. Sci. Comput., 36 (2014), 1731-1751.  doi: 10.1137/130941274.

[26]

W. ZhaoY. Li and Y. Fu, Second-order schemes for solving decoupled forward backward stochastic differential equations, Sci. China Math., 57 (2014), 665-686.  doi: 10.1007/s11425-013-4764-0.

[27]

W. ZhaoJ. Wang and S. Peng, Error estimates of the $\theta$-scheme for backward stochastic differential equations, Discrete Contin. Dyn. Syst. Ser. B, 12 (2009), 905-924.  doi: 10.3934/dcdsb.2009.12.905.

[28]

W. Zhao, W. Zhang and L. Ju, A numerical method and its error estimates for the decoupled forward-backward stochastic differential equations, Commun. Comput. Phys., 15 (2014), 618-646. doi: 10.4208/cicp.280113.190813a.

[29]

W. ZhaoT. Zhou and T. Kong, High order numerical schemes for second-order FBSDEs with applications to stochastic optimal control, Commun. Comput. Phys., 21 (2015), 808-834. 

Table 1.  The errors of Scheme 4.1 with $ b = 1 $, $ \sigma = 1 $ for Ex. 5.1
k = 1 Case 1: $ \bar{\bar{b}} = 0, \bar{\bar{\sigma}} = 0 $ Case 2: $ \bar{\bar{b}} = b, \bar{\bar{\sigma}} = 0 $ Case 3: $ \bar{\bar{b}} = 0, \bar{\bar{\sigma}} = \sigma $ Case 4: $ \bar{\bar{b}} = b, \bar{\bar{\sigma}} = \sigma $
$ \Delta t $ $ |Y_0-Y^0| $ $ |Z_0-Z^0| $ $ |Y_0-Y^0| $ $ |Z_0-Z^0| $ $ |Y_0-Y^0| $ $ |Z_0-Z^0| $ $ |Y_0-Y^0| $ $ |Z_0-Z^0| $
$ \frac{1}{15} $ 5.565e-02 1.152e-01 3.385e-02 6.502e-02 4.796e-02 1.732e-01 2.146e-02 1.537e-01
$ \frac{1}{20} $ 4.241e-02 9.257e-02 2.602e-02 4.585e-02 3.774e-02 1.336e-01 1.520e-02 1.175e-01
$ \frac{1}{25} $ 3.422e-02 7.616e-02 2.111e-02 3.400e-02 3.091e-02 1.092e-01 1.170e-02 9.380e-02
$ \frac{1}{30} $ 2.857e-02 6.377e-02 1.770e-02 2.692e-02 2.596e-02 9.268e-02 9.542e-03 7.724e-02
k = 2 Case 1: $ \bar{\bar{b}} = 0, \bar{\bar{\sigma}} = 0 $ Case 2: $ \bar{\bar{b}} = b, \bar{\bar{\sigma}} = 0 $ Case 3: $ \bar{\bar{b}} = 0, \bar{\bar{\sigma}} = \sigma $ Case 4: $ \bar{\bar{b}} = b, \bar{\bar{\sigma}} = \sigma $
$ \Delta t $ $ |Y_0-Y^0| $ $ |Z_0-Z^0| $ $ |Y_0-Y^0| $ $ |Z_0-Z^0| $ $ |Y_0-Y^0| $ $ |Z_0-Z^0| $ $ |Y_0-Y^0| $ $ |Z_0-Z^0| $
$ \frac{1}{15} $ 1.122e-02 9.952e-03 4.370e-03 1.590e-02 1.355e-02 3.715e-02 4.945e-03 3.855e-02
$ \frac{1}{20} $ 6.722e-03 5.644e-03 2.781e-03 8.588e-03 8.000e-03 2.395e-02 2.719e-03 2.267e-02
$ \frac{1}{25} $ 4.505e-03 3.557e-03 1.958e-03 5.055e-03 5.221e-03 1.648e-02 1.683e-03 1.455e-02
$ \frac{1}{30} $ 3.248e-03 2.443e-03 1.471e-03 3.671e-03 3.662e-03 1.197e-02 1.120e-03 9.942e-03
k = 3 Case 1: $ \bar{\bar{b}} = 0, \bar{\bar{\sigma}} = 0 $ Case 2: $ \bar{\bar{b}} = b, \bar{\bar{\sigma}} = 0 $ Case 3: $ \bar{\bar{b}} = 0, \bar{\bar{\sigma}} = \sigma $ Case 4: $ \bar{\bar{b}} = b, \bar{\bar{\sigma}} = \sigma $
$ \Delta t $ $ |Y_0-Y^0| $ $ |Z_0-Z^0| $ $ |Y_0-Y^0| $ $ |Z_0-Z^0| $ $ |Y_0-Y^0| $ $ |Z_0-Z^0| $ $ |Y_0-Y^0| $ $ |Z_0-Z^0| $
$ \frac{1}{15} $ 1.518e-03 5.198e-03 7.993e-04 5.353e-03 4.316e-03 1.266e-02 1.525e-03 1.234e-02
$ \frac{1}{20} $ 7.410e-04 2.531e-03 4.301e-04 2.404e-03 2.093e-03 6.538e-03 6.871e-04 5.892e-03
$ \frac{1}{25} $ 4.196e-04 1.308e-03 2.707e-04 1.164e-03 1.162e-03 3.760e-03 3.483e-04 3.119e-03
$ \frac{1}{30} $ 2.611e-04 7.228e-04 1.838e-04 6.463e-04 7.088e-04 2.337e-03 1.944e-04 1.798e-03
k = 4 Case 1: $ \bar{\bar{b}} = 0, \bar{\bar{\sigma}} = 0 $ Case 2: $ \bar{\bar{b}} = b, \bar{\bar{\sigma}} = 0 $ Case 3: $ \bar{\bar{b}} = 0, \bar{\bar{\sigma}} = \sigma $ Case 4: $ \bar{\bar{b}} = b, \bar{\bar{\sigma}} = \sigma $
$ \Delta t $ $ |Y_0-Y^0| $ $ |Z_0-Z^0| $ $ |Y_0-Y^0| $ $ |Z_0-Z^0| $ $ |Y_0-Y^0| $ $ |Z_0-Z^0| $ $ |Y_0-Y^0| $ $ |Z_0-Z^0| $
$ \frac{1}{15} $ 1.332e-04 2.656e-03 1.880e-04 2.142e-03 1.442e-03 4.931e-03 5.740e-04 4.726e-03
$ \frac{1}{20} $ 5.901e-05 1.020e-03 8.249e-05 8.381e-04 5.794e-04 1.943e-03 2.188e-04 1.914e-03
$ \frac{1}{25} $ 3.039e-05 4.278e-04 4.547e-05 3.501e-04 2.743e-04 9.415e-04 9.360e-05 8.531e-04
$ \frac{1}{30} $ 1.741e-05 2.000e-04 2.679e-05 1.623e-04 1.443e-04 4.965e-04 4.636e-05 4.259e-04
k = 5 Case 1: $ \bar{\bar{b}} = 0, \bar{\bar{\sigma}} = 0 $ Case 2: $ \bar{\bar{b}} = b, \bar{\bar{\sigma}} = 0 $ Case 3: $ \bar{\bar{b}} = 0, \bar{\bar{\sigma}} = \sigma $ Case 4: $ \bar{\bar{b}} = b, \bar{\bar{\sigma}} = \sigma $
$ \Delta t $ $ |Y_0-Y^0| $ $ |Z_0-Z^0| $ $ |Y_0-Y^0| $ $ |Z_0-Z^0| $ $ |Y_0-Y^0| $ $ |Z_0-Z^0| $ $ |Y_0-Y^0| $ $ |Z_0-Z^0| $
$ \frac{1}{15} $ 5.377e-05 9.346e-04 5.914e-05 8.994e-04 5.077e-04 2.010e-03 2.425e-04 2.011e-03
$ \frac{1}{20} $ 2.249e-05 3.264e-04 1.837e-05 3.228e-04 1.724e-04 6.706e-04 8.127e-05 7.200e-04
$ \frac{1}{25} $ 1.061e-05 1.220e-04 9.341e-06 1.209e-04 7.104e-05 2.676e-04 2.789e-05 2.708e-04
$ \frac{1}{30} $ 5.107e-06 5.131e-05 5.008e-06 4.895e-05 3.315e-05 1.223e-04 1.163e-05 1.153e-04
k = 6 Case 1: $ \bar{\bar{b}} = 0, \bar{\bar{\sigma}} = 0 $ Case 2: $ \bar{\bar{b}} = b, \bar{\bar{\sigma}} = 0 $ Case 3: $ \bar{\bar{b}} = 0, \bar{\bar{\sigma}} = \sigma $ Case 4: $ \bar{\bar{b}} = b, \bar{\bar{\sigma}} = \sigma $
$ \Delta t $ $ |Y_0-Y^0| $ $ |Z_0-Z^0| $ $ |Y_0-Y^0| $ $ |Z_0-Z^0| $ $ |Y_0-Y^0| $ $ |Z_0-Z^0| $ $ |Y_0-Y^0| $ $ |Z_0-Z^0| $
$ \frac{1}{15} $ 2.551e-05 3.935e-04 1.878e-05 4.139e-04 1.964e-04 8.533e-04 1.217e-04 9.782e-04
$ \frac{1}{20} $ 6.342e-06 1.318e-04 4.988e-06 1.370e-04 5.911e-05 2.457e-04 3.251e-05 2.984e-04
$ \frac{1}{25} $ 2.819e-06 4.485e-05 2.455e-06 4.635e-05 2.144e-05 8.782e-05 9.493e-06 9.755e-05
$ \frac{1}{30} $ 1.264e-06 1.644e-05 1.150e-06 1.682e-05 8.642e-06 3.459e-05 3.280e-06 3.570e-05
k = 1 Case 1: $ \bar{\bar{b}} = 0, \bar{\bar{\sigma}} = 0 $ Case 2: $ \bar{\bar{b}} = b, \bar{\bar{\sigma}} = 0 $ Case 3: $ \bar{\bar{b}} = 0, \bar{\bar{\sigma}} = \sigma $ Case 4: $ \bar{\bar{b}} = b, \bar{\bar{\sigma}} = \sigma $
$ \Delta t $ $ |Y_0-Y^0| $ $ |Z_0-Z^0| $ $ |Y_0-Y^0| $ $ |Z_0-Z^0| $ $ |Y_0-Y^0| $ $ |Z_0-Z^0| $ $ |Y_0-Y^0| $ $ |Z_0-Z^0| $
$ \frac{1}{15} $ 5.565e-02 1.152e-01 3.385e-02 6.502e-02 4.796e-02 1.732e-01 2.146e-02 1.537e-01
$ \frac{1}{20} $ 4.241e-02 9.257e-02 2.602e-02 4.585e-02 3.774e-02 1.336e-01 1.520e-02 1.175e-01
$ \frac{1}{25} $ 3.422e-02 7.616e-02 2.111e-02 3.400e-02 3.091e-02 1.092e-01 1.170e-02 9.380e-02
$ \frac{1}{30} $ 2.857e-02 6.377e-02 1.770e-02 2.692e-02 2.596e-02 9.268e-02 9.542e-03 7.724e-02
k = 2 Case 1: $ \bar{\bar{b}} = 0, \bar{\bar{\sigma}} = 0 $ Case 2: $ \bar{\bar{b}} = b, \bar{\bar{\sigma}} = 0 $ Case 3: $ \bar{\bar{b}} = 0, \bar{\bar{\sigma}} = \sigma $ Case 4: $ \bar{\bar{b}} = b, \bar{\bar{\sigma}} = \sigma $
$ \Delta t $ $ |Y_0-Y^0| $ $ |Z_0-Z^0| $ $ |Y_0-Y^0| $ $ |Z_0-Z^0| $ $ |Y_0-Y^0| $ $ |Z_0-Z^0| $ $ |Y_0-Y^0| $ $ |Z_0-Z^0| $
$ \frac{1}{15} $ 1.122e-02 9.952e-03 4.370e-03 1.590e-02 1.355e-02 3.715e-02 4.945e-03 3.855e-02
$ \frac{1}{20} $ 6.722e-03 5.644e-03 2.781e-03 8.588e-03 8.000e-03 2.395e-02 2.719e-03 2.267e-02
$ \frac{1}{25} $ 4.505e-03 3.557e-03 1.958e-03 5.055e-03 5.221e-03 1.648e-02 1.683e-03 1.455e-02
$ \frac{1}{30} $ 3.248e-03 2.443e-03 1.471e-03 3.671e-03 3.662e-03 1.197e-02 1.120e-03 9.942e-03
k = 3 Case 1: $ \bar{\bar{b}} = 0, \bar{\bar{\sigma}} = 0 $ Case 2: $ \bar{\bar{b}} = b, \bar{\bar{\sigma}} = 0 $ Case 3: $ \bar{\bar{b}} = 0, \bar{\bar{\sigma}} = \sigma $ Case 4: $ \bar{\bar{b}} = b, \bar{\bar{\sigma}} = \sigma $
$ \Delta t $ $ |Y_0-Y^0| $ $ |Z_0-Z^0| $ $ |Y_0-Y^0| $ $ |Z_0-Z^0| $ $ |Y_0-Y^0| $ $ |Z_0-Z^0| $ $ |Y_0-Y^0| $ $ |Z_0-Z^0| $
$ \frac{1}{15} $ 1.518e-03 5.198e-03 7.993e-04 5.353e-03 4.316e-03 1.266e-02 1.525e-03 1.234e-02
$ \frac{1}{20} $ 7.410e-04 2.531e-03 4.301e-04 2.404e-03 2.093e-03 6.538e-03 6.871e-04 5.892e-03
$ \frac{1}{25} $ 4.196e-04 1.308e-03 2.707e-04 1.164e-03 1.162e-03 3.760e-03 3.483e-04 3.119e-03
$ \frac{1}{30} $ 2.611e-04 7.228e-04 1.838e-04 6.463e-04 7.088e-04 2.337e-03 1.944e-04 1.798e-03
k = 4 Case 1: $ \bar{\bar{b}} = 0, \bar{\bar{\sigma}} = 0 $ Case 2: $ \bar{\bar{b}} = b, \bar{\bar{\sigma}} = 0 $ Case 3: $ \bar{\bar{b}} = 0, \bar{\bar{\sigma}} = \sigma $ Case 4: $ \bar{\bar{b}} = b, \bar{\bar{\sigma}} = \sigma $
$ \Delta t $ $ |Y_0-Y^0| $ $ |Z_0-Z^0| $ $ |Y_0-Y^0| $ $ |Z_0-Z^0| $ $ |Y_0-Y^0| $ $ |Z_0-Z^0| $ $ |Y_0-Y^0| $ $ |Z_0-Z^0| $
$ \frac{1}{15} $ 1.332e-04 2.656e-03 1.880e-04 2.142e-03 1.442e-03 4.931e-03 5.740e-04 4.726e-03
$ \frac{1}{20} $ 5.901e-05 1.020e-03 8.249e-05 8.381e-04 5.794e-04 1.943e-03 2.188e-04 1.914e-03
$ \frac{1}{25} $ 3.039e-05 4.278e-04 4.547e-05 3.501e-04 2.743e-04 9.415e-04 9.360e-05 8.531e-04
$ \frac{1}{30} $ 1.741e-05 2.000e-04 2.679e-05 1.623e-04 1.443e-04 4.965e-04 4.636e-05 4.259e-04
k = 5 Case 1: $ \bar{\bar{b}} = 0, \bar{\bar{\sigma}} = 0 $ Case 2: $ \bar{\bar{b}} = b, \bar{\bar{\sigma}} = 0 $ Case 3: $ \bar{\bar{b}} = 0, \bar{\bar{\sigma}} = \sigma $ Case 4: $ \bar{\bar{b}} = b, \bar{\bar{\sigma}} = \sigma $
$ \Delta t $ $ |Y_0-Y^0| $ $ |Z_0-Z^0| $ $ |Y_0-Y^0| $ $ |Z_0-Z^0| $ $ |Y_0-Y^0| $ $ |Z_0-Z^0| $ $ |Y_0-Y^0| $ $ |Z_0-Z^0| $
$ \frac{1}{15} $ 5.377e-05 9.346e-04 5.914e-05 8.994e-04 5.077e-04 2.010e-03 2.425e-04 2.011e-03
$ \frac{1}{20} $ 2.249e-05 3.264e-04 1.837e-05 3.228e-04 1.724e-04 6.706e-04 8.127e-05 7.200e-04
$ \frac{1}{25} $ 1.061e-05 1.220e-04 9.341e-06 1.209e-04 7.104e-05 2.676e-04 2.789e-05 2.708e-04
$ \frac{1}{30} $ 5.107e-06 5.131e-05 5.008e-06 4.895e-05 3.315e-05 1.223e-04 1.163e-05 1.153e-04
k = 6 Case 1: $ \bar{\bar{b}} = 0, \bar{\bar{\sigma}} = 0 $ Case 2: $ \bar{\bar{b}} = b, \bar{\bar{\sigma}} = 0 $ Case 3: $ \bar{\bar{b}} = 0, \bar{\bar{\sigma}} = \sigma $ Case 4: $ \bar{\bar{b}} = b, \bar{\bar{\sigma}} = \sigma $
$ \Delta t $ $ |Y_0-Y^0| $ $ |Z_0-Z^0| $ $ |Y_0-Y^0| $ $ |Z_0-Z^0| $ $ |Y_0-Y^0| $ $ |Z_0-Z^0| $ $ |Y_0-Y^0| $ $ |Z_0-Z^0| $
$ \frac{1}{15} $ 2.551e-05 3.935e-04 1.878e-05 4.139e-04 1.964e-04 8.533e-04 1.217e-04 9.782e-04
$ \frac{1}{20} $ 6.342e-06 1.318e-04 4.988e-06 1.370e-04 5.911e-05 2.457e-04 3.251e-05 2.984e-04
$ \frac{1}{25} $ 2.819e-06 4.485e-05 2.455e-06 4.635e-05 2.144e-05 8.782e-05 9.493e-06 9.755e-05
$ \frac{1}{30} $ 1.264e-06 1.644e-05 1.150e-06 1.682e-05 8.642e-06 3.459e-05 3.280e-06 3.570e-05
Table 2.  The errors of Scheme 4.1 with $ b = 5 $, $ \sigma = 1 $ for Ex. 5.1
k = 1 Case 1: $ \bar{\bar{b}} = 0, \bar{\bar{\sigma}} = 0 $ Case 2: $ \bar{\bar{b}} = b, \bar{\bar{\sigma}} = 0 $ Case 3: $ \bar{\bar{b}} = 0, \bar{\bar{\sigma}} = \sigma $ Case 4: $ \bar{\bar{b}} = b, \bar{\bar{\sigma}} = \sigma $
$ \Delta t $ $ |Y_0-Y^0| $ $ |Z_0-Z^0| $ $ |Y_0-Y^0| $ $ |Z_0-Z^0| $ $ |Y_0-Y^0| $ $ |Z_0-Z^0| $ $ |Y_0-Y^0| $ $ |Z_0-Z^0| $
$ \frac{1}{15} $ 2.182e-01 4.303e-01 3.216e-02 6.829e-02 1.971e-01 3.969e-01 2.207e-02 1.539e-01
$ \frac{1}{20} $ 1.651e-01 3.578e-01 2.661e-02 4.673e-02 1.557e-01 3.074e-01 1.508e-02 1.177e-01
$ \frac{1}{25} $ 1.302e-01 3.013e-01 2.216e-02 3.389e-02 1.269e-01 2.660e-01 1.147e-02 9.416e-02
$ \frac{1}{30} $ 1.058e-01 2.592e-01 1.909e-02 2.865e-02 1.061e-01 2.331e-01 8.924e-03 7.698e-02
k = 2 Case 1: $ \bar{\bar{b}} = 0, \bar{\bar{\sigma}} = 0 $ Case 2: $ \bar{\bar{b}} = b, \bar{\bar{\sigma}} = 0 $ Case 3: $ \bar{\bar{b}} = 0, \bar{\bar{\sigma}} = \sigma $ Case 4: $ \bar{\bar{b}} = b, \bar{\bar{\sigma}} = \sigma $
$ \Delta t $ $ |Y_0-Y^0| $ $ |Z_0-Z^0| $ $ |Y_0-Y^0| $ $ |Z_0-Z^0| $ $ |Y_0-Y^0| $ $ |Z_0-Z^0| $ $ |Y_0-Y^0| $ $ |Z_0-Z^0| $
$ \frac{1}{15} $ 1.586e-01 3.414e-01 4.025e-03 1.811e-02 1.457e-01 2.964e-01 5.299e-03 3.867e-02
$ \frac{1}{20} $ 8.731e-02 1.997e-01 3.250e-03 8.854e-03 8.955e-02 1.817e-01 2.694e-03 2.259e-02
$ \frac{1}{25} $ 5.620e-02 1.304e-01 2.467e-03 5.938e-03 5.992e-02 1.210e-01 1.573e-03 1.433e-02
$ \frac{1}{30} $ 4.126e-02 9.162e-02 1.828e-03 4.243e-03 4.276e-02 8.680e-02 1.039e-03 9.889e-03
k = 3 Case 1: $ \bar{\bar{b}} = 0, \bar{\bar{\sigma}} = 0 $ Case 2: $ \bar{\bar{b}} = b, \bar{\bar{\sigma}} = 0 $ Case 3: $ \bar{\bar{b}} = 0, \bar{\bar{\sigma}} = \sigma $ Case 4: $ \bar{\bar{b}} = b, \bar{\bar{\sigma}} = \sigma $
$ \Delta t $ $ |Y_0-Y^0| $ $ |Z_0-Z^0| $ $ |Y_0-Y^0| $ $ |Z_0-Z^0| $ $ |Y_0-Y^0| $ $ |Z_0-Z^0| $ $ |Y_0-Y^0| $ $ |Z_0-Z^0| $
$ \frac{1}{15} $ 1.148e-01 3.305e-01 7.149e-04 5.242e-03 8.573e-02 2.405e-01 1.398e-03 1.171e-02
$ \frac{1}{20} $ 4.955e-02 1.552e-01 5.287e-04 2.797e-03 4.227e-02 1.147e-01 6.942e-04 5.813e-03
$ \frac{1}{25} $ 2.661e-02 8.080e-02 4.105e-04 1.459e-03 2.400e-02 6.287e-02 3.479e-04 3.112e-03
$ \frac{1}{30} $ 1.577e-02 4.657e-02 3.057e-04 8.920e-04 1.486e-02 3.814e-02 1.722e-04 1.725e-03
k = 4 Case 1: $ \bar{\bar{b}} = 0, \bar{\bar{\sigma}} = 0 $ Case 2: $ \bar{\bar{b}} = b, \bar{\bar{\sigma}} = 0 $ Case 3: $ \bar{\bar{b}} = 0, \bar{\bar{\sigma}} = \sigma $ Case 4: $ \bar{\bar{b}} = b, \bar{\bar{\sigma}} = \sigma $
$ \Delta t $ $ |Y_0-Y^0| $ $ |Z_0-Z^0| $ $ |Y_0-Y^0| $ $ |Z_0-Z^0| $ $ |Y_0-Y^0| $ $ |Z_0-Z^0| $ $ |Y_0-Y^0| $ $ |Z_0-Z^0| $
$ \frac{1}{15} $ 1.003e-01 2.872e-01 2.383e-04 1.778e-03 6.024e-02 1.763e-01 4.832e-04 4.238e-03
$ \frac{1}{20} $ 3.491e-02 1.039e-01 1.013e-04 9.635e-04 2.676e-02 7.735e-02 2.074e-04 1.777e-03
$ \frac{1}{25} $ 1.614e-02 4.563e-02 9.230e-05 3.843e-04 1.304e-02 3.675e-02 8.852e-05 7.994e-04
$ \frac{1}{30} $ 8.115e-03 2.140e-02 5.403e-05 2.389e-04 6.938e-03 1.913e-02 5.136e-05 4.561e-04
k = 5 Case 1: $ \bar{\bar{b}} = 0, \bar{\bar{\sigma}} = 0 $ Case 2: $ \bar{\bar{b}} = b, \bar{\bar{\sigma}} = 0 $ Case 3: $ \bar{\bar{b}} = 0, \bar{\bar{\sigma}} = \sigma $ Case 4: $ \bar{\bar{b}} = b, \bar{\bar{\sigma}} = \sigma $
$ \Delta t $ $ |Y_0-Y^0| $ $ |Z_0-Z^0| $ $ |Y_0-Y^0| $ $ |Z_0-Z^0| $ $ |Y_0-Y^0| $ $ |Z_0-Z^0| $ $ |Y_0-Y^0| $ $ |Z_0-Z^0| $
$ \frac{1}{15} $ 9.551e-02 3.260e-01 8.368e-05 7.301e-04 5.461e-02 1.597e-01 1.600e-04 1.648e-03
$ \frac{1}{20} $ 2.574e-02 9.071e-02 1.891e-05 2.792e-04 1.689e-02 5.753e-02 7.795e-05 6.725e-04
$ \frac{1}{25} $ 9.002e-03 2.865e-02 2.061e-05 1.547e-04 6.151e-03 2.346e-02 1.876e-05 2.037e-04
$ \frac{1}{30} $ 3.744e-03 1.255e-02 1.359e-05 6.370e-05 2.847e-03 1.076e-02 1.232e-05 1.136e-04
k = 6 Case 1: $ \bar{\bar{b}} = 0, \bar{\bar{\sigma}} = 0 $ Case 2: $ \bar{\bar{b}} = b, \bar{\bar{\sigma}} = 0 $ Case 3: $ \bar{\bar{b}} = 0, \bar{\bar{\sigma}} = \sigma $ Case 4: $ \bar{\bar{b}} = b, \bar{\bar{\sigma}} = \sigma $
$ \Delta t $ $ |Y_0-Y^0| $ $ |Z_0-Z^0| $ $ |Y_0-Y^0| $ $ |Z_0-Z^0| $ $ |Y_0-Y^0| $ $ |Z_0-Z^0| $ $ |Y_0-Y^0| $ $ |Z_0-Z^0| $
$ \frac{1}{15} $ 9.859e-02 3.329e-01 3.338e-05 3.397e-04 4.665e-02 1.441e-01 7.870e-05 7.999e-04
$ \frac{1}{20} $ 2.129e-02 7.413e-02 1.435e-05 9.758e-05 1.260e-02 3.741e-02 3.018e-05 2.791e-04
$ \frac{1}{25} $ 6.412e-03 2.066e-02 6.947e-06 6.856e-05 4.122e-03 1.316e-02 9.675e-06 7.952e-05
$ \frac{1}{30} $ 2.180e-03 7.511e-03 3.173e-06 2.020e-05 1.615e-03 5.427e-03 3.225e-06 2.782e-05
k = 1 Case 1: $ \bar{\bar{b}} = 0, \bar{\bar{\sigma}} = 0 $ Case 2: $ \bar{\bar{b}} = b, \bar{\bar{\sigma}} = 0 $ Case 3: $ \bar{\bar{b}} = 0, \bar{\bar{\sigma}} = \sigma $ Case 4: $ \bar{\bar{b}} = b, \bar{\bar{\sigma}} = \sigma $
$ \Delta t $ $ |Y_0-Y^0| $ $ |Z_0-Z^0| $ $ |Y_0-Y^0| $ $ |Z_0-Z^0| $ $ |Y_0-Y^0| $ $ |Z_0-Z^0| $ $ |Y_0-Y^0| $ $ |Z_0-Z^0| $
$ \frac{1}{15} $ 2.182e-01 4.303e-01 3.216e-02 6.829e-02 1.971e-01 3.969e-01 2.207e-02 1.539e-01
$ \frac{1}{20} $ 1.651e-01 3.578e-01 2.661e-02 4.673e-02 1.557e-01 3.074e-01 1.508e-02 1.177e-01
$ \frac{1}{25} $ 1.302e-01 3.013e-01 2.216e-02 3.389e-02 1.269e-01 2.660e-01 1.147e-02 9.416e-02
$ \frac{1}{30} $ 1.058e-01 2.592e-01 1.909e-02 2.865e-02 1.061e-01 2.331e-01 8.924e-03 7.698e-02
k = 2 Case 1: $ \bar{\bar{b}} = 0, \bar{\bar{\sigma}} = 0 $ Case 2: $ \bar{\bar{b}} = b, \bar{\bar{\sigma}} = 0 $ Case 3: $ \bar{\bar{b}} = 0, \bar{\bar{\sigma}} = \sigma $ Case 4: $ \bar{\bar{b}} = b, \bar{\bar{\sigma}} = \sigma $
$ \Delta t $ $ |Y_0-Y^0| $ $ |Z_0-Z^0| $ $ |Y_0-Y^0| $ $ |Z_0-Z^0| $ $ |Y_0-Y^0| $ $ |Z_0-Z^0| $ $ |Y_0-Y^0| $ $ |Z_0-Z^0| $
$ \frac{1}{15} $ 1.586e-01 3.414e-01 4.025e-03 1.811e-02 1.457e-01 2.964e-01 5.299e-03 3.867e-02
$ \frac{1}{20} $ 8.731e-02 1.997e-01 3.250e-03 8.854e-03 8.955e-02 1.817e-01 2.694e-03 2.259e-02
$ \frac{1}{25} $ 5.620e-02 1.304e-01 2.467e-03 5.938e-03 5.992e-02 1.210e-01 1.573e-03 1.433e-02
$ \frac{1}{30} $ 4.126e-02 9.162e-02 1.828e-03 4.243e-03 4.276e-02 8.680e-02 1.039e-03 9.889e-03
k = 3 Case 1: $ \bar{\bar{b}} = 0, \bar{\bar{\sigma}} = 0 $ Case 2: $ \bar{\bar{b}} = b, \bar{\bar{\sigma}} = 0 $ Case 3: $ \bar{\bar{b}} = 0, \bar{\bar{\sigma}} = \sigma $ Case 4: $ \bar{\bar{b}} = b, \bar{\bar{\sigma}} = \sigma $
$ \Delta t $ $ |Y_0-Y^0| $ $ |Z_0-Z^0| $ $ |Y_0-Y^0| $ $ |Z_0-Z^0| $ $ |Y_0-Y^0| $ $ |Z_0-Z^0| $ $ |Y_0-Y^0| $ $ |Z_0-Z^0| $
$ \frac{1}{15} $ 1.148e-01 3.305e-01 7.149e-04 5.242e-03 8.573e-02 2.405e-01 1.398e-03 1.171e-02
$ \frac{1}{20} $ 4.955e-02 1.552e-01 5.287e-04 2.797e-03 4.227e-02 1.147e-01 6.942e-04 5.813e-03
$ \frac{1}{25} $ 2.661e-02 8.080e-02 4.105e-04 1.459e-03 2.400e-02 6.287e-02 3.479e-04 3.112e-03
$ \frac{1}{30} $ 1.577e-02 4.657e-02 3.057e-04 8.920e-04 1.486e-02 3.814e-02 1.722e-04 1.725e-03
k = 4 Case 1: $ \bar{\bar{b}} = 0, \bar{\bar{\sigma}} = 0 $ Case 2: $ \bar{\bar{b}} = b, \bar{\bar{\sigma}} = 0 $ Case 3: $ \bar{\bar{b}} = 0, \bar{\bar{\sigma}} = \sigma $ Case 4: $ \bar{\bar{b}} = b, \bar{\bar{\sigma}} = \sigma $
$ \Delta t $ $ |Y_0-Y^0| $ $ |Z_0-Z^0| $ $ |Y_0-Y^0| $ $ |Z_0-Z^0| $ $ |Y_0-Y^0| $ $ |Z_0-Z^0| $ $ |Y_0-Y^0| $ $ |Z_0-Z^0| $
$ \frac{1}{15} $ 1.003e-01 2.872e-01 2.383e-04 1.778e-03 6.024e-02 1.763e-01 4.832e-04 4.238e-03
$ \frac{1}{20} $ 3.491e-02 1.039e-01 1.013e-04 9.635e-04 2.676e-02 7.735e-02 2.074e-04 1.777e-03
$ \frac{1}{25} $ 1.614e-02 4.563e-02 9.230e-05 3.843e-04 1.304e-02 3.675e-02 8.852e-05 7.994e-04
$ \frac{1}{30} $ 8.115e-03 2.140e-02 5.403e-05 2.389e-04 6.938e-03 1.913e-02 5.136e-05 4.561e-04
k = 5 Case 1: $ \bar{\bar{b}} = 0, \bar{\bar{\sigma}} = 0 $ Case 2: $ \bar{\bar{b}} = b, \bar{\bar{\sigma}} = 0 $ Case 3: $ \bar{\bar{b}} = 0, \bar{\bar{\sigma}} = \sigma $ Case 4: $ \bar{\bar{b}} = b, \bar{\bar{\sigma}} = \sigma $
$ \Delta t $ $ |Y_0-Y^0| $ $ |Z_0-Z^0| $ $ |Y_0-Y^0| $ $ |Z_0-Z^0| $ $ |Y_0-Y^0| $ $ |Z_0-Z^0| $ $ |Y_0-Y^0| $ $ |Z_0-Z^0| $
$ \frac{1}{15} $ 9.551e-02 3.260e-01 8.368e-05 7.301e-04 5.461e-02 1.597e-01 1.600e-04 1.648e-03
$ \frac{1}{20} $ 2.574e-02 9.071e-02 1.891e-05 2.792e-04 1.689e-02 5.753e-02 7.795e-05 6.725e-04
$ \frac{1}{25} $ 9.002e-03 2.865e-02 2.061e-05 1.547e-04 6.151e-03 2.346e-02 1.876e-05 2.037e-04
$ \frac{1}{30} $ 3.744e-03 1.255e-02 1.359e-05 6.370e-05 2.847e-03 1.076e-02 1.232e-05 1.136e-04
k = 6 Case 1: $ \bar{\bar{b}} = 0, \bar{\bar{\sigma}} = 0 $ Case 2: $ \bar{\bar{b}} = b, \bar{\bar{\sigma}} = 0 $ Case 3: $ \bar{\bar{b}} = 0, \bar{\bar{\sigma}} = \sigma $ Case 4: $ \bar{\bar{b}} = b, \bar{\bar{\sigma}} = \sigma $
$ \Delta t $ $ |Y_0-Y^0| $ $ |Z_0-Z^0| $ $ |Y_0-Y^0| $ $ |Z_0-Z^0| $ $ |Y_0-Y^0| $ $ |Z_0-Z^0| $ $ |Y_0-Y^0| $ $ |Z_0-Z^0| $
$ \frac{1}{15} $ 9.859e-02 3.329e-01 3.338e-05 3.397e-04 4.665e-02 1.441e-01 7.870e-05 7.999e-04
$ \frac{1}{20} $ 2.129e-02 7.413e-02 1.435e-05 9.758e-05 1.260e-02 3.741e-02 3.018e-05 2.791e-04
$ \frac{1}{25} $ 6.412e-03 2.066e-02 6.947e-06 6.856e-05 4.122e-03 1.316e-02 9.675e-06 7.952e-05
$ \frac{1}{30} $ 2.180e-03 7.511e-03 3.173e-06 2.020e-05 1.615e-03 5.427e-03 3.225e-06 2.782e-05
Table 3.  The errors of Scheme 4.1 with $ b = 10 $, $ \sigma = 1 $ for Ex. 5.1
k = 1 Case 1: $ \bar{\bar{b}} = 0, \bar{\bar{\sigma}} = 0 $ Case 2: $ \bar{\bar{b}} = b, \bar{\bar{\sigma}} = 0 $ Case 3: $ \bar{\bar{b}} = 0, \bar{\bar{\sigma}} = \sigma $ Case 4: $ \bar{\bar{b}} = b, \bar{\bar{\sigma}} = \sigma $
$ \Delta t $ $ |Y_0-Y^0| $ $ |Z_0-Z^0| $ $ |Y_0-Y^0| $ $ |Z_0-Z^0| $ $ |Y_0-Y^0| $ $ |Z_0-Z^0| $ $ |Y_0-Y^0| $ $ |Z_0-Z^0| $
$ \frac{1}{15} $ 5.181e-02 1.250e-01 2.796e-02 6.815e-02 4.717e-02 1.944e-01 2.588e-02 1.576e-01
$ \frac{1}{20} $ 3.843e-02 9.541e-02 2.077e-02 5.144e-02 3.586e-02 1.524e-01 1.960e-02 1.232e-01
$ \frac{1}{25} $ 3.050e-02 7.720e-02 1.652e-02 4.127e-02 2.897e-02 1.251e-01 1.578e-02 1.006e-01
$ \frac{1}{30} $ 2.523e-02 6.484e-02 1.372e-02 3.472e-02 2.425e-02 1.059e-01 1.320e-02 8.584e-02
k = 2 Case 1: $ \bar{\bar{b}} = 0, \bar{\bar{\sigma}} = 0 $ Case 2: $ \bar{\bar{b}} = b, \bar{\bar{\sigma}} = 0 $ Case 3: $ \bar{\bar{b}} = 0, \bar{\bar{\sigma}} = \sigma $ Case 4: $ \bar{\bar{b}} = b, \bar{\bar{\sigma}} = \sigma $
$ \Delta t $ $ |Y_0-Y^0| $ $ |Z_0-Z^0| $ $ |Y_0-Y^0| $ $ |Z_0-Z^0| $ $ |Y_0-Y^0| $ $ |Z_0-Z^0| $ $ |Y_0-Y^0| $ $ |Z_0-Z^0| $
$ \frac{1}{15} $ 4.917e-01 1.039e+00 5.761e-03 1.401e-02 3.993e-01 8.410e-01 4.412e-03 3.729e-02
$ \frac{1}{20} $ 3.448e-01 7.170e-01 2.803e-03 8.561e-03 2.854e-01 6.164e-01 2.727e-03 2.274e-02
$ \frac{1}{25} $ 2.357e-01 4.800e-01 1.939e-03 4.953e-03 2.071e-01 4.352e-01 1.683e-03 1.438e-02
$ \frac{1}{30} $ 1.667e-01 3.541e-01 1.389e-03 3.686e-03 1.532e-01 3.099e-01 1.164e-03 1.004e-02
k = 3 Case 1: $ \bar{\bar{b}} = 0, \bar{\bar{\sigma}} = 0 $ Case 2: $ \bar{\bar{b}} = b, \bar{\bar{\sigma}} = 0 $ Case 3: $ \bar{\bar{b}} = 0, \bar{\bar{\sigma}} = \sigma $ Case 4: $ \bar{\bar{b}} = b, \bar{\bar{\sigma}} = \sigma $
$ \Delta t $ $ |Y_0-Y^0| $ $ |Z_0-Z^0| $ $ |Y_0-Y^0| $ $ |Z_0-Z^0| $ $ |Y_0-Y^0| $ $ |Z_0-Z^0| $ $ |Y_0-Y^0| $ $ |Z_0-Z^0| $
$ \frac{1}{15} $ 6.489e-01 1.674e+00 1.629e-03 5.857e-03 4.829e-01 1.088e+00 1.465e-03 1.198e-02
$ \frac{1}{20} $ 3.241e-01 9.866e-01 5.183e-04 2.023e-03 2.603e-01 7.271e-01 5.862e-04 5.554e-03
$ \frac{1}{25} $ 1.899e-01 5.217e-01 2.742e-04 1.135e-03 1.392e-01 4.518e-01 3.464e-04 3.092e-03
$ \frac{1}{30} $ 1.157e-01 3.160e-01 1.965e-04 6.306e-04 9.566e-02 2.844e-01 1.763e-04 1.700e-03
k = 4 Case 1: $ \bar{\bar{b}} = 0, \bar{\bar{\sigma}} = 0 $ Case 2: $ \bar{\bar{b}} = b, \bar{\bar{\sigma}} = 0 $ Case 3: $ \bar{\bar{b}} = 0, \bar{\bar{\sigma}} = \sigma $ Case 4: $ \bar{\bar{b}} = b, \bar{\bar{\sigma}} = \sigma $
$ \Delta t $ $ |Y_0-Y^0| $ $ |Z_0-Z^0| $ $ |Y_0-Y^0| $ $ |Z_0-Z^0| $ $ |Y_0-Y^0| $ $ |Z_0-Z^0| $ $ |Y_0-Y^0| $ $ |Z_0-Z^0| $
$ \frac{1}{15} $ 9.734e-01 2.587e+00 3.257e-04 2.688e-03 5.946e-01 1.525e+00 5.273e-04 4.400e-03
$ \frac{1}{20} $ 4.297e-01 1.193e+00 1.797e-04 4.954e-04 3.055e-01 7.672e-01 1.274e-04 1.489e-03
$ \frac{1}{25} $ 2.010e-01 5.811e-01 4.152e-05 3.768e-04 1.549e-01 3.857e-01 1.001e-04 8.915e-04
$ \frac{1}{30} $ 1.125e-01 2.918e-01 2.880e-05 1.476e-04 8.144e-02 2.288e-01 4.368e-05 4.084e-04
k = 5 Case 1: $ \bar{\bar{b}} = 0, \bar{\bar{\sigma}} = 0 $ Case 2: $ \bar{\bar{b}} = b, \bar{\bar{\sigma}} = 0 $ Case 3: $ \bar{\bar{b}} = 0, \bar{\bar{\sigma}} = \sigma $ Case 4: $ \bar{\bar{b}} = b, \bar{\bar{\sigma}} = \sigma $
$ \Delta t $ $ |Y_0-Y^0| $ $ |Z_0-Z^0| $ $ |Y_0-Y^0| $ $ |Z_0-Z^0| $ $ |Y_0-Y^0| $ $ |Z_0-Z^0| $ $ |Y_0-Y^0| $ $ |Z_0-Z^0| $
$ \frac{1}{15} $ 1.432e+00 4.053e+00 8.015e-05 8.045e-04 7.547e-01 2.122e+00 2.440e-04 2.041e-03
$ \frac{1}{20} $ 5.296e-01 1.611e+00 9.127e-05 2.672e-04 3.072e-01 9.677e-01 9.220e-05 7.931e-04
$ \frac{1}{25} $ 2.040e-01 7.428e-01 1.189e-05 9.266e-05 1.382e-01 4.365e-01 2.198e-05 2.226e-04
$ \frac{1}{30} $ 9.414e-02 3.360e-01 5.087e-06 4.835e-05 6.835e-02 1.994e-01 1.145e-05 1.142e-04
k = 6 Case 1: $ \bar{\bar{b}} = 0, \bar{\bar{\sigma}} = 0 $ Case 2: $ \bar{\bar{b}} = b, \bar{\bar{\sigma}} = 0 $ Case 3: $ \bar{\bar{b}} = 0, \bar{\bar{\sigma}} = \sigma $ Case 4: $ \bar{\bar{b}} = b, \bar{\bar{\sigma}} = \sigma $
$ \Delta t $ $ |Y_0-Y^0| $ $ |Z_0-Z^0| $ $ |Y_0-Y^0| $ $ |Z_0-Z^0| $ $ |Y_0-Y^0| $ $ |Z_0-Z^0| $ $ |Y_0-Y^0| $ $ |Z_0-Z^0| $
$ \frac{1}{15} $ 2.117e+00 6.804e+00 2.143e-05 4.101e-04 1.073e+00 2.566e+00 8.551e-05 8.529e-04
$ \frac{1}{20} $ 6.795e-01 2.344e+00 5.912e-05 2.660e-04 3.827e-01 1.115e+00 4.831e-05 3.871e-04
$ \frac{1}{25} $ 2.739e-01 8.312e-01 4.996e-06 4.322e-05 1.464e-01 4.481e-01 8.071e-06 5.051e-05
$ \frac{1}{30} $ 9.606e-02 3.503e-01 8.488e-07 1.913e-05 6.020e-02 2.035e-01 4.445e-06 4.193e-05
k = 1 Case 1: $ \bar{\bar{b}} = 0, \bar{\bar{\sigma}} = 0 $ Case 2: $ \bar{\bar{b}} = b, \bar{\bar{\sigma}} = 0 $ Case 3: $ \bar{\bar{b}} = 0, \bar{\bar{\sigma}} = \sigma $ Case 4: $ \bar{\bar{b}} = b, \bar{\bar{\sigma}} = \sigma $
$ \Delta t $ $ |Y_0-Y^0| $ $ |Z_0-Z^0| $ $ |Y_0-Y^0| $ $ |Z_0-Z^0| $ $ |Y_0-Y^0| $ $ |Z_0-Z^0| $ $ |Y_0-Y^0| $ $ |Z_0-Z^0| $
$ \frac{1}{15} $ 5.181e-02 1.250e-01 2.796e-02 6.815e-02 4.717e-02 1.944e-01 2.588e-02 1.576e-01
$ \frac{1}{20} $ 3.843e-02 9.541e-02 2.077e-02 5.144e-02 3.586e-02 1.524e-01 1.960e-02 1.232e-01
$ \frac{1}{25} $ 3.050e-02 7.720e-02 1.652e-02 4.127e-02 2.897e-02 1.251e-01 1.578e-02 1.006e-01
$ \frac{1}{30} $ 2.523e-02 6.484e-02 1.372e-02 3.472e-02 2.425e-02 1.059e-01 1.320e-02 8.584e-02
k = 2 Case 1: $ \bar{\bar{b}} = 0, \bar{\bar{\sigma}} = 0 $ Case 2: $ \bar{\bar{b}} = b, \bar{\bar{\sigma}} = 0 $ Case 3: $ \bar{\bar{b}} = 0, \bar{\bar{\sigma}} = \sigma $ Case 4: $ \bar{\bar{b}} = b, \bar{\bar{\sigma}} = \sigma $
$ \Delta t $ $ |Y_0-Y^0| $ $ |Z_0-Z^0| $ $ |Y_0-Y^0| $ $ |Z_0-Z^0| $ $ |Y_0-Y^0| $ $ |Z_0-Z^0| $ $ |Y_0-Y^0| $ $ |Z_0-Z^0| $
$ \frac{1}{15} $ 4.917e-01 1.039e+00 5.761e-03 1.401e-02 3.993e-01 8.410e-01 4.412e-03 3.729e-02
$ \frac{1}{20} $ 3.448e-01 7.170e-01 2.803e-03 8.561e-03 2.854e-01 6.164e-01 2.727e-03 2.274e-02
$ \frac{1}{25} $ 2.357e-01 4.800e-01 1.939e-03 4.953e-03 2.071e-01 4.352e-01 1.683e-03 1.438e-02
$ \frac{1}{30} $ 1.667e-01 3.541e-01 1.389e-03 3.686e-03 1.532e-01 3.099e-01 1.164e-03 1.004e-02
k = 3 Case 1: $ \bar{\bar{b}} = 0, \bar{\bar{\sigma}} = 0 $ Case 2: $ \bar{\bar{b}} = b, \bar{\bar{\sigma}} = 0 $ Case 3: $ \bar{\bar{b}} = 0, \bar{\bar{\sigma}} = \sigma $ Case 4: $ \bar{\bar{b}} = b, \bar{\bar{\sigma}} = \sigma $
$ \Delta t $ $ |Y_0-Y^0| $ $ |Z_0-Z^0| $ $ |Y_0-Y^0| $ $ |Z_0-Z^0| $ $ |Y_0-Y^0| $ $ |Z_0-Z^0| $ $ |Y_0-Y^0| $ $ |Z_0-Z^0| $
$ \frac{1}{15} $ 6.489e-01 1.674e+00 1.629e-03 5.857e-03 4.829e-01 1.088e+00 1.465e-03 1.198e-02
$ \frac{1}{20} $ 3.241e-01 9.866e-01 5.183e-04 2.023e-03 2.603e-01 7.271e-01 5.862e-04 5.554e-03
$ \frac{1}{25} $ 1.899e-01 5.217e-01 2.742e-04 1.135e-03 1.392e-01 4.518e-01 3.464e-04 3.092e-03
$ \frac{1}{30} $ 1.157e-01 3.160e-01 1.965e-04 6.306e-04 9.566e-02 2.844e-01 1.763e-04 1.700e-03
k = 4 Case 1: $ \bar{\bar{b}} = 0, \bar{\bar{\sigma}} = 0 $ Case 2: $ \bar{\bar{b}} = b, \bar{\bar{\sigma}} = 0 $ Case 3: $ \bar{\bar{b}} = 0, \bar{\bar{\sigma}} = \sigma $ Case 4: $ \bar{\bar{b}} = b, \bar{\bar{\sigma}} = \sigma $
$ \Delta t $ $ |Y_0-Y^0| $ $ |Z_0-Z^0| $ $ |Y_0-Y^0| $ $ |Z_0-Z^0| $ $ |Y_0-Y^0| $ $ |Z_0-Z^0| $ $ |Y_0-Y^0| $ $ |Z_0-Z^0| $
$ \frac{1}{15} $ 9.734e-01 2.587e+00 3.257e-04 2.688e-03 5.946e-01 1.525e+00 5.273e-04 4.400e-03
$ \frac{1}{20} $ 4.297e-01 1.193e+00 1.797e-04 4.954e-04 3.055e-01 7.672e-01 1.274e-04 1.489e-03
$ \frac{1}{25} $ 2.010e-01 5.811e-01 4.152e-05 3.768e-04 1.549e-01 3.857e-01 1.001e-04 8.915e-04
$ \frac{1}{30} $ 1.125e-01 2.918e-01 2.880e-05 1.476e-04 8.144e-02 2.288e-01 4.368e-05 4.084e-04
k = 5 Case 1: $ \bar{\bar{b}} = 0, \bar{\bar{\sigma}} = 0 $ Case 2: $ \bar{\bar{b}} = b, \bar{\bar{\sigma}} = 0 $ Case 3: $ \bar{\bar{b}} = 0, \bar{\bar{\sigma}} = \sigma $ Case 4: $ \bar{\bar{b}} = b, \bar{\bar{\sigma}} = \sigma $
$ \Delta t $ $ |Y_0-Y^0| $ $ |Z_0-Z^0| $ $ |Y_0-Y^0| $ $ |Z_0-Z^0| $ $ |Y_0-Y^0| $ $ |Z_0-Z^0| $ $ |Y_0-Y^0| $ $ |Z_0-Z^0| $
$ \frac{1}{15} $ 1.432e+00 4.053e+00 8.015e-05 8.045e-04 7.547e-01 2.122e+00 2.440e-04 2.041e-03
$ \frac{1}{20} $ 5.296e-01 1.611e+00 9.127e-05 2.672e-04 3.072e-01 9.677e-01 9.220e-05 7.931e-04
$ \frac{1}{25} $ 2.040e-01 7.428e-01 1.189e-05 9.266e-05 1.382e-01 4.365e-01 2.198e-05 2.226e-04
$ \frac{1}{30} $ 9.414e-02 3.360e-01 5.087e-06 4.835e-05 6.835e-02 1.994e-01 1.145e-05 1.142e-04
k = 6 Case 1: $ \bar{\bar{b}} = 0, \bar{\bar{\sigma}} = 0 $ Case 2: $ \bar{\bar{b}} = b, \bar{\bar{\sigma}} = 0 $ Case 3: $ \bar{\bar{b}} = 0, \bar{\bar{\sigma}} = \sigma $ Case 4: $ \bar{\bar{b}} = b, \bar{\bar{\sigma}} = \sigma $
$ \Delta t $ $ |Y_0-Y^0| $ $ |Z_0-Z^0| $ $ |Y_0-Y^0| $ $ |Z_0-Z^0| $ $ |Y_0-Y^0| $ $ |Z_0-Z^0| $ $ |Y_0-Y^0| $ $ |Z_0-Z^0| $
$ \frac{1}{15} $ 2.117e+00 6.804e+00 2.143e-05 4.101e-04 1.073e+00 2.566e+00 8.551e-05 8.529e-04
$ \frac{1}{20} $ 6.795e-01 2.344e+00 5.912e-05 2.660e-04 3.827e-01 1.115e+00 4.831e-05 3.871e-04
$ \frac{1}{25} $ 2.739e-01 8.312e-01 4.996e-06 4.322e-05 1.464e-01 4.481e-01 8.071e-06 5.051e-05
$ \frac{1}{30} $ 9.606e-02 3.503e-01 8.488e-07 1.913e-05 6.020e-02 2.035e-01 4.445e-06 4.193e-05
Table 4.  The errors of Scheme 4.1 with $ b = 1 $, $ \sigma = 1 $ for Ex. 5.2
k = 1 Case 1: $ \bar{\bar{b}} = 0, \bar{\bar{\sigma}} = 0 $ Case 2: $ \bar{\bar{b}} = b, \bar{\bar{\sigma}} = 0 $ Case 3: $ \bar{\bar{b}} = 0, \bar{\bar{\sigma}} = \sigma $ Case 4: $ \bar{\bar{b}} = b, \bar{\bar{\sigma}} = \sigma $
$ \Delta t $ $ |Y_0-Y^0| $ $ |Z_0-Z^0| $ $ |Y_0-Y^0| $ $ |Z_0-Z^0| $ $ |Y_0-Y^0| $ $ |Z_0-Z^0| $ $ |Y_0-Y^0| $ $ |Z_0-Z^0| $
$ \frac{1}{12} $ 4.105e-03 5.714e-03 8.365e-04 4.125e-03 3.933e-03 6.808e-03 8.581e-04 6.639e-03
$ \frac{1}{16} $ 3.060e-03 4.362e-03 6.273e-04 3.089e-03 2.975e-03 5.346e-03 6.503e-04 5.083e-03
$ \frac{1}{20} $ 2.436e-03 3.518e-03 4.977e-04 2.471e-03 2.399e-03 4.412e-03 5.380e-04 4.120e-03
$ \frac{1}{24} $ 2.024e-03 2.942e-03 4.181e-04 2.056e-03 2.007e-03 3.747e-03 4.556e-04 3.462e-03
k = 2 Case 1: $ \bar{\bar{b}} = 0, \bar{\bar{\sigma}} = 0 $ Case 2: $ \bar{\bar{b}} = b, \bar{\bar{\sigma}} = 0 $ Case 3: $ \bar{\bar{b}} = 0, \bar{\bar{\sigma}} = \sigma $ Case 4: $ \bar{\bar{b}} = b, \bar{\bar{\sigma}} = \sigma $
$ \Delta t $ $ |Y_0-Y^0| $ $ |Z_0-Z^0| $ $ |Y_0-Y^0| $ $ |Z_0-Z^0| $ $ |Y_0-Y^0| $ $ |Z_0-Z^0| $ $ |Y_0-Y^0| $ $ |Z_0-Z^0| $
$ \frac{1}{12} $ 3.154e-04 2.073e-04 5.813e-05 2.748e-04 3.920e-04 7.100e-04 7.049e-05 5.513e-04
$ \frac{1}{16} $ 1.830e-04 1.192e-04 3.494e-05 1.563e-04 2.300e-04 4.181e-04 3.991e-05 3.240e-04
$ \frac{1}{20} $ 1.197e-04 7.880e-05 2.311e-05 1.007e-04 1.507e-04 2.742e-04 2.538e-05 2.129e-04
$ \frac{1}{24} $ 8.374e-05 5.548e-05 1.628e-05 7.054e-05 1.063e-04 1.926e-04 1.848e-05 1.508e-04
k = 3 Case 1: $ \bar{\bar{b}} = 0, \bar{\bar{\sigma}} = 0 $ Case 2: $ \bar{\bar{b}} = b, \bar{\bar{\sigma}} = 0 $ Case 3: $ \bar{\bar{b}} = 0, \bar{\bar{\sigma}} = \sigma $ Case 4: $ \bar{\bar{b}} = b, \bar{\bar{\sigma}} = \sigma $
$ \Delta t $ $ |Y_0-Y^0| $ $ |Z_0-Z^0| $ $ |Y_0-Y^0| $ $ |Z_0-Z^0| $ $ |Y_0-Y^0| $ $ |Z_0-Z^0| $ $ |Y_0-Y^0| $ $ |Z_0-Z^0| $
$ \frac{1}{12} $ 3.329e-05 5.716e-05 6.428e-06 3.795e-05 5.813e-05 1.083e-04 1.030e-05 8.267e-05
$ \frac{1}{16} $ 1.476e-05 2.619e-05 3.175e-06 1.728e-05 2.841e-05 4.990e-05 4.649e-06 3.881e-05
$ \frac{1}{20} $ 7.830e-06 1.360e-05 1.766e-06 9.119e-06 1.577e-05 2.674e-05 2.424e-06 2.101e-05
$ \frac{1}{24} $ 4.653e-06 7.780e-06 1.081e-06 5.333e-06 9.548e-06 1.593e-05 1.403e-06 1.256e-05
k = 4 Case 1: $ \bar{\bar{b}} = 0, \bar{\bar{\sigma}} = 0 $ Case 2: $ \bar{\bar{b}} = b, \bar{\bar{\sigma}} = 0 $ Case 3: $ \bar{\bar{b}} = 0, \bar{\bar{\sigma}} = \sigma $ Case 4: $ \bar{\bar{b}} = b, \bar{\bar{\sigma}} = \sigma $
$ \Delta t $ $ |Y_0-Y^0| $ $ |Z_0-Z^0| $ $ |Y_0-Y^0| $ $ |Z_0-Z^0| $ $ |Y_0-Y^0| $ $ |Z_0-Z^0| $ $ |Y_0-Y^0| $ $ |Z_0-Z^0| $
$ \frac{1}{12} $ 2.616e-06 1.647e-05 1.002e-06 8.273e-06 1.283e-05 2.250e-05 2.193e-06 1.833e-05
$ \frac{1}{16} $ 7.797e-07 5.906e-06 4.216e-07 3.099e-06 4.719e-06 8.419e-06 8.240e-07 7.104e-06
$ \frac{1}{20} $ 3.360e-07 2.582e-06 2.034e-07 1.381e-06 2.095e-06 3.952e-06 3.552e-07 3.226e-06
$ \frac{1}{24} $ 1.662e-07 1.289e-06 1.082e-07 6.960e-07 1.059e-06 2.034e-06 1.752e-07 1.657e-06
k = 5 Case 1: $ \bar{\bar{b}} = 0, \bar{\bar{\sigma}} = 0 $ Case 2: $ \bar{\bar{b}} = b, \bar{\bar{\sigma}} = 0 $ Case 3: $ \bar{\bar{b}} = 0, \bar{\bar{\sigma}} = \sigma $ Case 4: $ \bar{\bar{b}} = b, \bar{\bar{\sigma}} = \sigma $
$ \Delta t $ $ |Y_0-Y^0| $ $ |Z_0-Z^0| $ $ |Y_0-Y^0| $ $ |Z_0-Z^0| $ $ |Y_0-Y^0| $ $ |Z_0-Z^0| $ $ |Y_0-Y^0| $ $ |Z_0-Z^0| $
$ \frac{1}{12} $ 5.529e-07 2.942e-06 2.335e-07 2.457e-06 2.788e-06 6.972e-06 6.110e-07 5.315e-06
$ \frac{1}{16} $ 1.562e-07 8.938e-07 7.126e-08 7.612e-07 8.694e-07 2.140e-06 1.973e-07 1.763e-06
$ \frac{1}{20} $ 5.644e-08 4.817e-07 3.114e-08 4.712e-07 3.377e-07 7.491e-07 7.317e-08 6.889e-07
$ \frac{1}{24} $ 2.403e-08 2.232e-07 1.485e-08 2.823e-07 1.515e-07 3.390e-07 3.036e-08 3.055e-07
k = 6 Case 1: $ \bar{\bar{b}} = 0, \bar{\bar{\sigma}} = 0 $ Case 2: $ \bar{\bar{b}} = b, \bar{\bar{\sigma}} = 0 $ Case 3: $ \bar{\bar{b}} = 0, \bar{\bar{\sigma}} = \sigma $ Case 4: $ \bar{\bar{b}} = b, \bar{\bar{\sigma}} = \sigma $
$ \Delta t $ $ |Y_0-Y^0| $ $ |Z_0-Z^0| $ $ |Y_0-Y^0| $ $ |Z_0-Z^0| $ $ |Y_0-Y^0| $ $ |Z_0-Z^0| $ $ |Y_0-Y^0| $ $ |Z_0-Z^0| $
$ \frac{1}{12} $ 1.913e-07 1.582e-06 1.170e-07 1.451e-06 8.519e-07 3.274e-06 2.211e-07 2.042e-06
$ \frac{1}{16} $ 9.004e-08 1.721e-06 1.030e-07 2.062e-06 2.131e-07 4.811e-07 5.975e-08 5.508e-07
$ \frac{1}{20} $ 1.019e-07 1.799e-06 1.008e-07 2.561e-06 6.204e-08 2.111e-07 1.929e-08 1.814e-07
$ \frac{1}{24} $ 1.070e-07 2.281e-06 1.181e-07 3.290e-06 2.545e-08 2.302e-07 6.984e-09 1.047e-07
k = 1 Case 1: $ \bar{\bar{b}} = 0, \bar{\bar{\sigma}} = 0 $ Case 2: $ \bar{\bar{b}} = b, \bar{\bar{\sigma}} = 0 $ Case 3: $ \bar{\bar{b}} = 0, \bar{\bar{\sigma}} = \sigma $ Case 4: $ \bar{\bar{b}} = b, \bar{\bar{\sigma}} = \sigma $
$ \Delta t $ $ |Y_0-Y^0| $ $ |Z_0-Z^0| $ $ |Y_0-Y^0| $ $ |Z_0-Z^0| $ $ |Y_0-Y^0| $ $ |Z_0-Z^0| $ $ |Y_0-Y^0| $ $ |Z_0-Z^0| $
$ \frac{1}{12} $ 4.105e-03 5.714e-03 8.365e-04 4.125e-03 3.933e-03 6.808e-03 8.581e-04 6.639e-03
$ \frac{1}{16} $ 3.060e-03 4.362e-03 6.273e-04 3.089e-03 2.975e-03 5.346e-03 6.503e-04 5.083e-03
$ \frac{1}{20} $ 2.436e-03 3.518e-03 4.977e-04 2.471e-03 2.399e-03 4.412e-03 5.380e-04 4.120e-03
$ \frac{1}{24} $ 2.024e-03 2.942e-03 4.181e-04 2.056e-03 2.007e-03 3.747e-03 4.556e-04 3.462e-03
k = 2 Case 1: $ \bar{\bar{b}} = 0, \bar{\bar{\sigma}} = 0 $ Case 2: $ \bar{\bar{b}} = b, \bar{\bar{\sigma}} = 0 $ Case 3: $ \bar{\bar{b}} = 0, \bar{\bar{\sigma}} = \sigma $ Case 4: $ \bar{\bar{b}} = b, \bar{\bar{\sigma}} = \sigma $
$ \Delta t $ $ |Y_0-Y^0| $ $ |Z_0-Z^0| $ $ |Y_0-Y^0| $ $ |Z_0-Z^0| $ $ |Y_0-Y^0| $ $ |Z_0-Z^0| $ $ |Y_0-Y^0| $ $ |Z_0-Z^0| $
$ \frac{1}{12} $ 3.154e-04 2.073e-04 5.813e-05 2.748e-04 3.920e-04 7.100e-04 7.049e-05 5.513e-04
$ \frac{1}{16} $ 1.830e-04 1.192e-04 3.494e-05 1.563e-04 2.300e-04 4.181e-04 3.991e-05 3.240e-04
$ \frac{1}{20} $ 1.197e-04 7.880e-05 2.311e-05 1.007e-04 1.507e-04 2.742e-04 2.538e-05 2.129e-04
$ \frac{1}{24} $ 8.374e-05 5.548e-05 1.628e-05 7.054e-05 1.063e-04 1.926e-04 1.848e-05 1.508e-04
k = 3 Case 1: $ \bar{\bar{b}} = 0, \bar{\bar{\sigma}} = 0 $ Case 2: $ \bar{\bar{b}} = b, \bar{\bar{\sigma}} = 0 $ Case 3: $ \bar{\bar{b}} = 0, \bar{\bar{\sigma}} = \sigma $ Case 4: $ \bar{\bar{b}} = b, \bar{\bar{\sigma}} = \sigma $
$ \Delta t $ $ |Y_0-Y^0| $ $ |Z_0-Z^0| $ $ |Y_0-Y^0| $ $ |Z_0-Z^0| $ $ |Y_0-Y^0| $ $ |Z_0-Z^0| $ $ |Y_0-Y^0| $ $ |Z_0-Z^0| $
$ \frac{1}{12} $ 3.329e-05 5.716e-05 6.428e-06 3.795e-05 5.813e-05 1.083e-04 1.030e-05 8.267e-05
$ \frac{1}{16} $ 1.476e-05 2.619e-05 3.175e-06 1.728e-05 2.841e-05 4.990e-05 4.649e-06 3.881e-05
$ \frac{1}{20} $ 7.830e-06 1.360e-05 1.766e-06 9.119e-06 1.577e-05 2.674e-05 2.424e-06 2.101e-05
$ \frac{1}{24} $ 4.653e-06 7.780e-06 1.081e-06 5.333e-06 9.548e-06 1.593e-05 1.403e-06 1.256e-05
k = 4 Case 1: $ \bar{\bar{b}} = 0, \bar{\bar{\sigma}} = 0 $ Case 2: $ \bar{\bar{b}} = b, \bar{\bar{\sigma}} = 0 $ Case 3: $ \bar{\bar{b}} = 0, \bar{\bar{\sigma}} = \sigma $ Case 4: $ \bar{\bar{b}} = b, \bar{\bar{\sigma}} = \sigma $
$ \Delta t $ $ |Y_0-Y^0| $ $ |Z_0-Z^0| $ $ |Y_0-Y^0| $ $ |Z_0-Z^0| $ $ |Y_0-Y^0| $ $ |Z_0-Z^0| $ $ |Y_0-Y^0| $ $ |Z_0-Z^0| $
$ \frac{1}{12} $ 2.616e-06 1.647e-05 1.002e-06 8.273e-06 1.283e-05 2.250e-05 2.193e-06 1.833e-05
$ \frac{1}{16} $ 7.797e-07 5.906e-06 4.216e-07 3.099e-06 4.719e-06 8.419e-06 8.240e-07 7.104e-06
$ \frac{1}{20} $ 3.360e-07 2.582e-06 2.034e-07 1.381e-06 2.095e-06 3.952e-06 3.552e-07 3.226e-06
$ \frac{1}{24} $ 1.662e-07 1.289e-06 1.082e-07 6.960e-07 1.059e-06 2.034e-06 1.752e-07 1.657e-06
k = 5 Case 1: $ \bar{\bar{b}} = 0, \bar{\bar{\sigma}} = 0 $ Case 2: $ \bar{\bar{b}} = b, \bar{\bar{\sigma}} = 0 $ Case 3: $ \bar{\bar{b}} = 0, \bar{\bar{\sigma}} = \sigma $ Case 4: $ \bar{\bar{b}} = b, \bar{\bar{\sigma}} = \sigma $
$ \Delta t $ $ |Y_0-Y^0| $ $ |Z_0-Z^0| $ $ |Y_0-Y^0| $ $ |Z_0-Z^0| $ $ |Y_0-Y^0| $ $ |Z_0-Z^0| $ $ |Y_0-Y^0| $ $ |Z_0-Z^0| $
$ \frac{1}{12} $ 5.529e-07 2.942e-06 2.335e-07 2.457e-06 2.788e-06 6.972e-06 6.110e-07 5.315e-06
$ \frac{1}{16} $ 1.562e-07 8.938e-07 7.126e-08 7.612e-07 8.694e-07 2.140e-06 1.973e-07 1.763e-06
$ \frac{1}{20} $ 5.644e-08 4.817e-07 3.114e-08 4.712e-07 3.377e-07 7.491e-07 7.317e-08 6.889e-07
$ \frac{1}{24} $ 2.403e-08 2.232e-07 1.485e-08 2.823e-07 1.515e-07 3.390e-07 3.036e-08 3.055e-07
k = 6 Case 1: $ \bar{\bar{b}} = 0, \bar{\bar{\sigma}} = 0 $ Case 2: $ \bar{\bar{b}} = b, \bar{\bar{\sigma}} = 0 $ Case 3: $ \bar{\bar{b}} = 0, \bar{\bar{\sigma}} = \sigma $ Case 4: $ \bar{\bar{b}} = b, \bar{\bar{\sigma}} = \sigma $
$ \Delta t $ $ |Y_0-Y^0| $ $ |Z_0-Z^0| $ $ |Y_0-Y^0| $ $ |Z_0-Z^0| $ $ |Y_0-Y^0| $ $ |Z_0-Z^0| $ $ |Y_0-Y^0| $ $ |Z_0-Z^0| $
$ \frac{1}{12} $ 1.913e-07 1.582e-06 1.170e-07 1.451e-06 8.519e-07 3.274e-06 2.211e-07 2.042e-06
$ \frac{1}{16} $ 9.004e-08 1.721e-06 1.030e-07 2.062e-06 2.131e-07 4.811e-07 5.975e-08 5.508e-07
$ \frac{1}{20} $ 1.019e-07 1.799e-06 1.008e-07 2.561e-06 6.204e-08 2.111e-07 1.929e-08 1.814e-07
$ \frac{1}{24} $ 1.070e-07 2.281e-06 1.181e-07 3.290e-06 2.545e-08 2.302e-07 6.984e-09 1.047e-07
Table 5.  The errors of Scheme 4.1 with $ b = 5 $, $ \sigma = 1 $ for Ex. 5.2
k = 1 Case 1: $ \bar{\bar{b}} = 0, \bar{\bar{\sigma}} = 0 $ Case 2: $ \bar{\bar{b}} = b, \bar{\bar{\sigma}} = 0 $ Case 3: $ \bar{\bar{b}} = 0, \bar{\bar{\sigma}} = \sigma $ Case 4: $ \bar{\bar{b}} = b, \bar{\bar{\sigma}} = \sigma $
$ \Delta t $ $ |Y_0-Y^0| $ $ |Z_0-Z^0| $ $ |Y_0-Y^0| $ $ |Z_0-Z^0| $ $ |Y_0-Y^0| $ $ |Z_0-Z^0| $ $ |Y_0-Y^0| $ $ |Z_0-Z^0| $
$ \frac{1}{12} $ 1.895e-02 1.840e-02 8.456e-04 4.131e-03 1.869e-02 1.852e-02 8.619e-04 6.649e-03
$ \frac{1}{16} $ 1.463e-02 1.393e-02 6.377e-04 3.078e-03 1.401e-02 1.423e-02 6.504e-04 5.080e-03
$ \frac{1}{20} $ 1.186e-02 1.128e-02 5.045e-04 2.460e-03 1.133e-02 1.162e-02 5.377e-04 4.117e-03
$ \frac{1}{24} $ 9.949e-03 9.466e-03 4.171e-04 2.048e-03 9.567e-03 9.785e-03 4.556e-04 3.461e-03
k = 2 Case 1: $ \bar{\bar{b}} = 0, \bar{\bar{\sigma}} = 0 $ Case 2: $ \bar{\bar{b}} = b, \bar{\bar{\sigma}} = 0 $ Case 3: $ \bar{\bar{b}} = 0, \bar{\bar{\sigma}} = \sigma $ Case 4: $ \bar{\bar{b}} = b, \bar{\bar{\sigma}} = \sigma $
$ \Delta t $ $ |Y_0-Y^0| $ $ |Z_0-Z^0| $ $ |Y_0-Y^0| $ $ |Z_0-Z^0| $ $ |Y_0-Y^0| $ $ |Z_0-Z^0| $ $ |Y_0-Y^0| $ $ |Z_0-Z^0| $
$ \frac{1}{12} $ 6.605e-03 1.234e-02 5.867e-05 2.740e-04 6.149e-03 1.067e-02 7.012e-05 5.487e-04
$ \frac{1}{16} $ 3.678e-03 6.963e-03 3.660e-05 1.554e-04 3.602e-03 6.584e-03 4.001e-05 3.244e-04
$ \frac{1}{20} $ 2.372e-03 4.374e-03 2.420e-05 9.806e-05 2.370e-03 4.370e-03 2.525e-05 2.125e-04
$ \frac{1}{24} $ 1.692e-03 2.994e-03 1.688e-05 6.797e-05 1.655e-03 3.089e-03 1.840e-05 1.502e-04
k = 3 Case 1: $ \bar{\bar{b}} = 0, \bar{\bar{\sigma}} = 0 $ Case 2: $ \bar{\bar{b}} = b, \bar{\bar{\sigma}} = 0 $ Case 3: $ \bar{\bar{b}} = 0, \bar{\bar{\sigma}} = \sigma $ Case 4: $ \bar{\bar{b}} = b, \bar{\bar{\sigma}} = \sigma $
$ \Delta t $ $ |Y_0-Y^0| $ $ |Z_0-Z^0| $ $ |Y_0-Y^0| $ $ |Z_0-Z^0| $ $ |Y_0-Y^0| $ $ |Z_0-Z^0| $ $ |Y_0-Y^0| $ $ |Z_0-Z^0| $
$ \frac{1}{12} $ 4.074e-03 7.196e-03 6.300e-06 3.728e-05 3.605e-03 5.281e-03 1.017e-05 8.182e-05
$ \frac{1}{16} $ 1.832e-03 2.785e-03 3.285e-06 1.720e-05 1.616e-03 2.493e-03 4.597e-06 3.834e-05
$ \frac{1}{20} $ 9.493e-04 1.456e-03 1.937e-06 8.953e-06 8.535e-04 1.320e-03 2.411e-06 2.089e-05
$ \frac{1}{24} $ 5.516e-04 8.496e-04 1.200e-06 4.954e-06 5.042e-04 7.793e-04 1.408e-06 1.257e-05
k = 4 Case 1: $ \bar{\bar{b}} = 0, \bar{\bar{\sigma}} = 0 $ Case 2: $ \bar{\bar{b}} = b, \bar{\bar{\sigma}} = 0 $ Case 3: $ \bar{\bar{b}} = 0, \bar{\bar{\sigma}} = \sigma $ Case 4: $ \bar{\bar{b}} = b, \bar{\bar{\sigma}} = \sigma $
$ \Delta t $ $ |Y_0-Y^0| $ $ |Z_0-Z^0| $ $ |Y_0-Y^0| $ $ |Z_0-Z^0| $ $ |Y_0-Y^0| $ $ |Z_0-Z^0| $ $ |Y_0-Y^0| $ $ |Z_0-Z^0| $
$ \frac{1}{12} $ 2.175e-03 5.736e-03 1.070e-06 8.069e-06 1.745e-03 3.847e-03 2.138e-06 1.805e-05
$ \frac{1}{16} $ 7.148e-04 1.670e-03 4.136e-07 2.920e-06 5.906e-04 1.404e-03 7.690e-07 6.788e-06
$ \frac{1}{20} $ 3.020e-04 7.275e-04 2.112e-07 1.415e-06 2.757e-04 6.430e-04 3.398e-07 3.093e-06
$ \frac{1}{24} $ 1.532e-04 3.651e-04 1.299e-07 6.663e-07 1.425e-04 3.321e-04 1.726e-07 1.613e-06
k = 5 Case 1: $ \bar{\bar{b}} = 0, \bar{\bar{\sigma}} = 0 $ Case 2: $ \bar{\bar{b}} = b, \bar{\bar{\sigma}} = 0 $ Case 3: $ \bar{\bar{b}} = 0, \bar{\bar{\sigma}} = \sigma $ Case 4: $ \bar{\bar{b}} = b, \bar{\bar{\sigma}} = \sigma $
$ \Delta t $ $ |Y_0-Y^0| $ $ |Z_0-Z^0| $ $ |Y_0-Y^0| $ $ |Z_0-Z^0| $ $ |Y_0-Y^0| $ $ |Z_0-Z^0| $ $ |Y_0-Y^0| $ $ |Z_0-Z^0| $
$ \frac{1}{12} $ 1.738e-03 4.156e-03 2.475e-07 2.509e-06 1.123e-03 2.525e-03 5.901e-07 5.241e-06
$ \frac{1}{16} $ 4.262e-04 1.137e-03 7.288e-08 7.014e-07 3.470e-04 6.684e-04 1.786e-07 1.642e-06
$ \frac{1}{20} $ 1.502e-04 3.231e-04 3.029e-08 5.094e-07 1.304e-04 2.659e-04 6.709e-08 6.430e-07
$ \frac{1}{24} $ 6.136e-05 1.315e-04 1.596e-08 3.320e-07 5.705e-05 1.186e-04 2.704e-08 2.734e-07
k = 6 Case 1: $ \bar{\bar{b}} = 0, \bar{\bar{\sigma}} = 0 $ Case 2: $ \bar{\bar{b}} = b, \bar{\bar{\sigma}} = 0 $ Case 3: $ \bar{\bar{b}} = 0, \bar{\bar{\sigma}} = \sigma $ Case 4: $ \bar{\bar{b}} = b, \bar{\bar{\sigma}} = \sigma $
$ \Delta t $ $ |Y_0-Y^0| $ $ |Z_0-Z^0| $ $ |Y_0-Y^0| $ $ |Z_0-Z^0| $ $ |Y_0-Y^0| $ $ |Z_0-Z^0| $ $ |Y_0-Y^0| $ $ |Z_0-Z^0| $
$ \frac{1}{12} $ 1.301e-03 8.282e-03 1.262e-07 1.682e-06 6.971e-04 1.946e-03 1.860e-07 1.964e-06
$ \frac{1}{16} $ 5.095e-04 4.004e-03 1.125e-07 2.145e-06 1.750e-04 4.942e-04 4.792e-08 4.750e-07
$ \frac{1}{20} $ 2.988e-04 8.144e-04 1.090e-07 2.759e-06 5.535e-05 1.594e-04 1.828e-08 1.708e-07
$ \frac{1}{24} $ 1.828e-04 2.020e-03 1.426e-07 4.105e-06 2.038e-05 1.826e-04 6.291e-09 1.087e-07
k = 1 Case 1: $ \bar{\bar{b}} = 0, \bar{\bar{\sigma}} = 0 $ Case 2: $ \bar{\bar{b}} = b, \bar{\bar{\sigma}} = 0 $ Case 3: $ \bar{\bar{b}} = 0, \bar{\bar{\sigma}} = \sigma $ Case 4: $ \bar{\bar{b}} = b, \bar{\bar{\sigma}} = \sigma $
$ \Delta t $ $ |Y_0-Y^0| $ $ |Z_0-Z^0| $ $ |Y_0-Y^0| $ $ |Z_0-Z^0| $ $ |Y_0-Y^0| $ $ |Z_0-Z^0| $ $ |Y_0-Y^0| $ $ |Z_0-Z^0| $
$ \frac{1}{12} $ 1.895e-02 1.840e-02 8.456e-04 4.131e-03 1.869e-02 1.852e-02 8.619e-04 6.649e-03
$ \frac{1}{16} $ 1.463e-02 1.393e-02 6.377e-04 3.078e-03 1.401e-02 1.423e-02 6.504e-04 5.080e-03
$ \frac{1}{20} $ 1.186e-02 1.128e-02 5.045e-04 2.460e-03 1.133e-02 1.162e-02 5.377e-04 4.117e-03
$ \frac{1}{24} $ 9.949e-03 9.466e-03 4.171e-04 2.048e-03 9.567e-03 9.785e-03 4.556e-04 3.461e-03
k = 2 Case 1: $ \bar{\bar{b}} = 0, \bar{\bar{\sigma}} = 0 $ Case 2: $ \bar{\bar{b}} = b, \bar{\bar{\sigma}} = 0 $ Case 3: $ \bar{\bar{b}} = 0, \bar{\bar{\sigma}} = \sigma $ Case 4: $ \bar{\bar{b}} = b, \bar{\bar{\sigma}} = \sigma $
$ \Delta t $ $ |Y_0-Y^0| $ $ |Z_0-Z^0| $ $ |Y_0-Y^0| $ $ |Z_0-Z^0| $ $ |Y_0-Y^0| $ $ |Z_0-Z^0| $ $ |Y_0-Y^0| $ $ |Z_0-Z^0| $
$ \frac{1}{12} $ 6.605e-03 1.234e-02 5.867e-05 2.740e-04 6.149e-03 1.067e-02 7.012e-05 5.487e-04
$ \frac{1}{16} $ 3.678e-03 6.963e-03 3.660e-05 1.554e-04 3.602e-03 6.584e-03 4.001e-05 3.244e-04
$ \frac{1}{20} $ 2.372e-03 4.374e-03 2.420e-05 9.806e-05 2.370e-03 4.370e-03 2.525e-05 2.125e-04
$ \frac{1}{24} $ 1.692e-03 2.994e-03 1.688e-05 6.797e-05 1.655e-03 3.089e-03 1.840e-05 1.502e-04
k = 3 Case 1: $ \bar{\bar{b}} = 0, \bar{\bar{\sigma}} = 0 $ Case 2: $ \bar{\bar{b}} = b, \bar{\bar{\sigma}} = 0 $ Case 3: $ \bar{\bar{b}} = 0, \bar{\bar{\sigma}} = \sigma $ Case 4: $ \bar{\bar{b}} = b, \bar{\bar{\sigma}} = \sigma $
$ \Delta t $ $ |Y_0-Y^0| $ $ |Z_0-Z^0| $ $ |Y_0-Y^0| $ $ |Z_0-Z^0| $ $ |Y_0-Y^0| $ $ |Z_0-Z^0| $ $ |Y_0-Y^0| $ $ |Z_0-Z^0| $
$ \frac{1}{12} $ 4.074e-03 7.196e-03 6.300e-06 3.728e-05 3.605e-03 5.281e-03 1.017e-05 8.182e-05
$ \frac{1}{16} $ 1.832e-03 2.785e-03 3.285e-06 1.720e-05 1.616e-03 2.493e-03 4.597e-06 3.834e-05
$ \frac{1}{20} $ 9.493e-04 1.456e-03 1.937e-06 8.953e-06 8.535e-04 1.320e-03 2.411e-06 2.089e-05
$ \frac{1}{24} $ 5.516e-04 8.496e-04 1.200e-06 4.954e-06 5.042e-04 7.793e-04 1.408e-06 1.257e-05
k = 4 Case 1: $ \bar{\bar{b}} = 0, \bar{\bar{\sigma}} = 0 $ Case 2: $ \bar{\bar{b}} = b, \bar{\bar{\sigma}} = 0 $ Case 3: $ \bar{\bar{b}} = 0, \bar{\bar{\sigma}} = \sigma $ Case 4: $ \bar{\bar{b}} = b, \bar{\bar{\sigma}} = \sigma $
$ \Delta t $ $ |Y_0-Y^0| $ $ |Z_0-Z^0| $ $ |Y_0-Y^0| $ $ |Z_0-Z^0| $ $ |Y_0-Y^0| $ $ |Z_0-Z^0| $ $ |Y_0-Y^0| $ $ |Z_0-Z^0| $
$ \frac{1}{12} $ 2.175e-03 5.736e-03 1.070e-06 8.069e-06 1.745e-03 3.847e-03 2.138e-06 1.805e-05
$ \frac{1}{16} $ 7.148e-04 1.670e-03 4.136e-07 2.920e-06 5.906e-04 1.404e-03 7.690e-07 6.788e-06
$ \frac{1}{20} $ 3.020e-04 7.275e-04 2.112e-07 1.415e-06 2.757e-04 6.430e-04 3.398e-07 3.093e-06
$ \frac{1}{24} $ 1.532e-04 3.651e-04 1.299e-07 6.663e-07 1.425e-04 3.321e-04 1.726e-07 1.613e-06
k = 5 Case 1: $ \bar{\bar{b}} = 0, \bar{\bar{\sigma}} = 0 $ Case 2: $ \bar{\bar{b}} = b, \bar{\bar{\sigma}} = 0 $ Case 3: $ \bar{\bar{b}} = 0, \bar{\bar{\sigma}} = \sigma $ Case 4: $ \bar{\bar{b}} = b, \bar{\bar{\sigma}} = \sigma $
$ \Delta t $ $ |Y_0-Y^0| $ $ |Z_0-Z^0| $ $ |Y_0-Y^0| $ $ |Z_0-Z^0| $ $ |Y_0-Y^0| $ $ |Z_0-Z^0| $ $ |Y_0-Y^0| $ $ |Z_0-Z^0| $
$ \frac{1}{12} $ 1.738e-03 4.156e-03 2.475e-07 2.509e-06 1.123e-03 2.525e-03 5.901e-07 5.241e-06
$ \frac{1}{16} $ 4.262e-04 1.137e-03 7.288e-08 7.014e-07 3.470e-04 6.684e-04 1.786e-07 1.642e-06
$ \frac{1}{20} $ 1.502e-04 3.231e-04 3.029e-08 5.094e-07 1.304e-04 2.659e-04 6.709e-08 6.430e-07
$ \frac{1}{24} $ 6.136e-05 1.315e-04 1.596e-08 3.320e-07 5.705e-05 1.186e-04 2.704e-08 2.734e-07
k = 6 Case 1: $ \bar{\bar{b}} = 0, \bar{\bar{\sigma}} = 0 $ Case 2: $ \bar{\bar{b}} = b, \bar{\bar{\sigma}} = 0 $ Case 3: $ \bar{\bar{b}} = 0, \bar{\bar{\sigma}} = \sigma $ Case 4: $ \bar{\bar{b}} = b, \bar{\bar{\sigma}} = \sigma $
$ \Delta t $ $ |Y_0-Y^0| $ $ |Z_0-Z^0| $ $ |Y_0-Y^0| $ $ |Z_0-Z^0| $ $ |Y_0-Y^0| $ $ |Z_0-Z^0| $ $ |Y_0-Y^0| $ $ |Z_0-Z^0| $
$ \frac{1}{12} $ 1.301e-03 8.282e-03 1.262e-07 1.682e-06 6.971e-04 1.946e-03 1.860e-07 1.964e-06
$ \frac{1}{16} $ 5.095e-04 4.004e-03 1.125e-07 2.145e-06 1.750e-04 4.942e-04 4.792e-08 4.750e-07
$ \frac{1}{20} $ 2.988e-04 8.144e-04 1.090e-07 2.759e-06 5.535e-05 1.594e-04 1.828e-08 1.708e-07
$ \frac{1}{24} $ 1.828e-04 2.020e-03 1.426e-07 4.105e-06 2.038e-05 1.826e-04 6.291e-09 1.087e-07
Table 6.  The errors of Scheme 4.1 with $ b = 10 $, $ \sigma = 1 $ for Ex. 5.2
k = 1 Case 1: $ \bar{\bar{b}} = 0, \bar{\bar{\sigma}} = 0 $ Case 2: $ \bar{\bar{b}} = b, \bar{\bar{\sigma}} = 0 $ Case 3: $ \bar{\bar{b}} = 0, \bar{\bar{\sigma}} = \sigma $ Case 4: $ \bar{\bar{b}} = b, \bar{\bar{\sigma}} = \sigma $
$ \Delta t $ $ |Y_0-Y^0| $ $ |Z_0-Z^0| $ $ |Y_0-Y^0| $ $ |Z_0-Z^0| $ $ |Y_0-Y^0| $ $ |Z_0-Z^0| $ $ |Y_0-Y^0| $ $ |Z_0-Z^0| $
$ \frac{1}{12} $ 3.768e-02 3.905e-02 8.467e-04 4.106e-03 3.672e-02 3.742e-02 8.551e-04 6.632e-03
$ \frac{1}{16} $ 2.879e-02 2.949e-02 6.256e-04 3.090e-03 2.831e-02 2.836e-02 6.505e-04 5.086e-03
$ \frac{1}{20} $ 2.311e-02 2.459e-02 5.001e-04 2.473e-03 2.287e-02 2.393e-02 5.382e-04 4.119e-03
$ \frac{1}{24} $ 1.920e-02 2.067e-02 4.173e-04 2.059e-03 1.911e-02 2.030e-02 4.571e-04 3.466e-03
k = 2 Case 1: $ \bar{\bar{b}} = 0, \bar{\bar{\sigma}} = 0 $ Case 2: $ \bar{\bar{b}} = b, \bar{\bar{\sigma}} = 0 $ Case 3: $ \bar{\bar{b}} = 0, \bar{\bar{\sigma}} = \sigma $ Case 4: $ \bar{\bar{b}} = b, \bar{\bar{\sigma}} = \sigma $
$ \Delta t $ $ |Y_0-Y^0| $ $ |Z_0-Z^0| $ $ |Y_0-Y^0| $ $ |Z_0-Z^0| $ $ |Y_0-Y^0| $ $ |Z_0-Z^0| $ $ |Y_0-Y^0| $ $ |Z_0-Z^0| $
$ \frac{1}{12} $ 2.710e-02 4.473e-02 6.102e-05 2.732e-04 2.402e-02 3.978e-02 7.074e-05 5.513e-04
$ \frac{1}{16} $ 1.502e-02 2.748e-02 3.425e-05 1.532e-04 1.444e-02 2.551e-02 3.974e-05 3.231e-04
$ \frac{1}{20} $ 1.003e-02 1.821e-02 2.308e-05 1.008e-04 9.176e-03 1.640e-02 2.540e-05 2.130e-04
$ \frac{1}{24} $ 6.951e-03 1.327e-02 1.635e-05 7.013e-05 6.451e-03 1.205e-02 1.841e-05 1.503e-04
k = 3 Case 1: $ \bar{\bar{b}} = 0, \bar{\bar{\sigma}} = 0 $ Case 2: $ \bar{\bar{b}} = b, \bar{\bar{\sigma}} = 0 $ Case 3: $ \bar{\bar{b}} = 0, \bar{\bar{\sigma}} = \sigma $ Case 4: $ \bar{\bar{b}} = b, \bar{\bar{\sigma}} = \sigma $
$ \Delta t $ $ |Y_0-Y^0| $ $ |Z_0-Z^0| $ $ |Y_0-Y^0| $ $ |Z_0-Z^0| $ $ |Y_0-Y^0| $ $ |Z_0-Z^0| $ $ |Y_0-Y^0| $ $ |Z_0-Z^0| $
$ \frac{1}{12} $ 3.127e-02 6.145e-02 6.569e-06 3.761e-05 2.586e-02 4.532e-02 1.013e-05 8.171e-05
$ \frac{1}{16} $ 1.452e-02 3.056e-02 3.294e-06 1.539e-05 1.207e-02 2.181e-02 4.627e-06 3.853e-05
$ \frac{1}{20} $ 7.818e-03 1.373e-02 1.671e-06 9.015e-06 6.879e-03 1.200e-02 2.398e-06 2.080e-05
$ \frac{1}{24} $ 4.342e-03 7.597e-03 1.081e-06 5.331e-06 4.134e-03 6.770e-03 1.403e-06 1.256e-05
k = 4 Case 1: $ \bar{\bar{b}} = 0, \bar{\bar{\sigma}} = 0 $ Case 2: $ \bar{\bar{b}} = b, \bar{\bar{\sigma}} = 0 $ Case 3: $ \bar{\bar{b}} = 0, \bar{\bar{\sigma}} = \sigma $ Case 4: $ \bar{\bar{b}} = b, \bar{\bar{\sigma}} = \sigma $
$ \Delta t $ $ |Y_0-Y^0| $ $ |Z_0-Z^0| $ $ |Y_0-Y^0| $ $ |Z_0-Z^0| $ $ |Y_0-Y^0| $ $ |Z_0-Z^0| $ $ |Y_0-Y^0| $ $ |Z_0-Z^0| $
$ \frac{1}{12} $ 4.507e-02 1.018e-01 1.073e-06 8.016e-06 2.971e-02 5.359e-02 2.129e-06 1.797e-05
$ \frac{1}{16} $ 1.422e-02 4.148e-02 5.479e-07 2.920e-06 1.011e-02 2.178e-02 7.949e-07 6.842e-06
$ \frac{1}{20} $ 4.966e-03 1.289e-02 2.020e-07 1.170e-06 4.130e-03 1.039e-02 3.214e-07 2.975e-06
$ \frac{1}{24} $ 2.407e-03 6.291e-03 1.076e-07 6.872e-07 2.034e-03 4.908e-03 1.724e-07 1.637e-06
k = 5 Case 1: $ \bar{\bar{b}} = 0, \bar{\bar{\sigma}} = 0 $ Case 2: $ \bar{\bar{b}} = b, \bar{\bar{\sigma}} = 0 $ Case 3: $ \bar{\bar{b}} = 0, \bar{\bar{\sigma}} = \sigma $ Case 4: $ \bar{\bar{b}} = b, \bar{\bar{\sigma}} = \sigma $
$ \Delta t $ $ |Y_0-Y^0| $ $ |Z_0-Z^0| $ $ |Y_0-Y^0| $ $ |Z_0-Z^0| $ $ |Y_0-Y^0| $ $ |Z_0-Z^0| $ $ |Y_0-Y^0| $ $ |Z_0-Z^0| $
$ \frac{1}{12} $ 9.803e-02 1.972e-01 2.484e-07 2.519e-06 3.445e-02 7.197e-02 5.880e-07 5.241e-06
$ \frac{1}{16} $ 6.435e-02 1.470e-01 6.926e-08 8.647e-07 9.862e-03 3.819e-02 1.770e-07 1.600e-06
$ \frac{1}{20} $ 2.926e-02 1.051e-01 4.978e-08 4.641e-07 3.456e-03 7.644e-03 7.673e-08 7.162e-07
$ \frac{1}{24} $ 1.005e-02 3.746e-02 1.496e-08 3.401e-07 1.410e-03 3.138e-03 2.581e-08 2.606e-07
k = 6 Case 1: $ \bar{\bar{b}} = 0, \bar{\bar{\sigma}} = 0 $ Case 2: $ \bar{\bar{b}} = b, \bar{\bar{\sigma}} = 0 $ Case 3: $ \bar{\bar{b}} = 0, \bar{\bar{\sigma}} = \sigma $ Case 4: $ \bar{\bar{b}} = b, \bar{\bar{\sigma}} = \sigma $
$ \Delta t $ $ |Y_0-Y^0| $ $ |Z_0-Z^0| $ $ |Y_0-Y^0| $ $ |Z_0-Z^0| $ $ |Y_0-Y^0| $ $ |Z_0-Z^0| $ $ |Y_0-Y^0| $ $ |Z_0-Z^0| $
$ \frac{1}{12} $ 1.846e-01 4.120e-01 1.227e-07 1.679e-06 4.826e-02 1.027e-01 1.849e-07 1.971e-06
$ \frac{1}{16} $ 1.899e-01 4.765e-01 1.111e-07 2.096e-06 2.091e-02 6.755e-02 5.838e-08 5.410e-07
$ \frac{1}{20} $ 1.910e-01 7.164e-01 9.555e-08 2.265e-06 4.921e-03 1.971e-02 2.450e-08 2.008e-07
$ \frac{1}{24} $ 1.524e-01 4.078e-01 1.704e-07 4.768e-06 1.082e-03 1.026e-02 4.522e-09 1.106e-07
k = 1 Case 1: $ \bar{\bar{b}} = 0, \bar{\bar{\sigma}} = 0 $ Case 2: $ \bar{\bar{b}} = b, \bar{\bar{\sigma}} = 0 $ Case 3: $ \bar{\bar{b}} = 0, \bar{\bar{\sigma}} = \sigma $ Case 4: $ \bar{\bar{b}} = b, \bar{\bar{\sigma}} = \sigma $
$ \Delta t $ $ |Y_0-Y^0| $ $ |Z_0-Z^0| $ $ |Y_0-Y^0| $ $ |Z_0-Z^0| $ $ |Y_0-Y^0| $ $ |Z_0-Z^0| $ $ |Y_0-Y^0| $ $ |Z_0-Z^0| $
$ \frac{1}{12} $ 3.768e-02 3.905e-02 8.467e-04 4.106e-03 3.672e-02 3.742e-02 8.551e-04 6.632e-03
$ \frac{1}{16} $ 2.879e-02 2.949e-02 6.256e-04 3.090e-03 2.831e-02 2.836e-02 6.505e-04 5.086e-03
$ \frac{1}{20} $ 2.311e-02 2.459e-02 5.001e-04 2.473e-03 2.287e-02 2.393e-02 5.382e-04 4.119e-03
$ \frac{1}{24} $ 1.920e-02 2.067e-02 4.173e-04 2.059e-03 1.911e-02 2.030e-02 4.571e-04 3.466e-03
k = 2 Case 1: $ \bar{\bar{b}} = 0, \bar{\bar{\sigma}} = 0 $ Case 2: $ \bar{\bar{b}} = b, \bar{\bar{\sigma}} = 0 $ Case 3: $ \bar{\bar{b}} = 0, \bar{\bar{\sigma}} = \sigma $ Case 4: $ \bar{\bar{b}} = b, \bar{\bar{\sigma}} = \sigma $
$ \Delta t $ $ |Y_0-Y^0| $ $ |Z_0-Z^0| $ $ |Y_0-Y^0| $ $ |Z_0-Z^0| $ $ |Y_0-Y^0| $ $ |Z_0-Z^0| $ $ |Y_0-Y^0| $ $ |Z_0-Z^0| $
$ \frac{1}{12} $ 2.710e-02 4.473e-02 6.102e-05 2.732e-04 2.402e-02 3.978e-02 7.074e-05 5.513e-04
$ \frac{1}{16} $ 1.502e-02 2.748e-02 3.425e-05 1.532e-04 1.444e-02 2.551e-02 3.974e-05 3.231e-04
$ \frac{1}{20} $ 1.003e-02 1.821e-02 2.308e-05 1.008e-04 9.176e-03 1.640e-02 2.540e-05 2.130e-04
$ \frac{1}{24} $ 6.951e-03 1.327e-02 1.635e-05 7.013e-05 6.451e-03 1.205e-02 1.841e-05 1.503e-04
k = 3 Case 1: $ \bar{\bar{b}} = 0, \bar{\bar{\sigma}} = 0 $ Case 2: $ \bar{\bar{b}} = b, \bar{\bar{\sigma}} = 0 $ Case 3: $ \bar{\bar{b}} = 0, \bar{\bar{\sigma}} = \sigma $ Case 4: $ \bar{\bar{b}} = b, \bar{\bar{\sigma}} = \sigma $
$ \Delta t $ $ |Y_0-Y^0| $ $ |Z_0-Z^0| $ $ |Y_0-Y^0| $ $ |Z_0-Z^0| $ $ |Y_0-Y^0| $ $ |Z_0-Z^0| $ $ |Y_0-Y^0| $ $ |Z_0-Z^0| $
$ \frac{1}{12} $ 3.127e-02 6.145e-02 6.569e-06 3.761e-05 2.586e-02 4.532e-02 1.013e-05 8.171e-05
$ \frac{1}{16} $ 1.452e-02 3.056e-02 3.294e-06 1.539e-05 1.207e-02 2.181e-02 4.627e-06 3.853e-05
$ \frac{1}{20} $ 7.818e-03 1.373e-02 1.671e-06 9.015e-06 6.879e-03 1.200e-02 2.398e-06 2.080e-05
$ \frac{1}{24} $ 4.342e-03 7.597e-03 1.081e-06 5.331e-06 4.134e-03 6.770e-03 1.403e-06 1.256e-05
k = 4 Case 1: $ \bar{\bar{b}} = 0, \bar{\bar{\sigma}} = 0 $ Case 2: $ \bar{\bar{b}} = b, \bar{\bar{\sigma}} = 0 $ Case 3: $ \bar{\bar{b}} = 0, \bar{\bar{\sigma}} = \sigma $ Case 4: $ \bar{\bar{b}} = b, \bar{\bar{\sigma}} = \sigma $
$ \Delta t $ $ |Y_0-Y^0| $ $ |Z_0-Z^0| $ $ |Y_0-Y^0| $ $ |Z_0-Z^0| $ $ |Y_0-Y^0| $ $ |Z_0-Z^0| $ $ |Y_0-Y^0| $ $ |Z_0-Z^0| $
$ \frac{1}{12} $ 4.507e-02 1.018e-01 1.073e-06 8.016e-06 2.971e-02 5.359e-02 2.129e-06 1.797e-05
$ \frac{1}{16} $ 1.422e-02 4.148e-02 5.479e-07 2.920e-06 1.011e-02 2.178e-02 7.949e-07 6.842e-06
$ \frac{1}{20} $ 4.966e-03 1.289e-02 2.020e-07 1.170e-06 4.130e-03 1.039e-02 3.214e-07 2.975e-06
$ \frac{1}{24} $ 2.407e-03 6.291e-03 1.076e-07 6.872e-07 2.034e-03 4.908e-03 1.724e-07 1.637e-06
k = 5 Case 1: $ \bar{\bar{b}} = 0, \bar{\bar{\sigma}} = 0 $ Case 2: $ \bar{\bar{b}} = b, \bar{\bar{\sigma}} = 0 $ Case 3: $ \bar{\bar{b}} = 0, \bar{\bar{\sigma}} = \sigma $ Case 4: $ \bar{\bar{b}} = b, \bar{\bar{\sigma}} = \sigma $
$ \Delta t $ $ |Y_0-Y^0| $ $ |Z_0-Z^0| $ $ |Y_0-Y^0| $ $ |Z_0-Z^0| $ $ |Y_0-Y^0| $ $ |Z_0-Z^0| $ $ |Y_0-Y^0| $ $ |Z_0-Z^0| $
$ \frac{1}{12} $ 9.803e-02 1.972e-01 2.484e-07 2.519e-06 3.445e-02 7.197e-02 5.880e-07 5.241e-06
$ \frac{1}{16} $ 6.435e-02 1.470e-01 6.926e-08 8.647e-07 9.862e-03 3.819e-02 1.770e-07 1.600e-06
$ \frac{1}{20} $ 2.926e-02 1.051e-01 4.978e-08 4.641e-07 3.456e-03 7.644e-03 7.673e-08 7.162e-07
$ \frac{1}{24} $ 1.005e-02 3.746e-02 1.496e-08 3.401e-07 1.410e-03 3.138e-03 2.581e-08 2.606e-07
k = 6 Case 1: $ \bar{\bar{b}} = 0, \bar{\bar{\sigma}} = 0 $ Case 2: $ \bar{\bar{b}} = b, \bar{\bar{\sigma}} = 0 $ Case 3: $ \bar{\bar{b}} = 0, \bar{\bar{\sigma}} = \sigma $ Case 4: $ \bar{\bar{b}} = b, \bar{\bar{\sigma}} = \sigma $
$ \Delta t $ $ |Y_0-Y^0| $ $ |Z_0-Z^0| $ $ |Y_0-Y^0| $ $ |Z_0-Z^0| $ $ |Y_0-Y^0| $ $ |Z_0-Z^0| $ $ |Y_0-Y^0| $ $ |Z_0-Z^0| $
$ \frac{1}{12} $ 1.846e-01 4.120e-01 1.227e-07 1.679e-06 4.826e-02 1.027e-01 1.849e-07 1.971e-06
$ \frac{1}{16} $ 1.899e-01 4.765e-01 1.111e-07 2.096e-06 2.091e-02 6.755e-02 5.838e-08 5.410e-07
$ \frac{1}{20} $ 1.910e-01 7.164e-01 9.555e-08 2.265e-06 4.921e-03 1.971e-02 2.450e-08 2.008e-07
$ \frac{1}{24} $ 1.524e-01 4.078e-01 1.704e-07 4.768e-06 1.082e-03 1.026e-02 4.522e-09 1.106e-07
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