May  2022, 27(5): 2587-2606. doi: 10.3934/dcdsb.2021149

Time splitting combined with exponential wave integrator Fourier pseudospectral method for quantum Zakharov system

South China Research Center for Applied Mathematics and Interdisciplinary Studies, South China Normal University, Guangzhou 510631, China

* Corresponding author: Gengen Zhang

Received  October 2020 Published  May 2022 Early access  May 2021

Fund Project: The author is supported by the National Natural Science Foundation of China (Grant No.11701110), the China Postdoctoral Science Foundation (Grant No.2020M682746)

In this paper we develop a time splitting combined with exponential wave integrator (EWI) Fourier pseudospectral (FP) method for the quantum Zakharov system (QZS), i.e. using the FP method for spatial derivatives, a time splitting technique and an EWI method for temporal derivatives in the Schrödinger-like equation and wave-type equations, respectively. The scheme is fully explicit and efficient due to fast Fourier transform. Numerical experiments for the QZS are presented to illustrate the accuracy and capability of the method, including accuracy tests, convergence of the QZS to the classical Zakharov system in the semi-classical limit, soliton-soliton collisions and pattern dynamics of the QZS in one-dimension, as well as the blow-up phenomena of QZS in two-dimension.

Citation: Gengen Zhang. Time splitting combined with exponential wave integrator Fourier pseudospectral method for quantum Zakharov system. Discrete and Continuous Dynamical Systems - B, 2022, 27 (5) : 2587-2606. doi: 10.3934/dcdsb.2021149
References:
[1]

W. Bao and X. Dong, Analysis and comparison of numerical methods for Klein-Gordon equation in nonrelativistic limit regime, Numer. Math., 120 (2012), 189-229.  doi: 10.1007/s00211-011-0411-2.

[2]

W. Bao, X. Dong and X. Zhao, An exponential wave integrator sine pseudospectral method for the Klein-Gordon-Zakharov system, SIAM J. Sci. Comput., 35 (2013), A2903–A2927. doi: 10.1137/110855004.

[3]

W. BaoX. Dong and X. Zhao, Uniformly accurate multiscale time integrators for highly oscillatory second order differential equations, J. Math. Study, 47 (2014), 111-150.  doi: 10.4208/jms.v47n2.14.01.

[4]

W. Bao and C. Su, Uniform error bounds of a finite difference method for the Zakharov system in the subsonic limit regime via an asymptotic consistent formulation, Multiscale Model. Simul., 15 (2017), 977-1002.  doi: 10.1137/16M1078112.

[5]

W. Bao and C. Su, A uniformly and optically accurate method for the Zakharov system in the subsonic limit regime, SIAM J. Sci. Comput., 40 (2018), A929–A953. doi: 10.1137/17M1113333.

[6]

W. Bao and F. Sun, Efficient and stable numerical methods for the generalized and vector Zakharov system, SIAM J. Sci. Comput., 26 (2005), 1057-1088.  doi: 10.1137/030600941.

[7]

W. BaoF. Sun and G. W. Wei, Numerical methods for the generalized Zakharov system, J. Comput. Phys., 190 (2003), 201-228.  doi: 10.1016/S0021-9991(03)00271-7.

[8]

W. Bao and X. Zhao, A uniformly accurate multiscale time integrator spectral method for the Klein-Gordon-Zakharov system in the high-plasma-frequency limit regime, J. Comput. Phys., 327 (2016), 270-293.  doi: 10.1016/j.jcp.2016.09.046.

[9]

W. Bao and X. Zhao, Comparison of numerical methods for the nonlinear Klein-Gordon equation in the nonrelativistic limit regime, J. Comput. Phys., 398 (2019), 108886, 30 pp. doi: 10.1016/j.jcp.2019.108886.

[10]

Y. Cai and Y. Yuan, Uniform error estimates of the conservative finite difference method for the Zakharov system in the subsonic limit regime, Math. Comp., 87 (2018), 1191-1225.  doi: 10.1090/mcom/3269.

[11]

Q. Chang, B. Guo and H. Jiang, Finite difference method for generalized Zakharov equations, Math. Comput., 64 (1995), 537–553, S7–S11. doi: 10.1090/S0025-5718-1995-1284664-5.

[12]

B. J. Choi, Global well-posedness of the adiabatic limit of quantum Zakharov system in 1D, preprint, (2019), arXiv: 1906.10807v2.

[13]

A. S. Davydov, Solitons in molecular systems, Phys. Scr., 20 (1979), 387-394.  doi: 10.1088/0031-8949/20/3-4/013.

[14]

L. M. DegtyarevV. G. Nakhankov and L. I. Rudakov, Dynamics of the formation and interaction of Langmuir solitons and strong turbulence, Sov. Phys. JETP, 40 (1974), 264-268. 

[15]

Y. FangH. Shih and K. Wang, Local well-posedness for the quantum Zakharov system in one spatial dimension, J. Hyperbolic Differ. Equ., 14 (2017), 157-192.  doi: 10.1142/S0219891617500059.

[16]

Y. FangJ. Segata and T. Wu, On the standing waves of quantum Zakharov system, J. Math. Anal. Appl., 458 (2018), 1427-1448.  doi: 10.1016/j.jmaa.2017.10.033.

[17]

Y. Fang and K. Nakanishi, Global well-posedness abd scattering for the quantum Zakharov system in $L^2$, Proc. Amer. Math. Soc., 6 (2019), 21-32.  doi: 10.1090/bproc/42.

[18]

Y. FangH. KuoH. Shih and K. Wang, Semi-classical limit for the quantum Zakharov system, Taiwan. J. Math., 23 (2019), 925-949.  doi: 10.11650/tjm/180806.

[19]

L. G. Garcia, F. Haas, L. P. L. de Oliveira and J. Goedert, Modified Zakharov equations for plasmas with a quantum correction, Phys. Plasmas, 12 (2005), 012302. doi: 10.1063/1.1819935.

[20]

L. Gauckler, On a splitting method for the Zakharov system, Numer. Math., 139 (2018), 349-379.  doi: 10.1007/s00211-017-0942-2.

[21]

R. T. Glassey, Approximate solutions to the Zakharov equations via finite differences, J. Comput. Phys., 100 (1992), 377-383.  doi: 10.1016/0021-9991(92)90243-R.

[22]

V. Grimm, On error bounds for the Gautschi-type exponential integrator applied to oscillatory second-order differential equations, Numer. Math., 100 (2005), 71-89.  doi: 10.1007/s00211-005-0583-8.

[23]

V. Grimm, A note on the Gautschi-type method for oscillatory second-order differential equations, Numer. Math., 102 (2005), 61-66.  doi: 10.1007/s00211-005-0639-9.

[24]

Y. GuoJ. Zhang and B. Guo, Global well-posedness and the classical limit of the solution for the quantum Zakharov system, Z. Angew. Math. Phys., 64 (2013), 53-68.  doi: 10.1007/s00033-012-0215-y.

[25] B. GuoZ. GanL. Kong and J. Zhang, The Zakharov System and its Soliton Solutions, Science Press, Beijing, 2016.  doi: 10.1007/978-981-10-2582-2.
[26]

F. Haas, Variational approach for the quantum Zakharov system, Phys. Plasmas, 14 (2007), 042309.

[27]

F. Haas and P. K. Shukla, Quantum and classical dynamics of Langmuir wave packets, Phys. Rev. E, 79 (2009), 066402.

[28]

F. Haas, Quantum Plasmas: An Hydrodynamic Approach, Springer Series on Atomic, Optical, and Plasma Physics, 65, Springer, New York, 2011. doi: 10.1007/978-1-4419-8201-8.

[29]

H. HofstätterO. Koch and M. Thalhammer, Convergence analysis of high-order time-splitting pseudo-spectral methods for rotational Gross-Pitaevskii equations, Numer. Math., 127 (2014), 315-364.  doi: 10.1007/s00211-013-0586-9.

[30]

M. Hochbruck and A. Ostermann, Exponential integrators, Acta Numer., 19 (2010), 209-286.  doi: 10.1017/S0962492910000048.

[31]

M. Hochbruck and C. H. Lubich, A Gautschi-type method for oscillatory second-order differential equations, Numer. Math., 83 (1999), 403-426.  doi: 10.1007/s002110050456.

[32]

J.-C. JiangC. K. Lin and S. Shao, On one dimensional quantum Zakharov system, Discrete Contin. Dyn. Syst., 36 (2016), 5445-5475.  doi: 10.3934/dcds.2016040.

[33]

S. JinP. A. Markowich and C. Zheng, Numerical simulation of a generalized Zakharov system, J. Comput. Phys., 201 (2004), 376-395.  doi: 10.1016/j.jcp.2004.06.001.

[34]

S. Jin and C. Zheng, A Time-splitting spectral method for the generalized Zakharov system in multi-dimensions, J. Sci. Comput., 26 (2006), 127-149.  doi: 10.1007/s10915-005-4929-2.

[35]

X. Li and L. Zhang, Error estimates of a trigonometric integrator sine pseudo-spectral method for the extended Fisher-Kolmogorov equation, Appl. Numer. Math., 131 (2018), 39-53.  doi: 10.1016/j.apnum.2018.04.010.

[36]

F. LiaoL. Zhang and S. Wang, Time-splitting combined with exponential wave integrator fourier pseudospectral method for Schrödinger-Boussinesq system, Commun. Nonlinear Sci. Numer. Simulat., 55 (2018), 93-104.  doi: 10.1016/j.cnsns.2017.06.033.

[37]

M. Marklund, Classical and quantum kinetics of the Zakharov system, Phys. Plasmas, 12 (2005), 082110, 5 pp. doi: 10.1063/1.2012147.

[38]

V. Masselin, A result of the blow-up rate for the Zakharov system in dimension 3, SIAM J. Math. Anal., 33 (2001), 440-447.  doi: 10.1137/S0036141099363687.

[39]

A. P. MisraD. Ghosh and A. R. Chowdhury, A novel hyperchaos in the quantum Zakharov system for plasmas, Phys. Lett. A, 372 (2008), 1469-1476.  doi: 10.1016/j.physleta.2007.09.054.

[40]

A. P. Misra and P. K. Shukla, Pattern dynamics and spatiotemporal chaos in the quantum Zakharov equations, Phys. Rev. E, 79 (2009), 056401. doi: 10.1103/PhysRevE.79.056401.

[41]

G. C. PapanicolaouC. SulemP. L. Sulem and X. P. Wang, Singular solutions of the Zakharov equations for Langmuir turbulence, Phys. Fluids B, 3 (1991), 969-980.  doi: 10.1063/1.859852.

[42]

C. Su and X. Zhao, A uniformly first-order accurate method for Klein-Gordon-Zakharov system in simultaneous high-plasma-frequency and subsonic limit regime, J. Comput. Phys., 428 (2021), 110064, 22 pp. doi: 10.1016/j.jcp.2020.110064.

[43]

A. Taleei and M. Dehghan, Time-splitting pseudo-spectral domain decomposition method for the soliton solutions of the one- and multi-dimensional nonlinear Schrödinger equations, Comput. Phys. Commun., 185 (2014), 1515-1528.  doi: 10.1016/j.cpc.2014.01.013.

[44]

S. WangT. Wang and L. Zhang, Numerical computations for N-coupled nonlinear Schrödinger equations by split step spectral methods, Appl. Math. Comput., 222 (2013), 438-452.  doi: 10.1016/j.amc.2013.07.060.

[45]

Y. Wang and X. Zhao, Symmetric high order Gautschi-type exponential wave integrators pseudospectral method for the nonlinear Klein-Gordon equation in the nonrelativistic limit regime, Int. J. Numer. Anal. Mod., 15 (2018), 405-427. 

[46]

Y. XiaY. Xu and C. Shu, Local discontinuous Galerkin methods for the generalized Zakharov system, J. Comput. Phys., 229 (2010), 1238-1259.  doi: 10.1016/j.jcp.2009.10.029.

[47]

A. XiaoC. Wang and J. Wang, Conservative linearly-implicit difference scheme for a class of modified Zakharov systems with high-order space fractional quantum correction, Appl. Numer. Math., 146 (2019), 379-399.  doi: 10.1016/j.apnum.2019.07.019.

[48]

S. Yao, J. Sun and T. Wu, Stationary quantum Zakharov systems involving a higher competing perturbation, Electron. J. Differential Equations, 2020 (2020), 18 pp.

[49]

V. E. Zakharov, Collapse of langmuir waves, Sov. Phys. JETP, 35 (1972), 908-914. 

[50]

G. Zhang and C. Su, A conservative linearly-implicit compact difference scheme for the quantum Zakharov system, J. Sci. Comput., 87 (2021), 71. doi: 10.1007/s10915-021-01482-3.

[51]

X. Zhao, On error estimates of an exponential wave integrator sine pseudospectral method for the Klein-Gordon-Zakharov system, Numer. Meth. Part. D. E., 32 (2016), 266-291.  doi: 10.1002/num.21994.

show all references

References:
[1]

W. Bao and X. Dong, Analysis and comparison of numerical methods for Klein-Gordon equation in nonrelativistic limit regime, Numer. Math., 120 (2012), 189-229.  doi: 10.1007/s00211-011-0411-2.

[2]

W. Bao, X. Dong and X. Zhao, An exponential wave integrator sine pseudospectral method for the Klein-Gordon-Zakharov system, SIAM J. Sci. Comput., 35 (2013), A2903–A2927. doi: 10.1137/110855004.

[3]

W. BaoX. Dong and X. Zhao, Uniformly accurate multiscale time integrators for highly oscillatory second order differential equations, J. Math. Study, 47 (2014), 111-150.  doi: 10.4208/jms.v47n2.14.01.

[4]

W. Bao and C. Su, Uniform error bounds of a finite difference method for the Zakharov system in the subsonic limit regime via an asymptotic consistent formulation, Multiscale Model. Simul., 15 (2017), 977-1002.  doi: 10.1137/16M1078112.

[5]

W. Bao and C. Su, A uniformly and optically accurate method for the Zakharov system in the subsonic limit regime, SIAM J. Sci. Comput., 40 (2018), A929–A953. doi: 10.1137/17M1113333.

[6]

W. Bao and F. Sun, Efficient and stable numerical methods for the generalized and vector Zakharov system, SIAM J. Sci. Comput., 26 (2005), 1057-1088.  doi: 10.1137/030600941.

[7]

W. BaoF. Sun and G. W. Wei, Numerical methods for the generalized Zakharov system, J. Comput. Phys., 190 (2003), 201-228.  doi: 10.1016/S0021-9991(03)00271-7.

[8]

W. Bao and X. Zhao, A uniformly accurate multiscale time integrator spectral method for the Klein-Gordon-Zakharov system in the high-plasma-frequency limit regime, J. Comput. Phys., 327 (2016), 270-293.  doi: 10.1016/j.jcp.2016.09.046.

[9]

W. Bao and X. Zhao, Comparison of numerical methods for the nonlinear Klein-Gordon equation in the nonrelativistic limit regime, J. Comput. Phys., 398 (2019), 108886, 30 pp. doi: 10.1016/j.jcp.2019.108886.

[10]

Y. Cai and Y. Yuan, Uniform error estimates of the conservative finite difference method for the Zakharov system in the subsonic limit regime, Math. Comp., 87 (2018), 1191-1225.  doi: 10.1090/mcom/3269.

[11]

Q. Chang, B. Guo and H. Jiang, Finite difference method for generalized Zakharov equations, Math. Comput., 64 (1995), 537–553, S7–S11. doi: 10.1090/S0025-5718-1995-1284664-5.

[12]

B. J. Choi, Global well-posedness of the adiabatic limit of quantum Zakharov system in 1D, preprint, (2019), arXiv: 1906.10807v2.

[13]

A. S. Davydov, Solitons in molecular systems, Phys. Scr., 20 (1979), 387-394.  doi: 10.1088/0031-8949/20/3-4/013.

[14]

L. M. DegtyarevV. G. Nakhankov and L. I. Rudakov, Dynamics of the formation and interaction of Langmuir solitons and strong turbulence, Sov. Phys. JETP, 40 (1974), 264-268. 

[15]

Y. FangH. Shih and K. Wang, Local well-posedness for the quantum Zakharov system in one spatial dimension, J. Hyperbolic Differ. Equ., 14 (2017), 157-192.  doi: 10.1142/S0219891617500059.

[16]

Y. FangJ. Segata and T. Wu, On the standing waves of quantum Zakharov system, J. Math. Anal. Appl., 458 (2018), 1427-1448.  doi: 10.1016/j.jmaa.2017.10.033.

[17]

Y. Fang and K. Nakanishi, Global well-posedness abd scattering for the quantum Zakharov system in $L^2$, Proc. Amer. Math. Soc., 6 (2019), 21-32.  doi: 10.1090/bproc/42.

[18]

Y. FangH. KuoH. Shih and K. Wang, Semi-classical limit for the quantum Zakharov system, Taiwan. J. Math., 23 (2019), 925-949.  doi: 10.11650/tjm/180806.

[19]

L. G. Garcia, F. Haas, L. P. L. de Oliveira and J. Goedert, Modified Zakharov equations for plasmas with a quantum correction, Phys. Plasmas, 12 (2005), 012302. doi: 10.1063/1.1819935.

[20]

L. Gauckler, On a splitting method for the Zakharov system, Numer. Math., 139 (2018), 349-379.  doi: 10.1007/s00211-017-0942-2.

[21]

R. T. Glassey, Approximate solutions to the Zakharov equations via finite differences, J. Comput. Phys., 100 (1992), 377-383.  doi: 10.1016/0021-9991(92)90243-R.

[22]

V. Grimm, On error bounds for the Gautschi-type exponential integrator applied to oscillatory second-order differential equations, Numer. Math., 100 (2005), 71-89.  doi: 10.1007/s00211-005-0583-8.

[23]

V. Grimm, A note on the Gautschi-type method for oscillatory second-order differential equations, Numer. Math., 102 (2005), 61-66.  doi: 10.1007/s00211-005-0639-9.

[24]

Y. GuoJ. Zhang and B. Guo, Global well-posedness and the classical limit of the solution for the quantum Zakharov system, Z. Angew. Math. Phys., 64 (2013), 53-68.  doi: 10.1007/s00033-012-0215-y.

[25] B. GuoZ. GanL. Kong and J. Zhang, The Zakharov System and its Soliton Solutions, Science Press, Beijing, 2016.  doi: 10.1007/978-981-10-2582-2.
[26]

F. Haas, Variational approach for the quantum Zakharov system, Phys. Plasmas, 14 (2007), 042309.

[27]

F. Haas and P. K. Shukla, Quantum and classical dynamics of Langmuir wave packets, Phys. Rev. E, 79 (2009), 066402.

[28]

F. Haas, Quantum Plasmas: An Hydrodynamic Approach, Springer Series on Atomic, Optical, and Plasma Physics, 65, Springer, New York, 2011. doi: 10.1007/978-1-4419-8201-8.

[29]

H. HofstätterO. Koch and M. Thalhammer, Convergence analysis of high-order time-splitting pseudo-spectral methods for rotational Gross-Pitaevskii equations, Numer. Math., 127 (2014), 315-364.  doi: 10.1007/s00211-013-0586-9.

[30]

M. Hochbruck and A. Ostermann, Exponential integrators, Acta Numer., 19 (2010), 209-286.  doi: 10.1017/S0962492910000048.

[31]

M. Hochbruck and C. H. Lubich, A Gautschi-type method for oscillatory second-order differential equations, Numer. Math., 83 (1999), 403-426.  doi: 10.1007/s002110050456.

[32]

J.-C. JiangC. K. Lin and S. Shao, On one dimensional quantum Zakharov system, Discrete Contin. Dyn. Syst., 36 (2016), 5445-5475.  doi: 10.3934/dcds.2016040.

[33]

S. JinP. A. Markowich and C. Zheng, Numerical simulation of a generalized Zakharov system, J. Comput. Phys., 201 (2004), 376-395.  doi: 10.1016/j.jcp.2004.06.001.

[34]

S. Jin and C. Zheng, A Time-splitting spectral method for the generalized Zakharov system in multi-dimensions, J. Sci. Comput., 26 (2006), 127-149.  doi: 10.1007/s10915-005-4929-2.

[35]

X. Li and L. Zhang, Error estimates of a trigonometric integrator sine pseudo-spectral method for the extended Fisher-Kolmogorov equation, Appl. Numer. Math., 131 (2018), 39-53.  doi: 10.1016/j.apnum.2018.04.010.

[36]

F. LiaoL. Zhang and S. Wang, Time-splitting combined with exponential wave integrator fourier pseudospectral method for Schrödinger-Boussinesq system, Commun. Nonlinear Sci. Numer. Simulat., 55 (2018), 93-104.  doi: 10.1016/j.cnsns.2017.06.033.

[37]

M. Marklund, Classical and quantum kinetics of the Zakharov system, Phys. Plasmas, 12 (2005), 082110, 5 pp. doi: 10.1063/1.2012147.

[38]

V. Masselin, A result of the blow-up rate for the Zakharov system in dimension 3, SIAM J. Math. Anal., 33 (2001), 440-447.  doi: 10.1137/S0036141099363687.

[39]

A. P. MisraD. Ghosh and A. R. Chowdhury, A novel hyperchaos in the quantum Zakharov system for plasmas, Phys. Lett. A, 372 (2008), 1469-1476.  doi: 10.1016/j.physleta.2007.09.054.

[40]

A. P. Misra and P. K. Shukla, Pattern dynamics and spatiotemporal chaos in the quantum Zakharov equations, Phys. Rev. E, 79 (2009), 056401. doi: 10.1103/PhysRevE.79.056401.

[41]

G. C. PapanicolaouC. SulemP. L. Sulem and X. P. Wang, Singular solutions of the Zakharov equations for Langmuir turbulence, Phys. Fluids B, 3 (1991), 969-980.  doi: 10.1063/1.859852.

[42]

C. Su and X. Zhao, A uniformly first-order accurate method for Klein-Gordon-Zakharov system in simultaneous high-plasma-frequency and subsonic limit regime, J. Comput. Phys., 428 (2021), 110064, 22 pp. doi: 10.1016/j.jcp.2020.110064.

[43]

A. Taleei and M. Dehghan, Time-splitting pseudo-spectral domain decomposition method for the soliton solutions of the one- and multi-dimensional nonlinear Schrödinger equations, Comput. Phys. Commun., 185 (2014), 1515-1528.  doi: 10.1016/j.cpc.2014.01.013.

[44]

S. WangT. Wang and L. Zhang, Numerical computations for N-coupled nonlinear Schrödinger equations by split step spectral methods, Appl. Math. Comput., 222 (2013), 438-452.  doi: 10.1016/j.amc.2013.07.060.

[45]

Y. Wang and X. Zhao, Symmetric high order Gautschi-type exponential wave integrators pseudospectral method for the nonlinear Klein-Gordon equation in the nonrelativistic limit regime, Int. J. Numer. Anal. Mod., 15 (2018), 405-427. 

[46]

Y. XiaY. Xu and C. Shu, Local discontinuous Galerkin methods for the generalized Zakharov system, J. Comput. Phys., 229 (2010), 1238-1259.  doi: 10.1016/j.jcp.2009.10.029.

[47]

A. XiaoC. Wang and J. Wang, Conservative linearly-implicit difference scheme for a class of modified Zakharov systems with high-order space fractional quantum correction, Appl. Numer. Math., 146 (2019), 379-399.  doi: 10.1016/j.apnum.2019.07.019.

[48]

S. Yao, J. Sun and T. Wu, Stationary quantum Zakharov systems involving a higher competing perturbation, Electron. J. Differential Equations, 2020 (2020), 18 pp.

[49]

V. E. Zakharov, Collapse of langmuir waves, Sov. Phys. JETP, 35 (1972), 908-914. 

[50]

G. Zhang and C. Su, A conservative linearly-implicit compact difference scheme for the quantum Zakharov system, J. Sci. Comput., 87 (2021), 71. doi: 10.1007/s10915-021-01482-3.

[51]

X. Zhao, On error estimates of an exponential wave integrator sine pseudospectral method for the Klein-Gordon-Zakharov system, Numer. Meth. Part. D. E., 32 (2016), 266-291.  doi: 10.1002/num.21994.

Figure 1.  Convergence of $ E $ (left) and $ N $ (right) between the QZS and classical ZS in Example $ 4.1 $
Figure 2.  Inelastic collision between two solitons in Example $ 4.2 $ under case (i)
Figure 3.  Inelastic collision between two solitons in Example $ 4.2 $ under case (ii)
Figure 4.  Inelastic collision between two solitons in Example $ 4.2 $ under case (iii)
Figure 5.  Pattern dynamics of QZS in Example $ 4.3 $, contours of $ |E| $
Figure 6.  Convergence of $ E $ (left) and $ N $ (right) between the QZS and classical ZS in Example $ 4.4 $
Figure 7.  Plot of energy density $ |E|^2 $ and ion density fluctuation $ N $ in Example $ 4.5 $, $ \mu = \nu = 20 $, $ \varepsilon = \frac{1}{2^5} $
Table 1.  Spatial errors of the scheme at $ T = 1 $ for Example 4.1 with different $ \varepsilon $, $ \tau = 10^{-5} $
$ e_{\varepsilon} $ $ h = 1 $ $ 1/2 $ $ 1/2^2 $ $ 1/2^3 $
$ \varepsilon =\frac{1}{2^1} $ 1.41e-2 6.83e-5 3.78e-9 4.77e-12
$ \varepsilon =\frac{1}{2^3} $ 4.27e-2 1.34e-4 4.80e-9 5.02e-12
$ \varepsilon =\frac{1}{2^5} $ 5.01e-2 2.49e-4 4.94e-9 5.22e-12
$ \varepsilon =\frac{1}{2^{7}} $ 5.06e-2 2.60e-4 7.75e-9 5.25e-12
$ \varepsilon =\frac{1}{2^9} $ 5.06e-2 2.61e-4 8.17e-9 5.17e-12
$ \varepsilon =\frac{1}{2^{11}} $ 5.06e-2 2.61e-4 8.20e-9 5.25e-12
$ n_{\varepsilon} $ $ h = 1 $ $ 1/2 $ $ 1/2^2 $ $ 1/2^3 $
$ \varepsilon =\frac{1}{2^1} $ 3.11e-2 7.41e-4 6.27e-8 3.30e-12
$ \varepsilon =\frac{1}{2^3} $ 1.07e-1 8.36e-4 4.98e-8 1.83e-12
$ \varepsilon =\frac{1}{2^5} $ 1.07e-1 6.86e-4 1.67e-8 2.38e-12
$ \varepsilon =\frac{1}{2^{7}} $ 1.07e-1 8.50e-4 1.66e-8 2.19e-12
$ \varepsilon =\frac{1}{2^9} $ 1.07e-1 8.61e-4 1.75e-8 2.38e-12
$ \varepsilon =\frac{1}{2^{11}} $ 1.07e-1 8.62e-4 1.75e-8 3.62e-12
$ e_{\varepsilon} $ $ h = 1 $ $ 1/2 $ $ 1/2^2 $ $ 1/2^3 $
$ \varepsilon =\frac{1}{2^1} $ 1.41e-2 6.83e-5 3.78e-9 4.77e-12
$ \varepsilon =\frac{1}{2^3} $ 4.27e-2 1.34e-4 4.80e-9 5.02e-12
$ \varepsilon =\frac{1}{2^5} $ 5.01e-2 2.49e-4 4.94e-9 5.22e-12
$ \varepsilon =\frac{1}{2^{7}} $ 5.06e-2 2.60e-4 7.75e-9 5.25e-12
$ \varepsilon =\frac{1}{2^9} $ 5.06e-2 2.61e-4 8.17e-9 5.17e-12
$ \varepsilon =\frac{1}{2^{11}} $ 5.06e-2 2.61e-4 8.20e-9 5.25e-12
$ n_{\varepsilon} $ $ h = 1 $ $ 1/2 $ $ 1/2^2 $ $ 1/2^3 $
$ \varepsilon =\frac{1}{2^1} $ 3.11e-2 7.41e-4 6.27e-8 3.30e-12
$ \varepsilon =\frac{1}{2^3} $ 1.07e-1 8.36e-4 4.98e-8 1.83e-12
$ \varepsilon =\frac{1}{2^5} $ 1.07e-1 6.86e-4 1.67e-8 2.38e-12
$ \varepsilon =\frac{1}{2^{7}} $ 1.07e-1 8.50e-4 1.66e-8 2.19e-12
$ \varepsilon =\frac{1}{2^9} $ 1.07e-1 8.61e-4 1.75e-8 2.38e-12
$ \varepsilon =\frac{1}{2^{11}} $ 1.07e-1 8.62e-4 1.75e-8 3.62e-12
Table 2.  Temporal errors of the scheme at $ T = 1 $ for Example 4.1 with different $ \varepsilon $, $ h = 1/2^4 $
$ e_{\varepsilon} $ $ \tau_0 = 1/20 $ $ \tau_0/2 $ $ \tau_0 /2^2 $ $ \tau_0 /2^3 $ $ \tau_0 /2^4 $ $ \tau_0 /2^5 $
$ \varepsilon =\frac{1}{2^1} $ 1.85e-3 3.80e-4 7.31e-5 1.38e-5 3.55e-6 8.03e-7
rate - 2.28 2.38 2.41 1.96 2.14
$ \varepsilon =\frac{1}{2^3} $ 6.81e-4 1.69e-4 4.22e-5 1.05e-5 2.63e-6 6.52e-7
rate - 2.01 2.00 2.00 2.00 2.01
$ \varepsilon =\frac{1}{2^5} $ 7.56e-4 1.85e-4 4.61e-5 1.15e-5 2.87e-6 7.13e-7
rate - 2.03 2.01 2.00 2.00 2.01
$ \varepsilon =\frac{1}{2^{7}} $ 7.63e-4 1.87e-4 4.66e-5 1.16e-5 2.90e-6 7.20e-7
rate - 2.03 2.01 2.00 2.00 2.01
$ \varepsilon =\frac{1}{2^9} $ 7.64e-4 1.87e-4 4.66e-5 1.16e-5 2.90e-6 7.20e-7
rate - 2.03 2.01 2.00 2.00 2.01
$ \varepsilon =\frac{1}{2^{11}} $ 7.64e-4 1.87e-4 4.66e-5 1.16e-5 2.90e-6 7.20e-7
rate - 2.03 2.01 2.00 2.00 2.01
$ n_{\varepsilon} $ $ \tau_0 =1/20 $ $ \tau_0/2 $ $ \tau_0 /2^2 $ $ \tau_0 /2^3 $ $ \tau_0 /2^4 $ $ \tau_0 /2^5 $
$ \varepsilon =\frac{1}{2^1} $ 1.14e-3 3.26e-4 7.08e-5 1.56e-5 3.86e-6 9.46e-7
rate - 1.80 2.20 2.18 2.01 2.03
$ \varepsilon =\frac{1}{2^3} $ 2.16e-3 5.15e-4 1.28e-4 3.20e-5 7.98e-6 1.98e-6
rate - 2.07 2.00 2.00 2.00 2.01
$ \varepsilon =\frac{1}{2^5} $ 2.33e-3 5.71e-4 1.42e-4 3.54e-5 8.84e-6 2.19e-6
rate - 2.03 2.01 2.00 2.00 2.01
$ \varepsilon =\frac{1}{2^{7}} $ 2.33e-3 5.74e-4 1.43e-4 3.56e-5 8.89e-6 2.21e-6
rate - 2.02 2.01 2.00 2.00 2.01
$ \varepsilon =\frac{1}{2^9} $ 2.33e-3 5.74e-4 1.43e-4 3.57e-5 8.89e-6 2.21e-6
rate - 2.02 2.01 2.00 2.00 2.01
$ \varepsilon =\frac{1}{2^{11}} $ 2.33e-3 5.74e-4 1.43e-4 3.57e-5 8.89e-6 2.21e-6
rate - 2.02 2.01 2.00 2.00 2.01
$ e_{\varepsilon} $ $ \tau_0 = 1/20 $ $ \tau_0/2 $ $ \tau_0 /2^2 $ $ \tau_0 /2^3 $ $ \tau_0 /2^4 $ $ \tau_0 /2^5 $
$ \varepsilon =\frac{1}{2^1} $ 1.85e-3 3.80e-4 7.31e-5 1.38e-5 3.55e-6 8.03e-7
rate - 2.28 2.38 2.41 1.96 2.14
$ \varepsilon =\frac{1}{2^3} $ 6.81e-4 1.69e-4 4.22e-5 1.05e-5 2.63e-6 6.52e-7
rate - 2.01 2.00 2.00 2.00 2.01
$ \varepsilon =\frac{1}{2^5} $ 7.56e-4 1.85e-4 4.61e-5 1.15e-5 2.87e-6 7.13e-7
rate - 2.03 2.01 2.00 2.00 2.01
$ \varepsilon =\frac{1}{2^{7}} $ 7.63e-4 1.87e-4 4.66e-5 1.16e-5 2.90e-6 7.20e-7
rate - 2.03 2.01 2.00 2.00 2.01
$ \varepsilon =\frac{1}{2^9} $ 7.64e-4 1.87e-4 4.66e-5 1.16e-5 2.90e-6 7.20e-7
rate - 2.03 2.01 2.00 2.00 2.01
$ \varepsilon =\frac{1}{2^{11}} $ 7.64e-4 1.87e-4 4.66e-5 1.16e-5 2.90e-6 7.20e-7
rate - 2.03 2.01 2.00 2.00 2.01
$ n_{\varepsilon} $ $ \tau_0 =1/20 $ $ \tau_0/2 $ $ \tau_0 /2^2 $ $ \tau_0 /2^3 $ $ \tau_0 /2^4 $ $ \tau_0 /2^5 $
$ \varepsilon =\frac{1}{2^1} $ 1.14e-3 3.26e-4 7.08e-5 1.56e-5 3.86e-6 9.46e-7
rate - 1.80 2.20 2.18 2.01 2.03
$ \varepsilon =\frac{1}{2^3} $ 2.16e-3 5.15e-4 1.28e-4 3.20e-5 7.98e-6 1.98e-6
rate - 2.07 2.00 2.00 2.00 2.01
$ \varepsilon =\frac{1}{2^5} $ 2.33e-3 5.71e-4 1.42e-4 3.54e-5 8.84e-6 2.19e-6
rate - 2.03 2.01 2.00 2.00 2.01
$ \varepsilon =\frac{1}{2^{7}} $ 2.33e-3 5.74e-4 1.43e-4 3.56e-5 8.89e-6 2.21e-6
rate - 2.02 2.01 2.00 2.00 2.01
$ \varepsilon =\frac{1}{2^9} $ 2.33e-3 5.74e-4 1.43e-4 3.57e-5 8.89e-6 2.21e-6
rate - 2.02 2.01 2.00 2.00 2.01
$ \varepsilon =\frac{1}{2^{11}} $ 2.33e-3 5.74e-4 1.43e-4 3.57e-5 8.89e-6 2.21e-6
rate - 2.02 2.01 2.00 2.00 2.01
Table 3.  Temporal errors of the scheme at $ T = 1 $ for Example $ 4.4 $ with different $ \varepsilon $, $ h = 1/2^4 $
$ e_{\varepsilon} $ $ \tau_0 = 1/20 $ $ \tau_0/2 $ $ \tau_0 /2^2 $ $ \tau_0 /2^3 $ $ \tau_0 /2^4 $ $ \tau_0 /2^5 $
$ \varepsilon =\frac{1}{2^1} $ 3.23e-2 8.06e-3 2.01e-3 5.02e-4 1.25e-4 3.11e-5
rate - 2.00 2.00 2.00 2.00 2.01
$ \varepsilon =\frac{1}{2^3} $ 2.01e-2 5.03e-3 1.26e-3 3.14e-4 7.83e-5 1.94e-5
rate - 2.00 2.00 2.00 2.00 2.01
$ \varepsilon =\frac{1}{2^5} $ 1.92e-2 4.80e-3 1.20e-3 3.00e-4 7.49e-5 1.86e-5
rate - 2.00 2.00 2.00 2.00 2.01
$ \varepsilon =\frac{1}{2^{7}} $ 1.92e-2 4.79e-3 1.20e-3 2.99e-4 7.47e-5 1.85e-5
rate - 2.00 2.00 2.00 2.00 2.01
$ \varepsilon =\frac{1}{2^9} $ 1.92e-2 4.79e-3 1.20e-3 2.99e-4 7.47e-5 1.85e-5
rate - 2.00 2.00 2.00 2.00 2.01
$ \varepsilon =\frac{1}{2^{11}} $ 1.92e-2 4.79e-3 1.20e-3 2.99e-04 7.47e-5 1.85e-5
rate - 2.00 2.00 2.00 2.00 2.01
$ n_{\varepsilon} $ $ \tau_0 =1/20 $ $ \tau_0/2 $ $ \tau_0 /2^2 $ $ \tau_0 /2^3 $ $ \tau_0 /2^4 $ $ \tau_0 /2^5 $
$ \varepsilon =\frac{1}{2^1} $ 1.01e-2 2.51e-3 6.26e-4 1.56e-4 3.90e-5 9.71e-6
rate - 2.01 2.00 2.00 2.00 2.01
$ \varepsilon =\frac{1}{2^3} $ 6.06e-3 1.52e-3 3.80e-4 9.49e-5 2.37e-5 5.92e-6
rate - 2.00 2.00 2.00 2.00 2.00
$ \varepsilon =\frac{1}{2^5} $ 6.27e-3 1.57e-3 3.93e-4 9.82e-5 2.45e-5 6.11e-6
rate - 2.00 2.00 2.00 2.00 2.01
$ \varepsilon =\frac{1}{2^{7}} $ 6.28e-3 1.57e-3 3.94e-4 9.84e-5 2.46e-5 6.10e-6
rate - 2.00 2.00 2.00 2.00 2.01
$ \varepsilon =\frac{1}{2^9} $ 6.28e-3 1.57e-3 3.94e-4 9.84e-5 2.45e-5 6.04e-6
rate - 2.00 2.00 2.00 2.00 2.02
$ \varepsilon =\frac{1}{2^{11}} $ 6.28e-3 1.57e-3 3.94e-4 9.84e-5 2.45e-5 6.03e-6
rate - 2.00 2.00 2.00 2.01 2.02
$ e_{\varepsilon} $ $ \tau_0 = 1/20 $ $ \tau_0/2 $ $ \tau_0 /2^2 $ $ \tau_0 /2^3 $ $ \tau_0 /2^4 $ $ \tau_0 /2^5 $
$ \varepsilon =\frac{1}{2^1} $ 3.23e-2 8.06e-3 2.01e-3 5.02e-4 1.25e-4 3.11e-5
rate - 2.00 2.00 2.00 2.00 2.01
$ \varepsilon =\frac{1}{2^3} $ 2.01e-2 5.03e-3 1.26e-3 3.14e-4 7.83e-5 1.94e-5
rate - 2.00 2.00 2.00 2.00 2.01
$ \varepsilon =\frac{1}{2^5} $ 1.92e-2 4.80e-3 1.20e-3 3.00e-4 7.49e-5 1.86e-5
rate - 2.00 2.00 2.00 2.00 2.01
$ \varepsilon =\frac{1}{2^{7}} $ 1.92e-2 4.79e-3 1.20e-3 2.99e-4 7.47e-5 1.85e-5
rate - 2.00 2.00 2.00 2.00 2.01
$ \varepsilon =\frac{1}{2^9} $ 1.92e-2 4.79e-3 1.20e-3 2.99e-4 7.47e-5 1.85e-5
rate - 2.00 2.00 2.00 2.00 2.01
$ \varepsilon =\frac{1}{2^{11}} $ 1.92e-2 4.79e-3 1.20e-3 2.99e-04 7.47e-5 1.85e-5
rate - 2.00 2.00 2.00 2.00 2.01
$ n_{\varepsilon} $ $ \tau_0 =1/20 $ $ \tau_0/2 $ $ \tau_0 /2^2 $ $ \tau_0 /2^3 $ $ \tau_0 /2^4 $ $ \tau_0 /2^5 $
$ \varepsilon =\frac{1}{2^1} $ 1.01e-2 2.51e-3 6.26e-4 1.56e-4 3.90e-5 9.71e-6
rate - 2.01 2.00 2.00 2.00 2.01
$ \varepsilon =\frac{1}{2^3} $ 6.06e-3 1.52e-3 3.80e-4 9.49e-5 2.37e-5 5.92e-6
rate - 2.00 2.00 2.00 2.00 2.00
$ \varepsilon =\frac{1}{2^5} $ 6.27e-3 1.57e-3 3.93e-4 9.82e-5 2.45e-5 6.11e-6
rate - 2.00 2.00 2.00 2.00 2.01
$ \varepsilon =\frac{1}{2^{7}} $ 6.28e-3 1.57e-3 3.94e-4 9.84e-5 2.46e-5 6.10e-6
rate - 2.00 2.00 2.00 2.00 2.01
$ \varepsilon =\frac{1}{2^9} $ 6.28e-3 1.57e-3 3.94e-4 9.84e-5 2.45e-5 6.04e-6
rate - 2.00 2.00 2.00 2.00 2.02
$ \varepsilon =\frac{1}{2^{11}} $ 6.28e-3 1.57e-3 3.94e-4 9.84e-5 2.45e-5 6.03e-6
rate - 2.00 2.00 2.00 2.01 2.02
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