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October  2019, 24(10): 5523-5538. doi: 10.3934/dcdsb.2019069

Fully decoupled schemes for the coupled Schrödinger-KdV system

1. 

School of Mathematical Science, Huaiyin Normal University, Huaian, Jiangsu 223300, China

2. 

Department of Basis Education, Jiangsu Vocational College of Finance & Economics, Huaian, Jiangsu, 223003, China

* Corresponding author: cjx1981@hytc.edu.cn (J. Cai)

Received  July 2018 Revised  December 2018 Published  October 2019 Early access  April 2019

Fund Project: The first author is supported by the Natural Science Foundation of Jiangsu Province of China grant BK20181482, Qing Lan Project of Jiangsu Province of China and Jiangsu Overseas Visiting Scholar Program for University Prominent Young & Middle-aged Teachers and President.

The coupled numerical schemes are inefficient for the time-dependent coupled Schrödinger-KdV system. In this study, some splitting schemes are proposed for the system based on the operator splitting method and coordinate increment discrete gradient method. The schemes are decoupled, so that each of the variables can be solved separately at each time level. Ample numerical experiments are carried out to demonstrate the efficiency and accuracy of our schemes.

Citation: Jiaxiang Cai, Juan Chen, Bin Yang. Fully decoupled schemes for the coupled Schrödinger-KdV system. Discrete and Continuous Dynamical Systems - B, 2019, 24 (10) : 5523-5538. doi: 10.3934/dcdsb.2019069
References:
[1]

K. O. Aiyesimoju and R. J. Sobey, Process splitting of the boundary conditions for the advection-dispersion equation, Int. J. Numer. Methods Fluids, 9 (1989), 235-244.  doi: 10.1002/fld.1650090208.

[2]

P. Amorim and M. Figueira, Convergence of a numerical scheme for a coupled Schrödinger-KdV system, Rev. Mat. Complut., 26 (2013), 409-426.  doi: 10.1007/s13163-012-0097-8.

[3]

K. Appert and J. Vaclavik, Dynamics of coupled solitons, Phys. Fluids, 20 (1977), 1845-1849.  doi: 10.1063/1.861802.

[4]

U. M. Ascher and R. I. McLachlan, Multisymplectic box schemes and the Korteweg-de Vries equation, Appl. Numer. Math., 48 (2004), 255-269.  doi: 10.1016/j.apnum.2003.09.002.

[5]

D. M. Bai and L. M. Zhang, The finite element method for the coupled Schrödinger-KdV equations, Phys. Lett. A, 373 (2009), 2237-2244.  doi: 10.1016/j.physleta.2009.04.043.

[6]

J. CaiC. Bai and H. Zhang, Efficient schemes for the coupled Schrödinger-KdV equations: Decoupled and conserving three invariants, Appl. Math. Lett., 86 (2018), 200-207.  doi: 10.1016/j.aml.2018.06.038.

[7]

J. CaiY. Wang and C. Jiang, Local structure-preserving algorithms for general multi-symplectic Hamiltonian PDEs, Comput. Phys. Commun., 235 (2019), 210-220.  doi: 10.1016/j.cpc.2018.08.015.

[8]

J. X. CaiC. Z. Bai and H. H. Zhang, Decoupled local/global energy-preserving schemes for the $N$-coupled nonlinear Schrödinger equations, J. Comput. Phys., 374 (2018), 281-299.  doi: 10.1016/j.jcp.2018.07.050.

[9]

J. CaiB. Yang and C. Zhang, Efficient mass- and energy-preserving schemes for the coupled nonlinear Schrödinger-Boussinesq system, Appl. Math. Lett., 91 (2019), 76-82.  doi: 10.1016/j.aml.2018.11.024.

[10]

J. X. CaiJ. L. HongY. S. Wang and Y. Z. Gong, Two energy-conserved splitting methods for three-dimensional time-domain Maxwell's equations and the convergence analysis, SIAM J. Numer. Anal., 53 (2015), 1918-1940.  doi: 10.1137/140971609.

[11]

E. CelledoniV. GrimmR. I. McLachlanD. I. McLarenD. O'NealeB. Owren and G. R. W. Quispel, Preserving energy resp. dissipation in numerical PDEs using the ``Average Vector Field" method, J. Comput. Phys., 231 (2012), 6770-6789.  doi: 10.1016/j.jcp.2012.06.022.

[12]

E. Fan, Multiple travelling wave solutions of nonlinear evolution equations using a unified algebraic method, J. Phys. A: Math. Gen., 35 (2002), 6853-6872.  doi: 10.1088/0305-4470/35/32/306.

[13]

A. Golbabai and A. S. Vaighani, A meshless method for numerical solution of the coupled Schrödinger-KdV equations, Computing, 92 (2011), 225-242.  doi: 10.1007/s00607-010-0138-4.

[14]

Y. Z. GongJ. Q. Gao and Y. S. Wang, High order Gauss-Seidel schemes for charged particle dynamics, Discrete Cont. Dyn. B, 23 (2018), 573-585.  doi: 10.3934/dcdsb.2018034.

[15]

O. Gonzalez and J. C. Simo, On the stability of symplectic and energy-momentum algorithms for nonlinear Hamiltonian systems with symmetry, Comput. Methods Appl. Mech. Eng., 134 (1996), 197-222.  doi: 10.1016/0045-7825(96)01009-2.

[16]

E. Hairer, C. Lubich and G. Wanner, Geometric Numerical Integration: Structure-preserving Algorithms for Ordinary Differential Equations, 2nd edition, Springer-Verlag, Berlin, 2006.

[17]

M. S. Ismail, F. M. Mosally and K. M. Alamoudi, Petrov-Galerkin method for the coupled nonlinear Schödinger-KdV equation, Abstr. Appl. Anal., 2014 (2014), Art. ID 705204, 8 pp. doi: 10.1155/2014/705204.

[18]

T. Itoh and K. Abe, Hamiltonian-conserving discrete canonical equations based on variational difference quotients, J. Comput. Phys., 76 (1998), 85-102.  doi: 10.1016/0021-9991(88)90132-5.

[19]

R. J. LeVeque, Intermediate boundary conditions for time-split methods applied to hyperbolic partial differential equations, Math. Comput., 47 (1986), 37-54.  doi: 10.1090/S0025-5718-1986-0842122-8.

[20]

Y. Q. Liu, R. J. Cheng and H. X. Ge, An element-free Galerkin (EFG) method for numerical solution of the coupled Schrödinger-KdV equations, Chin. Phys. B, 22 (2013), 100204, 9pp. doi: 10.1088/1674-1056/22/10/100204.

[21]

J. E. Marsden and A. Weinstein, The Hamiltonian structure of the Maxwell-Vlasov equations, Phys. D, 4 (1982), 394-406.  doi: 10.1016/0167-2789(82)90043-4.

[22]

Ö. Oruc and A. Esen, A Haar wavelet collocation method for coupled nonlinear Schödinger-KdV equations, Int. J. Modern Phys. C, 27 (2016), 1650103, 16pp. doi: 10.1142/S0129183116501035.

[23]

G. R. W. Quispel and D. I. McLaren, A new class of energy-preserving numerical integration methods, J. Phys. A, 41 (2008), 045206, 7pp. doi: 10.1088/1751-8113/41/4/045206.

[24]

M. Suzuki, Fractal decomposition of exponential operators with applications to many-body theories and Monte Carolo simulations, Phys. Lett. A, 146 (1990), 319-323.  doi: 10.1016/0375-9601(90)90962-N.

[25]

X. P. WangC. J. García-Cervera and W. N. E, A Gauss-Seidel projection method for micromagnetics simulations, J. Comput. Phys., 171 (2001), 357-372.  doi: 10.1006/jcph.2001.6793.

[26]

H. Yoshida, Construction of higher order symplectic integrators, Phys. Lett. A, 150 (1990), 262-268.  doi: 10.1016/0375-9601(90)90092-3.

[27]

Z. Zhang, S. S. Song, X. D. Chen and W. E. Zhou, Average vector field methods for the coupled Schrödinger-KdV equations, Chin. Phys. B, 23 (2014), 070208, 9pp. doi: 10.1088/1674-1056/23/7/070208.

show all references

References:
[1]

K. O. Aiyesimoju and R. J. Sobey, Process splitting of the boundary conditions for the advection-dispersion equation, Int. J. Numer. Methods Fluids, 9 (1989), 235-244.  doi: 10.1002/fld.1650090208.

[2]

P. Amorim and M. Figueira, Convergence of a numerical scheme for a coupled Schrödinger-KdV system, Rev. Mat. Complut., 26 (2013), 409-426.  doi: 10.1007/s13163-012-0097-8.

[3]

K. Appert and J. Vaclavik, Dynamics of coupled solitons, Phys. Fluids, 20 (1977), 1845-1849.  doi: 10.1063/1.861802.

[4]

U. M. Ascher and R. I. McLachlan, Multisymplectic box schemes and the Korteweg-de Vries equation, Appl. Numer. Math., 48 (2004), 255-269.  doi: 10.1016/j.apnum.2003.09.002.

[5]

D. M. Bai and L. M. Zhang, The finite element method for the coupled Schrödinger-KdV equations, Phys. Lett. A, 373 (2009), 2237-2244.  doi: 10.1016/j.physleta.2009.04.043.

[6]

J. CaiC. Bai and H. Zhang, Efficient schemes for the coupled Schrödinger-KdV equations: Decoupled and conserving three invariants, Appl. Math. Lett., 86 (2018), 200-207.  doi: 10.1016/j.aml.2018.06.038.

[7]

J. CaiY. Wang and C. Jiang, Local structure-preserving algorithms for general multi-symplectic Hamiltonian PDEs, Comput. Phys. Commun., 235 (2019), 210-220.  doi: 10.1016/j.cpc.2018.08.015.

[8]

J. X. CaiC. Z. Bai and H. H. Zhang, Decoupled local/global energy-preserving schemes for the $N$-coupled nonlinear Schrödinger equations, J. Comput. Phys., 374 (2018), 281-299.  doi: 10.1016/j.jcp.2018.07.050.

[9]

J. CaiB. Yang and C. Zhang, Efficient mass- and energy-preserving schemes for the coupled nonlinear Schrödinger-Boussinesq system, Appl. Math. Lett., 91 (2019), 76-82.  doi: 10.1016/j.aml.2018.11.024.

[10]

J. X. CaiJ. L. HongY. S. Wang and Y. Z. Gong, Two energy-conserved splitting methods for three-dimensional time-domain Maxwell's equations and the convergence analysis, SIAM J. Numer. Anal., 53 (2015), 1918-1940.  doi: 10.1137/140971609.

[11]

E. CelledoniV. GrimmR. I. McLachlanD. I. McLarenD. O'NealeB. Owren and G. R. W. Quispel, Preserving energy resp. dissipation in numerical PDEs using the ``Average Vector Field" method, J. Comput. Phys., 231 (2012), 6770-6789.  doi: 10.1016/j.jcp.2012.06.022.

[12]

E. Fan, Multiple travelling wave solutions of nonlinear evolution equations using a unified algebraic method, J. Phys. A: Math. Gen., 35 (2002), 6853-6872.  doi: 10.1088/0305-4470/35/32/306.

[13]

A. Golbabai and A. S. Vaighani, A meshless method for numerical solution of the coupled Schrödinger-KdV equations, Computing, 92 (2011), 225-242.  doi: 10.1007/s00607-010-0138-4.

[14]

Y. Z. GongJ. Q. Gao and Y. S. Wang, High order Gauss-Seidel schemes for charged particle dynamics, Discrete Cont. Dyn. B, 23 (2018), 573-585.  doi: 10.3934/dcdsb.2018034.

[15]

O. Gonzalez and J. C. Simo, On the stability of symplectic and energy-momentum algorithms for nonlinear Hamiltonian systems with symmetry, Comput. Methods Appl. Mech. Eng., 134 (1996), 197-222.  doi: 10.1016/0045-7825(96)01009-2.

[16]

E. Hairer, C. Lubich and G. Wanner, Geometric Numerical Integration: Structure-preserving Algorithms for Ordinary Differential Equations, 2nd edition, Springer-Verlag, Berlin, 2006.

[17]

M. S. Ismail, F. M. Mosally and K. M. Alamoudi, Petrov-Galerkin method for the coupled nonlinear Schödinger-KdV equation, Abstr. Appl. Anal., 2014 (2014), Art. ID 705204, 8 pp. doi: 10.1155/2014/705204.

[18]

T. Itoh and K. Abe, Hamiltonian-conserving discrete canonical equations based on variational difference quotients, J. Comput. Phys., 76 (1998), 85-102.  doi: 10.1016/0021-9991(88)90132-5.

[19]

R. J. LeVeque, Intermediate boundary conditions for time-split methods applied to hyperbolic partial differential equations, Math. Comput., 47 (1986), 37-54.  doi: 10.1090/S0025-5718-1986-0842122-8.

[20]

Y. Q. Liu, R. J. Cheng and H. X. Ge, An element-free Galerkin (EFG) method for numerical solution of the coupled Schrödinger-KdV equations, Chin. Phys. B, 22 (2013), 100204, 9pp. doi: 10.1088/1674-1056/22/10/100204.

[21]

J. E. Marsden and A. Weinstein, The Hamiltonian structure of the Maxwell-Vlasov equations, Phys. D, 4 (1982), 394-406.  doi: 10.1016/0167-2789(82)90043-4.

[22]

Ö. Oruc and A. Esen, A Haar wavelet collocation method for coupled nonlinear Schödinger-KdV equations, Int. J. Modern Phys. C, 27 (2016), 1650103, 16pp. doi: 10.1142/S0129183116501035.

[23]

G. R. W. Quispel and D. I. McLaren, A new class of energy-preserving numerical integration methods, J. Phys. A, 41 (2008), 045206, 7pp. doi: 10.1088/1751-8113/41/4/045206.

[24]

M. Suzuki, Fractal decomposition of exponential operators with applications to many-body theories and Monte Carolo simulations, Phys. Lett. A, 146 (1990), 319-323.  doi: 10.1016/0375-9601(90)90962-N.

[25]

X. P. WangC. J. García-Cervera and W. N. E, A Gauss-Seidel projection method for micromagnetics simulations, J. Comput. Phys., 171 (2001), 357-372.  doi: 10.1006/jcph.2001.6793.

[26]

H. Yoshida, Construction of higher order symplectic integrators, Phys. Lett. A, 150 (1990), 262-268.  doi: 10.1016/0375-9601(90)90092-3.

[27]

Z. Zhang, S. S. Song, X. D. Chen and W. E. Zhou, Average vector field methods for the coupled Schrödinger-KdV equations, Chin. Phys. B, 23 (2014), 070208, 9pp. doi: 10.1088/1674-1056/23/7/070208.

Figure 1.  The solutions for the CS-KdV system at $ T = 50 $. Solid line: exact solution; Star: numerical solutions
Figure 2.  Top: the errors in solution; Bottom: the changes in invariants
Figure 3.  Left: the maximal error in solution Vs. time step (Red: S-CI-1; Blue: S-CI-2$ \hat{b} $; Square: $ E $; Circle: $ N $); Right: the changes in invariants Vs. time step (Red: S-CI-1; Blue: S-CI-2$ \hat{b} $; Square: $ \mathcal{I}_1 $; Star: $ \mathcal{I}_3 $)
Figure 4.  Left: the maximal error in solution Vs. CPU time (Circle: S-CI-1; Star: S-AVF-2; Square: S-CI-2$ \hat{a} $; Diamond: S-CI-2$ \underline{a} $; Red triangle: AVFS [27])
Figure 5.  The numerical (Star) and exact (solid line) solutions at $ T = 1 $ for the case $ \gamma = 0.1 $
Figure 6.  The numerical (Star) and exact (solid line) solutions at $ T = 1 $ for the case $ \gamma = 1 $
Figure 7.  The errors in solution (top) and the relative changes in invariants (bottom) for the cases $ \gamma = 1 $ (left) and $ \gamma = 10 $ (right), respectively
Figure 8.  The numerical (circle) and exact solutions (solid line) for the case $ \gamma = 10 $
Table 1.  The solution errors for the CS-KdV system (1): $ x\in[-30,30] $, $ \Delta x = 0.5 $, $ \tau = 0.1 $ and $ T = 10 $
Method e2,p e2,q e2,N ${{\rm{e}}_{\infty ,p}}$ ${{\rm{e}}_{\infty ,q}}$ ${{\rm{e}}_{\infty ,N}}$
$\;{\rm{S-CI}}-2\hat a$ 7.16e-3 7.81e-3 1.27e-4 2.98e-3 6.02e-3 1.80e-4
${\rm{S-CI}}-2\underset{\raise0.3em\hbox{$\smash{\scriptscriptstyle-}$}}{a}$ 7.16e-3 7.81e-3 1.21e-4 2.98e-3 6.01e-3 1.71e-4
${\rm{S-CI}}-2\hat b$ 7.12e-3 7.75e-3 1.35e-4 3.08e-3 5.95e-3 1.91e-4
${\rm{S-CI}}-2\underset{\raise0.3em\hbox{$\smash{\scriptscriptstyle-}$}}{b}$ 7.12e-3 7.75e-3 1.38e-4 3.08e-3 5.95e-3 1.95e-4
AVF[27] 7.13e-3 7.80e-3 3.27e-4 2.95e-3 5.98e-3 1.13e-4
AVFS[27] 7.16e-3 7.81e-3 5.05e-4 2.99e-3 6.01e-3 1.71e-4
EFG[20] 9.28e-3 1.42e-2 2.09e-3 3.45e-3 9.53e-3 7.74e-4
Method e2,p e2,q e2,N ${{\rm{e}}_{\infty ,p}}$ ${{\rm{e}}_{\infty ,q}}$ ${{\rm{e}}_{\infty ,N}}$
$\;{\rm{S-CI}}-2\hat a$ 7.16e-3 7.81e-3 1.27e-4 2.98e-3 6.02e-3 1.80e-4
${\rm{S-CI}}-2\underset{\raise0.3em\hbox{$\smash{\scriptscriptstyle-}$}}{a}$ 7.16e-3 7.81e-3 1.21e-4 2.98e-3 6.01e-3 1.71e-4
${\rm{S-CI}}-2\hat b$ 7.12e-3 7.75e-3 1.35e-4 3.08e-3 5.95e-3 1.91e-4
${\rm{S-CI}}-2\underset{\raise0.3em\hbox{$\smash{\scriptscriptstyle-}$}}{b}$ 7.12e-3 7.75e-3 1.38e-4 3.08e-3 5.95e-3 1.95e-4
AVF[27] 7.13e-3 7.80e-3 3.27e-4 2.95e-3 5.98e-3 1.13e-4
AVFS[27] 7.16e-3 7.81e-3 5.05e-4 2.99e-3 6.01e-3 1.71e-4
EFG[20] 9.28e-3 1.42e-2 2.09e-3 3.45e-3 9.53e-3 7.74e-4
Table 2.  The maximal solution errors for the CS-KdV system (1): $ x\in[-50,50] $, $ \Delta x = 0.1 $, $ \tau = 0.1 $ and $ T = 8 $
Method ${{\rm{e}}_{\infty ,E}}$ ${{\rm{e}}_{\infty ,N}}$
${\rm{S-CI}}-2\hat a$ 2.15e-4 1.69e-4
${\rm{S-CI}}-2\underset{\raise0.3em\hbox{$\smash{\scriptscriptstyle-}$}}{a}$ 1.98e-4 1.61e-4
${\rm{S-CI}}-2\hat b$ 7.41e-5 2.88e-5
${\rm{S-CI}}-2\underset{\raise0.3em\hbox{$\smash{\scriptscriptstyle-}$}}{b}$ 7.40e-5 2.68e-5
HW[22] 1.21e-4 1.14e-4
2-order PGM[17] 9.41e-5 2.92e-5
Method ${{\rm{e}}_{\infty ,E}}$ ${{\rm{e}}_{\infty ,N}}$
${\rm{S-CI}}-2\hat a$ 2.15e-4 1.69e-4
${\rm{S-CI}}-2\underset{\raise0.3em\hbox{$\smash{\scriptscriptstyle-}$}}{a}$ 1.98e-4 1.61e-4
${\rm{S-CI}}-2\hat b$ 7.41e-5 2.88e-5
${\rm{S-CI}}-2\underset{\raise0.3em\hbox{$\smash{\scriptscriptstyle-}$}}{b}$ 7.40e-5 2.68e-5
HW[22] 1.21e-4 1.14e-4
2-order PGM[17] 9.41e-5 2.92e-5
Table 3.  The maximal solution errors for CS-KdV system (1): $ x\in[-50,50] $, $ \Delta x = 0.1 $, $ \tau = 0.0001 $ and $ T = 0.1 $
Method ${{\rm{e}}_{\infty ,E}}$ ${{\rm{e}}_{\infty ,N}}$
${\rm{S-CI}}-2\hat b$ 1.73e-5 2.57e-10
${\rm{S-CI}}-2\underset{\raise0.3em\hbox{$\smash{\scriptscriptstyle-}$}}{b}$ 1.73e-5 2.57e-10
4-order RK-PGM[17] 4.73e-5 5.65e-8
Method ${{\rm{e}}_{\infty ,E}}$ ${{\rm{e}}_{\infty ,N}}$
${\rm{S-CI}}-2\hat b$ 1.73e-5 2.57e-10
${\rm{S-CI}}-2\underset{\raise0.3em\hbox{$\smash{\scriptscriptstyle-}$}}{b}$ 1.73e-5 2.57e-10
4-order RK-PGM[17] 4.73e-5 5.65e-8
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