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doi: 10.3934/dcdss.2021044
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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 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 & Continuous Dynamical Systems - S, doi: 10.3934/dcdss.2021044
References:
[1]

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

[2]

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

[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. Google Scholar

[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.  Google Scholar

[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.  Google Scholar

[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.  Google Scholar

[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.  Google Scholar

[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.  Google Scholar

[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.  Google Scholar

[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.  Google Scholar

[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.  Google Scholar

[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.  Google Scholar

[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.  Google Scholar

[14]

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

[15]

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

[16]

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

[17]

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

[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.  Google Scholar

[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.  Google Scholar

[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.  Google Scholar

[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.  Google Scholar

[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.  Google Scholar

[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.  Google Scholar

[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.  Google Scholar

[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.  Google Scholar

[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.  Google Scholar

[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.  Google Scholar

[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.  Google Scholar

[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.   Google Scholar

show all references

References:
[1]

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

[2]

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

[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. Google Scholar

[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.  Google Scholar

[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.  Google Scholar

[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.  Google Scholar

[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.  Google Scholar

[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.  Google Scholar

[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.  Google Scholar

[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.  Google Scholar

[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.  Google Scholar

[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.  Google Scholar

[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.  Google Scholar

[14]

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

[15]

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

[16]

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

[17]

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

[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.  Google Scholar

[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.  Google Scholar

[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.  Google Scholar

[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.  Google Scholar

[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.  Google Scholar

[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.  Google Scholar

[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.  Google Scholar

[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.  Google Scholar

[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.  Google Scholar

[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.  Google Scholar

[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.  Google Scholar

[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.   Google Scholar

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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