doi: 10.3934/dcdsb.2021233
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High order one-step methods for backward stochastic differential equations via Itô-Taylor expansion

1. 

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

2. 

School of Mathematics, Shandong University, Jinan 250100, China

* Corresponding author

Received  November 2020 Revised  July 2021 Early access September 2021

Fund Project: This research is partially supported by the NSF of China (No. 12001539), the NSF of Hunan Province (No. 2020JJ5647) and China Postdoctoral Science Foundation (No. 2019TQ0073)

In this work, by combining the Feynman-Kac formula with an Itô-Taylor expansion, we propose a class of high order one-step schemes for backward stochastic differential equations, which can achieve at most six order rate of convergence and only need the terminal conditions on the last one step. Numerical experiments are carried out to show the efficiency and high order accuracy of the proposed schemes.

Citation: Quan Zhou, Yabing Sun. High order one-step methods for backward stochastic differential equations via Itô-Taylor expansion. Discrete & Continuous Dynamical Systems - B, doi: 10.3934/dcdsb.2021233
References:
[1]

M. Abramowitz and I. A. Stegun, Handbook of Mathematical Functions, Dover, New York, 1966. doi: 10.2307/2314682.  Google Scholar

[2]

E. Bayraktar and S. Yao, Quadratic reflected BSDEs with unbounded obstacles, Stochastic Process. Appl., 122 (2012), 1155-1203.  doi: 10.1016/j.spa.2011.12.013.  Google Scholar

[3]

C. BeckW. E and A. Jentzen, Machine learning approximation algorithms for high-dimensional fully nonlinear partial differential equations and second-order backward stochastic differential equations, J. Nonlinear Sci., 29 (2019), 1563-1619.  doi: 10.1007/s00332-018-9525-3.  Google Scholar

[4]

B. BouchardX. Tan and X. Warin, Numerical approximation of general Lipschitz BSDEs with branching processes, ESAIM Proc. Surveys, 65 (2019), 309-329.  doi: 10.1051/proc/201965309.  Google Scholar

[5]

B. BouchardX. TanX. Warin and Y. Zou, Numerical approximation of BSDEs using local polynomial drivers and branching processes, Monte Carlo Methods Appl., 23 (2017), 241-263.  doi: 10.1515/mcma-2017-0116.  Google Scholar

[6]

B. Bouchard and N. Touzi, Discrete-time approximation and Monte-Carlo simulation of backward stochastic differential equations, Stochastic Process. Appl., 111 (2014), 175-206.  doi: 10.1016/j.spa.2004.01.001.  Google Scholar

[7]

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

[8]

P. CheriditoH. M. SonerN. Touzi and N. Victoir, Second-order backward stochastic differential equations and fully nonlinear parabolic PDEs, Comm. Pure Appl. Math., 60 (2007), 1081-1110.  doi: 10.1002/cpa.20168.  Google Scholar

[9] G. Da Prato and J. Zabczyk, Second Order Partial Differential Equations in Hilbert Spaces, Cambridge University Press, London, 2002.  doi: 10.1017/CBO9780511543210.  Google Scholar
[10]

W. EJ. Han and A. Jentzen, Deep learning-based numerical methods for high-dimensional parabolic partial differential equations and backward stochastic differential equations, Commun. Math. Stat., 5 (2017), 349-380.  doi: 10.1007/s40304-017-0117-6.  Google Scholar

[11]

W. EM. HutzenthalerA. Jentzen and T. Kruse, On multilevel picard numerical approximations for high-dimensional nonlinear parabolic partial differential equations and high-dimensional nonlinear backward stochastic differential equations, J. Sci. Comput., 79 (2019), 1534-1571.  doi: 10.1007/s10915-018-00903-0.  Google Scholar

[12]

N. El KarouiC. KapoudjianE. PardouxS. Peng and M. C. Quenez, Reflected solutions of backward SDE's, and related obstacle problems for PDE's, Ann. Probab., 25 (1997), 702-737.  doi: 10.1214/aop/1024404416.  Google Scholar

[13]

N. El KarouiS. Peng and M. C. Quenez, Backward stochastic differential equations in finance, Math. Finance, 7 (1997), 1-71.  doi: 10.1111/1467-9965.00022.  Google Scholar

[14]

Y. FuW. Zhao and T. Zhou, Efficient spectral sparse grid approximations for solving multi-dimensional forward backward SDEs, Discrete Contin. Dyn. Syst. Ser. B, 22 (2017), 3439-3458.  doi: 10.3934/dcdsb.2017174.  Google Scholar

[15]

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

[16]

E. GobetJ.-P. Lemor and X. Warin, A regression-based Monte Carlo method to solve backward stochastic differential equations, Ann. Appl. Probab., 15 (2005), 2172-2202.  doi: 10.1214/105051605000000412.  Google Scholar

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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

[18]

P. Henry-LabordèreX. Tan and N. Touzi, A numerical algorithm for a class of BSDEs via the branching process, Stochastic Process. Appl., 124 (2014), 1112-1140.  doi: 10.1016/j.spa.2013.10.005.  Google Scholar

[19]

M. Hutzenthaler, A. Jentzen, T. Kruse and T. A. Nguyen, Multilevel picard approximations for high-dimensional semilinear second-order PDEs with lipschitz nonlinearities, preprint, arXiv: 2009.02484v4. Google Scholar

[20]

L. Kapllani and L. Teng, Deep learning algorithms for solving high dimensional nonlinear backward stochastic differential equations, preprint, arXiv: 2010.01319. Google Scholar

[21]

P. E. Kloeden and E. Platen, Numerical Solution of Stochastic Differential Equations, Springer-Verlag, Berlin, 1992. doi: 10.1007/978-3-662-12616-5.  Google Scholar

[22]

A. LionnetG. dos Reis and L. Szpruch, Time discretization of FBSDE with polynomial growth drivers and reaction-diffusion PDEs, Ann. Appl. Probab., 25 (2015), 2563-2625.  doi: 10.1214/14-AAP1056.  Google Scholar

[23]

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

[24]

C.-K. PakM.-C. Kim and C.-H. Rim, An efficient third-order scheme for BSDEs based on nonequidistant difference scheme, Numer. Algorithms, 85 (2020), 467-483.  doi: 10.1007/s11075-019-00822-7.  Google Scholar

[25]

É. Pardoux and S. G. 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

[26]

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

[27]

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

[28]

E. Rosazza Gianin, Risk measures via $g$-expectations, Insurance Math. Econom., 39 (2006), 19-34.  doi: 10.1016/j.insmatheco.2006.01.002.  Google Scholar

[29]

Y. Sun and W. Zhao, New second-order schemes for forward backward stochastic differential equations, East Asian J. Appl. Math., 8 (2018), 399-421.  doi: 10.4208/eajam.100118.070318.  Google Scholar

[30]

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

[31]

Y. Sun and W. Zhao, An explicit second order scheme for decoupled anticipated forward backward stochastic differential equations, East Asian J. Appl. Math., 10 (2020), 566-593.  doi: 10.4208/eajam.271119.200220.  Google Scholar

[32]

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

[33]

L. TengA. Lapitckii and M. Güenther, A multi-step scheme based on cubic spline for solving backward stochastic differential equations, Appl. Numer. Math., 150 (2020), 117-138.  doi: 10.1016/j.apnum.2019.09.016.  Google Scholar

[34]

C. ZhangJ. Wu and W. Zhao, One-step multi-derivative methods for backward stochastic differential equations, Numer. Math. Theor. Meth. Appl., 12 (2019), 1213-1230.  doi: 10.4208/nmtma.OA-2018-0122.  Google Scholar

[35]

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

[36]

W. Zhao, Y. Fu and T. Zhou, New kinds of high-order multistep schemes for coupled forward backward stochastic differential equations, SIAM J. Sci. Comput., 36 (2014), A1731–A1751. doi: 10.1137/130941274.  Google Scholar

[37]

W. ZhaoY. Li and G. Zhang, A generalized $\theta$-scheme for solving backward stochastic differential equations, Discrete Contin. Dyn. Syst. Ser. B, 17 (2012), 1585-1603.  doi: 10.3934/dcdsb.2012.17.1585.  Google Scholar

[38]

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

[39]

W. ZhaoG. Zhang and L. Ju, A stable multistep scheme for solving backward stochastic differential equations, SIAM J. Numer. Anal., 48 (2010), 1369-1394.  doi: 10.1137/09076979X.  Google Scholar

[40]

Q. Zhou and Y. Sun, Explicit high order one-step methods for decoupled forward backward stochastic differential equations, Adv. Appl. Math. Mech., 13 (2021), 1293-1317.  doi: 10.4208/aamm.OA-2020-0133.  Google Scholar

show all references

References:
[1]

M. Abramowitz and I. A. Stegun, Handbook of Mathematical Functions, Dover, New York, 1966. doi: 10.2307/2314682.  Google Scholar

[2]

E. Bayraktar and S. Yao, Quadratic reflected BSDEs with unbounded obstacles, Stochastic Process. Appl., 122 (2012), 1155-1203.  doi: 10.1016/j.spa.2011.12.013.  Google Scholar

[3]

C. BeckW. E and A. Jentzen, Machine learning approximation algorithms for high-dimensional fully nonlinear partial differential equations and second-order backward stochastic differential equations, J. Nonlinear Sci., 29 (2019), 1563-1619.  doi: 10.1007/s00332-018-9525-3.  Google Scholar

[4]

B. BouchardX. Tan and X. Warin, Numerical approximation of general Lipschitz BSDEs with branching processes, ESAIM Proc. Surveys, 65 (2019), 309-329.  doi: 10.1051/proc/201965309.  Google Scholar

[5]

B. BouchardX. TanX. Warin and Y. Zou, Numerical approximation of BSDEs using local polynomial drivers and branching processes, Monte Carlo Methods Appl., 23 (2017), 241-263.  doi: 10.1515/mcma-2017-0116.  Google Scholar

[6]

B. Bouchard and N. Touzi, Discrete-time approximation and Monte-Carlo simulation of backward stochastic differential equations, Stochastic Process. Appl., 111 (2014), 175-206.  doi: 10.1016/j.spa.2004.01.001.  Google Scholar

[7]

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

[8]

P. CheriditoH. M. SonerN. Touzi and N. Victoir, Second-order backward stochastic differential equations and fully nonlinear parabolic PDEs, Comm. Pure Appl. Math., 60 (2007), 1081-1110.  doi: 10.1002/cpa.20168.  Google Scholar

[9] G. Da Prato and J. Zabczyk, Second Order Partial Differential Equations in Hilbert Spaces, Cambridge University Press, London, 2002.  doi: 10.1017/CBO9780511543210.  Google Scholar
[10]

W. EJ. Han and A. Jentzen, Deep learning-based numerical methods for high-dimensional parabolic partial differential equations and backward stochastic differential equations, Commun. Math. Stat., 5 (2017), 349-380.  doi: 10.1007/s40304-017-0117-6.  Google Scholar

[11]

W. EM. HutzenthalerA. Jentzen and T. Kruse, On multilevel picard numerical approximations for high-dimensional nonlinear parabolic partial differential equations and high-dimensional nonlinear backward stochastic differential equations, J. Sci. Comput., 79 (2019), 1534-1571.  doi: 10.1007/s10915-018-00903-0.  Google Scholar

[12]

N. El KarouiC. KapoudjianE. PardouxS. Peng and M. C. Quenez, Reflected solutions of backward SDE's, and related obstacle problems for PDE's, Ann. Probab., 25 (1997), 702-737.  doi: 10.1214/aop/1024404416.  Google Scholar

[13]

N. El KarouiS. Peng and M. C. Quenez, Backward stochastic differential equations in finance, Math. Finance, 7 (1997), 1-71.  doi: 10.1111/1467-9965.00022.  Google Scholar

[14]

Y. FuW. Zhao and T. Zhou, Efficient spectral sparse grid approximations for solving multi-dimensional forward backward SDEs, Discrete Contin. Dyn. Syst. Ser. B, 22 (2017), 3439-3458.  doi: 10.3934/dcdsb.2017174.  Google Scholar

[15]

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

[16]

E. GobetJ.-P. Lemor and X. Warin, A regression-based Monte Carlo method to solve backward stochastic differential equations, Ann. Appl. Probab., 15 (2005), 2172-2202.  doi: 10.1214/105051605000000412.  Google Scholar

[17]

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

[18]

P. Henry-LabordèreX. Tan and N. Touzi, A numerical algorithm for a class of BSDEs via the branching process, Stochastic Process. Appl., 124 (2014), 1112-1140.  doi: 10.1016/j.spa.2013.10.005.  Google Scholar

[19]

M. Hutzenthaler, A. Jentzen, T. Kruse and T. A. Nguyen, Multilevel picard approximations for high-dimensional semilinear second-order PDEs with lipschitz nonlinearities, preprint, arXiv: 2009.02484v4. Google Scholar

[20]

L. Kapllani and L. Teng, Deep learning algorithms for solving high dimensional nonlinear backward stochastic differential equations, preprint, arXiv: 2010.01319. Google Scholar

[21]

P. E. Kloeden and E. Platen, Numerical Solution of Stochastic Differential Equations, Springer-Verlag, Berlin, 1992. doi: 10.1007/978-3-662-12616-5.  Google Scholar

[22]

A. LionnetG. dos Reis and L. Szpruch, Time discretization of FBSDE with polynomial growth drivers and reaction-diffusion PDEs, Ann. Appl. Probab., 25 (2015), 2563-2625.  doi: 10.1214/14-AAP1056.  Google Scholar

[23]

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

[24]

C.-K. PakM.-C. Kim and C.-H. Rim, An efficient third-order scheme for BSDEs based on nonequidistant difference scheme, Numer. Algorithms, 85 (2020), 467-483.  doi: 10.1007/s11075-019-00822-7.  Google Scholar

[25]

É. Pardoux and S. G. 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

[26]

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

[27]

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

[28]

E. Rosazza Gianin, Risk measures via $g$-expectations, Insurance Math. Econom., 39 (2006), 19-34.  doi: 10.1016/j.insmatheco.2006.01.002.  Google Scholar

[29]

Y. Sun and W. Zhao, New second-order schemes for forward backward stochastic differential equations, East Asian J. Appl. Math., 8 (2018), 399-421.  doi: 10.4208/eajam.100118.070318.  Google Scholar

[30]

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

[31]

Y. Sun and W. Zhao, An explicit second order scheme for decoupled anticipated forward backward stochastic differential equations, East Asian J. Appl. Math., 10 (2020), 566-593.  doi: 10.4208/eajam.271119.200220.  Google Scholar

[32]

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

[33]

L. TengA. Lapitckii and M. Güenther, A multi-step scheme based on cubic spline for solving backward stochastic differential equations, Appl. Numer. Math., 150 (2020), 117-138.  doi: 10.1016/j.apnum.2019.09.016.  Google Scholar

[34]

C. ZhangJ. Wu and W. Zhao, One-step multi-derivative methods for backward stochastic differential equations, Numer. Math. Theor. Meth. Appl., 12 (2019), 1213-1230.  doi: 10.4208/nmtma.OA-2018-0122.  Google Scholar

[35]

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

[36]

W. Zhao, Y. Fu and T. Zhou, New kinds of high-order multistep schemes for coupled forward backward stochastic differential equations, SIAM J. Sci. Comput., 36 (2014), A1731–A1751. doi: 10.1137/130941274.  Google Scholar

[37]

W. ZhaoY. Li and G. Zhang, A generalized $\theta$-scheme for solving backward stochastic differential equations, Discrete Contin. Dyn. Syst. Ser. B, 17 (2012), 1585-1603.  doi: 10.3934/dcdsb.2012.17.1585.  Google Scholar

[38]

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

[39]

W. ZhaoG. Zhang and L. Ju, A stable multistep scheme for solving backward stochastic differential equations, SIAM J. Numer. Anal., 48 (2010), 1369-1394.  doi: 10.1137/09076979X.  Google Scholar

[40]

Q. Zhou and Y. Sun, Explicit high order one-step methods for decoupled forward backward stochastic differential equations, Adv. Appl. Math. Mech., 13 (2021), 1293-1317.  doi: 10.4208/aamm.OA-2020-0133.  Google Scholar

Figure 1.  The plots of $ \log_2(|Y_0-Y^0|) $ and $ \log_2(|Z_0-Z^0|) $ w.r.t. $ \log_2(\triangle t) $ for Sch. 3.1.
Figure 2.  The plots of $ \log_2(|Y_0-Y^0|) $ and $ \log_2(|Z_0-Z^0|) $ w.r.t. $ \log_2(\triangle t) $ for Sch. 3.2.
Table 1.  Errors and convergence rates of Scheme 3.1
$ \theta_2=0 $ $ \theta_2=\frac{3}{10} $ $ \theta_2=\frac{2}{5} $
$ N $ $ |Y_0-Y^{0}| $ $ |Z_0-Z^{0}| $ $ |Y_0-Y^{0}| $ $ |Z_0-Z^{0}| $ $ |Y_0-Y^{0}| $ $ |Z_0-Z^{0}| $
$ 30 $ 1.900E-10 4.870E-10 2.571E-10 1.352E-09 4.061E-10 1.640E-09
$ 40 $ 3.904E-11 1.136E-10 8.297E-11 3.120E-10 1.236E-10 3.781E-10
$ 50 $ 1.115E-11 3.660E-11 3.300E-11 9.929E-11 4.771E-11 1.202E-10
$ 60 $ 4.883E-12 1.478E-11 1.163E-11 4.090E-11 1.713E-11 4.962E-11
$ 70 $ 2.558E-12 6.903E-12 4.084E-12 1.961E-11 6.298E-12 2.384E-11
CR 5.128 5.028 4.810 5.008 4.850 5.006
$\theta_2=\frac{1}{2}$ $\theta_2=\frac{7}{10}$ $\theta_2=1$
$N$ $|Y_0-Y^{0}|$ $|Z_0-Z^{0}|$ $|Y_0-Y^{0}|$ $|Z_0-Z^{0}|$ $|Y_0-Y^{0}|$ $|Z_0-Z^{0}|$
$30$ 5.551E-10 1.929E-09 8.532E-10 2.505E-09 1.300E-09 3.370E-09
$40$ 1.643E-10 4.442E-10 2.457E-10 5.765E-10 3.677E-10 7.748E-10
$50$ 6.243E-11 1.411E-10 9.186E-11 1.829E-10 1.360E-10 2.456E-10
$60$ 2.264E-11 5.833E-11 3.365E-11 7.575E-11 5.016E-11 1.019E-10
$70$ 8.513E-12 2.808E-11 1.294E-11 3.654E-11 1.958E-11 4.925E-11
CR 4.870 5.005 4.889 5.003 4.901 5.001
$ \theta_2=0 $ $ \theta_2=\frac{3}{10} $ $ \theta_2=\frac{2}{5} $
$ N $ $ |Y_0-Y^{0}| $ $ |Z_0-Z^{0}| $ $ |Y_0-Y^{0}| $ $ |Z_0-Z^{0}| $ $ |Y_0-Y^{0}| $ $ |Z_0-Z^{0}| $
$ 30 $ 1.900E-10 4.870E-10 2.571E-10 1.352E-09 4.061E-10 1.640E-09
$ 40 $ 3.904E-11 1.136E-10 8.297E-11 3.120E-10 1.236E-10 3.781E-10
$ 50 $ 1.115E-11 3.660E-11 3.300E-11 9.929E-11 4.771E-11 1.202E-10
$ 60 $ 4.883E-12 1.478E-11 1.163E-11 4.090E-11 1.713E-11 4.962E-11
$ 70 $ 2.558E-12 6.903E-12 4.084E-12 1.961E-11 6.298E-12 2.384E-11
CR 5.128 5.028 4.810 5.008 4.850 5.006
$\theta_2=\frac{1}{2}$ $\theta_2=\frac{7}{10}$ $\theta_2=1$
$N$ $|Y_0-Y^{0}|$ $|Z_0-Z^{0}|$ $|Y_0-Y^{0}|$ $|Z_0-Z^{0}|$ $|Y_0-Y^{0}|$ $|Z_0-Z^{0}|$
$30$ 5.551E-10 1.929E-09 8.532E-10 2.505E-09 1.300E-09 3.370E-09
$40$ 1.643E-10 4.442E-10 2.457E-10 5.765E-10 3.677E-10 7.748E-10
$50$ 6.243E-11 1.411E-10 9.186E-11 1.829E-10 1.360E-10 2.456E-10
$60$ 2.264E-11 5.833E-11 3.365E-11 7.575E-11 5.016E-11 1.019E-10
$70$ 8.513E-12 2.808E-11 1.294E-11 3.654E-11 1.958E-11 4.925E-11
CR 4.870 5.005 4.889 5.003 4.901 5.001
Table 2.  Errors and convergence rates of Scheme 3.2
$ \theta_2=0 $ $ \theta_2=\frac{3}{10} $ $ \theta_2=\frac{2}{5} $
$ N $ $ |Y_0-Y^{0}| $ $ |Z_0-Z^{0}| $ $ |Y_0-Y^{0}| $ $ |Z_0-Z^{0}| $ $ |Y_0-Y^{0}| $ $ |Z_0-Z^{0}| $
$ 30 $ 6.977E-12 1.462E-11 1.584E-11 3.048E-11 1.880E-11 3.576E-11
$ 40 $ 1.238E-12 2.621E-12 2.920E-12 5.410E-12 3.481E-12 6.340E-12
$ 50 $ 2.914E-13 7.099E-13 7.284E-13 1.448E-12 8.737E-13 1.690E-12
$ 60 $ 7.283E-14 2.430E-13 2.172E-13 4.964E-13 2.661E-13 5.821E-13
$ 70 $ 2.698E-14 2.542E-14 7.605E-14 2.017E-13 8.982E-14 2.398E-13
CR 6.612 7.045 6.298 5.921 6.286 5.910
$\theta_2=\frac{1}{2}$ $\theta_2=\frac{7}{10}$ $\theta_2=1$
$N$ $|Y_0-Y^{0}|$ $|Z_0-Z^{0}|$ $|Y_0-Y^{0}|$ $|Z_0-Z^{0}|$ $|Y_0-Y^{0}|$ $|Z_0-Z^{0}|$
$30$ 2.176E-11 4.106E-11 2.767E-11 5.163E-11 3.653E-11 6.749E-11
$40$ 4.042E-12 7.276E-12 5.163E-12 9.138E-12 6.846E-12 1.191E-11
$50$ 1.019E-12 1.942E-12 1.309E-12 2.429E-12 1.746E-12 3.170E-12
$60$ 3.140E-13 6.706E-13 4.107E-13 8.249E-13 5.551E-13 1.078E-12
$70$ 1.161E-13 2.445E-13 1.448E-13 3.592E-13 2.083E-13 4.128E-13
CR 6.183 6.008 6.187 5.888 6.103 5.996
$ \theta_2=0 $ $ \theta_2=\frac{3}{10} $ $ \theta_2=\frac{2}{5} $
$ N $ $ |Y_0-Y^{0}| $ $ |Z_0-Z^{0}| $ $ |Y_0-Y^{0}| $ $ |Z_0-Z^{0}| $ $ |Y_0-Y^{0}| $ $ |Z_0-Z^{0}| $
$ 30 $ 6.977E-12 1.462E-11 1.584E-11 3.048E-11 1.880E-11 3.576E-11
$ 40 $ 1.238E-12 2.621E-12 2.920E-12 5.410E-12 3.481E-12 6.340E-12
$ 50 $ 2.914E-13 7.099E-13 7.284E-13 1.448E-12 8.737E-13 1.690E-12
$ 60 $ 7.283E-14 2.430E-13 2.172E-13 4.964E-13 2.661E-13 5.821E-13
$ 70 $ 2.698E-14 2.542E-14 7.605E-14 2.017E-13 8.982E-14 2.398E-13
CR 6.612 7.045 6.298 5.921 6.286 5.910
$\theta_2=\frac{1}{2}$ $\theta_2=\frac{7}{10}$ $\theta_2=1$
$N$ $|Y_0-Y^{0}|$ $|Z_0-Z^{0}|$ $|Y_0-Y^{0}|$ $|Z_0-Z^{0}|$ $|Y_0-Y^{0}|$ $|Z_0-Z^{0}|$
$30$ 2.176E-11 4.106E-11 2.767E-11 5.163E-11 3.653E-11 6.749E-11
$40$ 4.042E-12 7.276E-12 5.163E-12 9.138E-12 6.846E-12 1.191E-11
$50$ 1.019E-12 1.942E-12 1.309E-12 2.429E-12 1.746E-12 3.170E-12
$60$ 3.140E-13 6.706E-13 4.107E-13 8.249E-13 5.551E-13 1.078E-12
$70$ 1.161E-13 2.445E-13 1.448E-13 3.592E-13 2.083E-13 4.128E-13
CR 6.183 6.008 6.187 5.888 6.103 5.996
Table 3.  Errors and convergence rates of the Euler, C-N and multi-step schemes
Euler Crank-Nicolson 3-step
$ N $ $ |Y_0-Y^{0}| $ $ |Z_0-Z^{0}| $ $ |Y_0-Y^{0}| $ $ |Z_0-Z^{0}| $ $ |Y_0-Y^{0}| $ $ |Z_0-Z^{0}| $
$ 30 $ 5.533E-02 4.186E-02 1.714E-04 2.541E-04 3.355E-05 3.992E-05
$ 40 $ 4.176E-02 3.130E-02 9.883E-05 1.348E-04 1.452E-05 1.662E-05
$ 50 $ 3.356E-02 2.448E-02 6.315E-05 8.619E-05 7.549E-06 8.439E-06
$ 60 $ 2.799E-02 2.092E-02 4.513E-05 5.539E-05 4.411E-06 4.855E-06
$ 70 $ 2.403E-02 1.796E-02 3.356E-05 3.969E-05 2.797E-06 3.044E-06
CR 0.985 1.000 1.927 2.188 2.932 3.037
4-step 5-step 6-step
$N$ $|Y_0-Y^{0}|$ $|Z_0-Z^{0}|$ $|Y_0-Y^{0}|$ $|Z_0-Z^{0}|$ $|Y_0-Y^{0}|$ $|Z_0-Z^{0}|$
$30$ 2.764E-07 5.529E-07 3.497E-08 7.795E-08 1.683E-09 1.863E-09
$40$ 8.634E-08 2.089E-07 9.029E-09 1.918E-08 3.175E-10 3.188E-10
$50$ 3.494E-08 9.405E-08 3.107E-09 6.418E-09 8.611E-11 8.138E-11
$60$ 1.668E-08 4.811E-08 1.289E-09 2.614E-09 2.943E-11 2.671E-11
$70$ 8.927E-09 2.709E-08 6.099E-10 1.221E-09 1.179E-11 1.033E-11
CR 4.052 3.563 4.779 4.906 5.853 6.128
Euler Crank-Nicolson 3-step
$ N $ $ |Y_0-Y^{0}| $ $ |Z_0-Z^{0}| $ $ |Y_0-Y^{0}| $ $ |Z_0-Z^{0}| $ $ |Y_0-Y^{0}| $ $ |Z_0-Z^{0}| $
$ 30 $ 5.533E-02 4.186E-02 1.714E-04 2.541E-04 3.355E-05 3.992E-05
$ 40 $ 4.176E-02 3.130E-02 9.883E-05 1.348E-04 1.452E-05 1.662E-05
$ 50 $ 3.356E-02 2.448E-02 6.315E-05 8.619E-05 7.549E-06 8.439E-06
$ 60 $ 2.799E-02 2.092E-02 4.513E-05 5.539E-05 4.411E-06 4.855E-06
$ 70 $ 2.403E-02 1.796E-02 3.356E-05 3.969E-05 2.797E-06 3.044E-06
CR 0.985 1.000 1.927 2.188 2.932 3.037
4-step 5-step 6-step
$N$ $|Y_0-Y^{0}|$ $|Z_0-Z^{0}|$ $|Y_0-Y^{0}|$ $|Z_0-Z^{0}|$ $|Y_0-Y^{0}|$ $|Z_0-Z^{0}|$
$30$ 2.764E-07 5.529E-07 3.497E-08 7.795E-08 1.683E-09 1.863E-09
$40$ 8.634E-08 2.089E-07 9.029E-09 1.918E-08 3.175E-10 3.188E-10
$50$ 3.494E-08 9.405E-08 3.107E-09 6.418E-09 8.611E-11 8.138E-11
$60$ 1.668E-08 4.811E-08 1.289E-09 2.614E-09 2.943E-11 2.671E-11
$70$ 8.927E-09 2.709E-08 6.099E-10 1.221E-09 1.179E-11 1.033E-11
CR 4.052 3.563 4.779 4.906 5.853 6.128
Table 4.  Errors and convergence rates of Scheme 3.1
$ \theta_2=0 $ $ \theta_2=\frac{3}{10} $ $ \theta_2=\frac{2}{5} $
$ N $ $ |Y_0-Y^{0}| $ $ |Z_0-Z^{0}| $ $ |Y_0-Y^{0}| $ $ |Z_0-Z^{0}| $ $ |Y_0-Y^{0}| $ $ |Z_0-Z^{0}| $
$ 20 $ 1.826E-10 1.007E-09 1.912E-10 1.391E-09 1.914E-10 1.578E-09
$ 30 $ 2.753E-11 1.342E-10 2.860E-11 1.937E-10 2.871E-11 2.186E-10
$ 40 $ 6.742E-12 3.115E-11 7.027E-12 4.862E-11 7.076E-12 5.457E-11
$ 50 $ 2.003E-12 8.484E-12 2.109E-12 1.795E-11 2.132E-12 1.985E-11
$ 60 $ 7.782E-13 3.367E-12 8.229E-13 7.359E-12 8.341E-13 8.109E-12
CR 4.975 5.212 4.964 4.756 4.953 4.784
$\theta_2=\frac{1}{2}$ $\theta_2=\frac{7}{10}$ $\theta_2=1$
$N$ $|Y_0-Y^{0}|$ $|Z_0-Z^{0}|$ $|Y_0-Y^{0}|$ $|Z_0-Z^{0}|$ $|Y_0-Y^{0}|$ $|Z_0-Z^{0}|$
$20$ 1.917E-10 1.765E-09 1.921E-10 2.139E-09 1.928E-10 2.701E-09
$30$ 2.882E-11 2.436E-10 2.904E-11 2.935E-10 2.937E-11 3.683E-10
$40$ 7.125E-12 6.052E-11 7.223E-12 7.243E-11 7.371E-12 9.028E-11
$50$ 2.155E-12 2.175E-11 2.200E-12 2.554E-11 2.269E-12 3.124E-11
$60$ 8.448E-13 8.866E-12 8.664E-13 1.037E-11 8.988E-13 1.265E-11
CR 4.942 4.807 4.921 4.841 4.890 4.874
$ \theta_2=0 $ $ \theta_2=\frac{3}{10} $ $ \theta_2=\frac{2}{5} $
$ N $ $ |Y_0-Y^{0}| $ $ |Z_0-Z^{0}| $ $ |Y_0-Y^{0}| $ $ |Z_0-Z^{0}| $ $ |Y_0-Y^{0}| $ $ |Z_0-Z^{0}| $
$ 20 $ 1.826E-10 1.007E-09 1.912E-10 1.391E-09 1.914E-10 1.578E-09
$ 30 $ 2.753E-11 1.342E-10 2.860E-11 1.937E-10 2.871E-11 2.186E-10
$ 40 $ 6.742E-12 3.115E-11 7.027E-12 4.862E-11 7.076E-12 5.457E-11
$ 50 $ 2.003E-12 8.484E-12 2.109E-12 1.795E-11 2.132E-12 1.985E-11
$ 60 $ 7.782E-13 3.367E-12 8.229E-13 7.359E-12 8.341E-13 8.109E-12
CR 4.975 5.212 4.964 4.756 4.953 4.784
$\theta_2=\frac{1}{2}$ $\theta_2=\frac{7}{10}$ $\theta_2=1$
$N$ $|Y_0-Y^{0}|$ $|Z_0-Z^{0}|$ $|Y_0-Y^{0}|$ $|Z_0-Z^{0}|$ $|Y_0-Y^{0}|$ $|Z_0-Z^{0}|$
$20$ 1.917E-10 1.765E-09 1.921E-10 2.139E-09 1.928E-10 2.701E-09
$30$ 2.882E-11 2.436E-10 2.904E-11 2.935E-10 2.937E-11 3.683E-10
$40$ 7.125E-12 6.052E-11 7.223E-12 7.243E-11 7.371E-12 9.028E-11
$50$ 2.155E-12 2.175E-11 2.200E-12 2.554E-11 2.269E-12 3.124E-11
$60$ 8.448E-13 8.866E-12 8.664E-13 1.037E-11 8.988E-13 1.265E-11
CR 4.942 4.807 4.921 4.841 4.890 4.874
Table 5.  Errors and convergence rates of Scheme 3.2
$ \theta_2=0 $ $ \theta_2=\frac{3}{10} $ $ \theta_2=\frac{2}{5} $
$ N $ $ |Y_0-Y^{0}| $ $ |Z_0-Z^{0}| $ $ |Y_0-Y^{0}| $ $ |Z_0-Z^{0}| $ $ |Y_0-Y^{0}| $ $ |Z_0-Z^{0}| $
$ 20 $ 2.785E-11 1.610E-10 2.690E-11 1.632E-10 2.659E-11 1.639E-10
$ 30 $ 4.607E-12 1.151E-11 4.522E-12 1.170E-11 4.494E-12 1.176E-11
$ 40 $ 8.240E-13 2.929E-12 8.087E-13 2.953E-12 8.034E-13 2.968E-12
$ 50 $ 2.065E-13 9.485E-13 2.025E-13 9.570E-13 2.011E-13 9.599E-13
$ 60 $ 5.729E-14 3.326E-13 5.607E-14 3.395E-13 5.573E-14 3.423E-13
CR 5.635 5.526 5.624 5.524 5.620 5.523
$\theta_2=\frac{1}{2}$ $\theta_2=\frac{7}{10}$ $\theta_2=1$
$N$ $|Y_0-Y^{0}|$ $|Z_0-Z^{0}|$ $|Y_0-Y^{0}|$ $|Z_0-Z^{0}|$ $|Y_0-Y^{0}|$ $|Z_0-Z^{0}|$
$20$ 2.628E-11 1.646E-10 2.565E-11 1.660E-10 2.471E-11 1.681E-10
$30$ 4.465E-12 1.182E-11 4.408E-12 1.194E-11 4.322E-12 1.213E-11
$40$ 7.985E-13 2.973E-12 7.884E-13 2.997E-12 7.730E-13 3.029E-12
$50$ 1.998E-13 9.618E-13 1.971E-13 9.737E-13 1.933E-13 9.786E-13
$60$ 5.485E-14 3.362E-13 5.396E-14 3.354E-13 5.251E-14 3.469E-13
CR 5.621 5.538 5.615 5.544 5.605 5.535
$ \theta_2=0 $ $ \theta_2=\frac{3}{10} $ $ \theta_2=\frac{2}{5} $
$ N $ $ |Y_0-Y^{0}| $ $ |Z_0-Z^{0}| $ $ |Y_0-Y^{0}| $ $ |Z_0-Z^{0}| $ $ |Y_0-Y^{0}| $ $ |Z_0-Z^{0}| $
$ 20 $ 2.785E-11 1.610E-10 2.690E-11 1.632E-10 2.659E-11 1.639E-10
$ 30 $ 4.607E-12 1.151E-11 4.522E-12 1.170E-11 4.494E-12 1.176E-11
$ 40 $ 8.240E-13 2.929E-12 8.087E-13 2.953E-12 8.034E-13 2.968E-12
$ 50 $ 2.065E-13 9.485E-13 2.025E-13 9.570E-13 2.011E-13 9.599E-13
$ 60 $ 5.729E-14 3.326E-13 5.607E-14 3.395E-13 5.573E-14 3.423E-13
CR 5.635 5.526 5.624 5.524 5.620 5.523
$\theta_2=\frac{1}{2}$ $\theta_2=\frac{7}{10}$ $\theta_2=1$
$N$ $|Y_0-Y^{0}|$ $|Z_0-Z^{0}|$ $|Y_0-Y^{0}|$ $|Z_0-Z^{0}|$ $|Y_0-Y^{0}|$ $|Z_0-Z^{0}|$
$20$ 2.628E-11 1.646E-10 2.565E-11 1.660E-10 2.471E-11 1.681E-10
$30$ 4.465E-12 1.182E-11 4.408E-12 1.194E-11 4.322E-12 1.213E-11
$40$ 7.985E-13 2.973E-12 7.884E-13 2.997E-12 7.730E-13 3.029E-12
$50$ 1.998E-13 9.618E-13 1.971E-13 9.737E-13 1.933E-13 9.786E-13
$60$ 5.485E-14 3.362E-13 5.396E-14 3.354E-13 5.251E-14 3.469E-13
CR 5.621 5.538 5.615 5.544 5.605 5.535
Table 6.  Errors and convergence rates of multi-step schemes
5-step 6-step
$ N $ $ |Y_0-Y^{0}| $ $ |Z_0-Z^{0}| $ $ |Y_0-Y^{0}| $ $ |Z_0-Z^{0}| $
$ 20 $ 2.50E-09 5.12E-07 7.75E-10 1.25E-07
$ 30 $ 3.49E-10 8.50E-08 6.74E-11 1.56E-08
$ 40 $ 7.99E-11 2.29E-08 1.18E-11 3.37E-09
$ 50 $ 2.49E-11 8.10E-09 3.03E-12 9.99E-10
$ 60 $ 1.06E-11 3.43E-09 1.01E-12 3.64E-10
CR 5.009 4.555 6.051 5.313
5-step 6-step
$ N $ $ |Y_0-Y^{0}| $ $ |Z_0-Z^{0}| $ $ |Y_0-Y^{0}| $ $ |Z_0-Z^{0}| $
$ 20 $ 2.50E-09 5.12E-07 7.75E-10 1.25E-07
$ 30 $ 3.49E-10 8.50E-08 6.74E-11 1.56E-08
$ 40 $ 7.99E-11 2.29E-08 1.18E-11 3.37E-09
$ 50 $ 2.49E-11 8.10E-09 3.03E-12 9.99E-10
$ 60 $ 1.06E-11 3.43E-09 1.01E-12 3.64E-10
CR 5.009 4.555 6.051 5.313
Table 7.  Errors and convergence rates of Scheme 3.1
$ \theta_2=0 $ $ \theta_2=\frac{3}{10} $ $ \theta_2=\frac{2}{5} $
$ N $ $ |Y_0-Y^{0}| $ $ |Z_0-Z^{0}| $ $ |Y_0-Y^{0}| $ $ |Z_0-Z^{0}| $ $ |Y_0-Y^{0}| $ $ |Z_0-Z^{0}| $
$ 30 $ 1.179E-04 2.343E-05 4.801E-04 6.426E-05 6.009E-04 9.333E-05
$ 40 $ 2.545E-05 6.052E-06 1.289E-04 1.786E-05 1.634E-04 2.582E-05
$ 50 $ 7.607E-06 2.115E-06 4.550E-05 6.408E-06 5.814E-05 9.247E-06
$ 60 $ 2.816E-06 8.941E-07 1.921E-05 2.730E-06 2.468E-05 3.938E-06
$ 70 $ 1.211E-06 4.310E-07 9.207E-06 1.316E-06 1.187E-05 1.898E-06
CR 5.405 4.715 4.667 4.590 4.632 4.598
$\theta_2=\frac{1}{2}$ $\theta_2=\frac{7}{10}$ $\theta_2=1$
$N$ $|Y_0-Y^{0}|$ $|Z_0-Z^{0}|$ $|Y_0-Y^{0}|$ $|Z_0-Z^{0}|$ $|Y_0-Y^{0}|$ $|Z_0-Z^{0}|$
$30$ 7.216E-04 1.223E-04 9.632E-04 1.801E-04 1.326E-03 2.661E-04
$40$ 1.980E-04 3.377E-05 2.670E-04 4.966E-05 3.705E-04 7.344E-05
$50$ 7.077E-05 1.209E-05 9.604E-05 1.776E-05 1.339E-04 2.627E-05
$60$ 3.014E-05 5.148E-06 4.107E-05 7.562E-06 5.747E-05 1.118E-05
$70$ 1.454E-05 2.481E-06 1.987E-05 3.644E-06 2.786E-05 5.390E-06
CR 4.609 4.601 4.581 4.604 4.559 4.603
$ \theta_2=0 $ $ \theta_2=\frac{3}{10} $ $ \theta_2=\frac{2}{5} $
$ N $ $ |Y_0-Y^{0}| $ $ |Z_0-Z^{0}| $ $ |Y_0-Y^{0}| $ $ |Z_0-Z^{0}| $ $ |Y_0-Y^{0}| $ $ |Z_0-Z^{0}| $
$ 30 $ 1.179E-04 2.343E-05 4.801E-04 6.426E-05 6.009E-04 9.333E-05
$ 40 $ 2.545E-05 6.052E-06 1.289E-04 1.786E-05 1.634E-04 2.582E-05
$ 50 $ 7.607E-06 2.115E-06 4.550E-05 6.408E-06 5.814E-05 9.247E-06
$ 60 $ 2.816E-06 8.941E-07 1.921E-05 2.730E-06 2.468E-05 3.938E-06
$ 70 $ 1.211E-06 4.310E-07 9.207E-06 1.316E-06 1.187E-05 1.898E-06
CR 5.405 4.715 4.667 4.590 4.632 4.598
$\theta_2=\frac{1}{2}$ $\theta_2=\frac{7}{10}$ $\theta_2=1$
$N$ $|Y_0-Y^{0}|$ $|Z_0-Z^{0}|$ $|Y_0-Y^{0}|$ $|Z_0-Z^{0}|$ $|Y_0-Y^{0}|$ $|Z_0-Z^{0}|$
$30$ 7.216E-04 1.223E-04 9.632E-04 1.801E-04 1.326E-03 2.661E-04
$40$ 1.980E-04 3.377E-05 2.670E-04 4.966E-05 3.705E-04 7.344E-05
$50$ 7.077E-05 1.209E-05 9.604E-05 1.776E-05 1.339E-04 2.627E-05
$60$ 3.014E-05 5.148E-06 4.107E-05 7.562E-06 5.747E-05 1.118E-05
$70$ 1.454E-05 2.481E-06 1.987E-05 3.644E-06 2.786E-05 5.390E-06
CR 4.609 4.601 4.581 4.604 4.559 4.603
Table 8.  Errors and convergence rates of Scheme 3.2
$ \theta_2=0 $ $ \theta_2=\frac{3}{10} $ $ \theta_2=\frac{2}{5} $
$ N $ $ |Y_0-Y^{0}| $ $ |Z_0-Z^{0}| $ $ |Y_0-Y^{0}| $ $ |Z_0-Z^{0}| $ $ |Y_0-Y^{0}| $ $ |Z_0-Z^{0}| $
$ 30 $ 2.412E-05 5.723E-06 9.919E-05 1.822E-05 1.242E-04 2.238E-05
$ 40 $ 3.982E-06 9.642E-07 2.045E-05 3.726E-06 2.594E-05 4.646E-06
$ 50 $ 9.623E-07 2.367E-07 5.855E-06 1.059E-06 7.485E-06 1.334E-06
$ 60 $ 2.981E-07 7.128E-08 2.078E-06 3.720E-07 2.672E-06 4.715E-07
$ 70 $ 1.095E-07 2.891E-08 8.586E-07 1.581E-07 1.108E-06 2.001E-07
CR 6.367 6.277 5.606 5.614 5.570 5.577
$\theta_2=\frac{1}{2}$ $\theta_2=\frac{7}{10}$ $\theta_2=1$
N |Y0-Y0| |Z0-Z0| |Y0-Y0| |Z0-Z0| |Y0-Y0| |Z0-Z0|
30 1.492E-04 2.653E-05 1.993E-04 3.483E-05 2.744E-04 4.729E-05
40 3.143E-05 5.566E-06 4.242E-05 7.413E-06 5.888E-05 1.016E-05
50 9.116E-06 1.608E-06 1.238E-05 2.161E-06 1.727E-05 2.981E-06
60 3.265E-06 5.727E-07 4.452E-06 7.756E-07 6.232E-06 1.073E-06
70 1.358E-06 2.418E-07 1.857E-06 3.220E-07 2.606E-06 4.511E-07
CR 5.547 5.552 5.519 5.527 5.496 5.494
$ \theta_2=0 $ $ \theta_2=\frac{3}{10} $ $ \theta_2=\frac{2}{5} $
$ N $ $ |Y_0-Y^{0}| $ $ |Z_0-Z^{0}| $ $ |Y_0-Y^{0}| $ $ |Z_0-Z^{0}| $ $ |Y_0-Y^{0}| $ $ |Z_0-Z^{0}| $
$ 30 $ 2.412E-05 5.723E-06 9.919E-05 1.822E-05 1.242E-04 2.238E-05
$ 40 $ 3.982E-06 9.642E-07 2.045E-05 3.726E-06 2.594E-05 4.646E-06
$ 50 $ 9.623E-07 2.367E-07 5.855E-06 1.059E-06 7.485E-06 1.334E-06
$ 60 $ 2.981E-07 7.128E-08 2.078E-06 3.720E-07 2.672E-06 4.715E-07
$ 70 $ 1.095E-07 2.891E-08 8.586E-07 1.581E-07 1.108E-06 2.001E-07
CR 6.367 6.277 5.606 5.614 5.570 5.577
$\theta_2=\frac{1}{2}$ $\theta_2=\frac{7}{10}$ $\theta_2=1$
N |Y0-Y0| |Z0-Z0| |Y0-Y0| |Z0-Z0| |Y0-Y0| |Z0-Z0|
30 1.492E-04 2.653E-05 1.993E-04 3.483E-05 2.744E-04 4.729E-05
40 3.143E-05 5.566E-06 4.242E-05 7.413E-06 5.888E-05 1.016E-05
50 9.116E-06 1.608E-06 1.238E-05 2.161E-06 1.727E-05 2.981E-06
60 3.265E-06 5.727E-07 4.452E-06 7.756E-07 6.232E-06 1.073E-06
70 1.358E-06 2.418E-07 1.857E-06 3.220E-07 2.606E-06 4.511E-07
CR 5.547 5.552 5.519 5.527 5.496 5.494
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Arnulf Jentzen. Taylor expansions of solutions of stochastic partial differential equations. Discrete & Continuous Dynamical Systems - B, 2010, 14 (2) : 515-557. doi: 10.3934/dcdsb.2010.14.515

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