# American Institute of Mathematical Sciences

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 and 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. [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. [3] C. Beck, W. 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. [4] B. Bouchard, X. 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. [5] B. Bouchard, X. Tan, X. 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. [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. [7] R. Buckdahn, J. 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. [8] P. Cheridito, H. M. Soner, N. 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. [9] G. Da Prato and J. Zabczyk, Second Order Partial Differential Equations in Hilbert Spaces, Cambridge University Press, London, 2002.  doi: 10.1017/CBO9780511543210. [10] W. E, J. 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. [11] W. E, M. Hutzenthaler, A. 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. [12] N. El Karoui, C. Kapoudjian, E. Pardoux, S. 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. [13] N. El Karoui, S. Peng and M. C. Quenez, Backward stochastic differential equations in finance, Math. Finance, 7 (1997), 1-71.  doi: 10.1111/1467-9965.00022. [14] Y. Fu, W. 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. [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. [16] E. Gobet, J.-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. [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. [18] P. Henry-Labordère, X. 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. [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. [20] L. Kapllani and L. Teng, Deep learning algorithms for solving high dimensional nonlinear backward stochastic differential equations, preprint, arXiv: 2010.01319. [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. [22] A. Lionnet, G. 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. [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. [24] C.-K. Pak, M.-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. [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. [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. [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. [28] E. Rosazza Gianin, Risk measures via $g$-expectations, Insurance Math. Econom., 39 (2006), 19-34.  doi: 10.1016/j.insmatheco.2006.01.002. [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. [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. [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. [32] Y. Sun, W. 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. [33] L. Teng, A. 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. [34] C. Zhang, J. 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. [35] W. Zhao, L. 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. [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. [37] W. Zhao, Y. 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. [38] W. Zhao, J. 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. [39] W. Zhao, G. 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. [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.

show all references

##### References:
 [1] M. Abramowitz and I. A. Stegun, Handbook of Mathematical Functions, Dover, New York, 1966. doi: 10.2307/2314682. [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. [3] C. Beck, W. 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. [4] B. Bouchard, X. 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. [5] B. Bouchard, X. Tan, X. 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. [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. [7] R. Buckdahn, J. 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. [8] P. Cheridito, H. M. Soner, N. 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. [9] G. Da Prato and J. Zabczyk, Second Order Partial Differential Equations in Hilbert Spaces, Cambridge University Press, London, 2002.  doi: 10.1017/CBO9780511543210. [10] W. E, J. 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. [11] W. E, M. Hutzenthaler, A. 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. [12] N. El Karoui, C. Kapoudjian, E. Pardoux, S. 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. [13] N. El Karoui, S. Peng and M. C. Quenez, Backward stochastic differential equations in finance, Math. Finance, 7 (1997), 1-71.  doi: 10.1111/1467-9965.00022. [14] Y. Fu, W. 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. [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. [16] E. Gobet, J.-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. [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. [18] P. Henry-Labordère, X. 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. [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. [20] L. Kapllani and L. Teng, Deep learning algorithms for solving high dimensional nonlinear backward stochastic differential equations, preprint, arXiv: 2010.01319. [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. [22] A. Lionnet, G. 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. [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. [24] C.-K. Pak, M.-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. [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. [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. [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. [28] E. Rosazza Gianin, Risk measures via $g$-expectations, Insurance Math. Econom., 39 (2006), 19-34.  doi: 10.1016/j.insmatheco.2006.01.002. [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. [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. [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. [32] Y. Sun, W. 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. [33] L. Teng, A. 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. [34] C. Zhang, J. 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. [35] W. Zhao, L. 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. [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. [37] W. Zhao, Y. 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. [38] W. Zhao, J. 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. [39] W. Zhao, G. 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. [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.
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.
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.
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
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
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
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
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
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
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
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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