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2019, Volume 24, Issue 10: 5523-5538. Doi: 10.3934/dcdsb.2019069

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

Received: July 2018

Revised: December 2018

Early access: April 2019

Published: August 2019

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.

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Abstract

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.

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Mathematics Subject Classification: 65P10, 65N35, 65N06.

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Figure 1. The solutions for the CS-KdV system at $ T = 50 $. Solid line: exact solution; Star: numerical solutions

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Figure 2. Top: the errors in solution; Bottom: the changes in invariants

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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 $)

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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])

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Figure 5. The numerical (Star) and exact (solid line) solutions at $ T = 1 $ for the case $ \gamma = 0.1 $

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Figure 6. The numerical (Star) and exact (solid line) solutions at $ T = 1 $ for the case $ \gamma = 1 $

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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

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Figure 8. The numerical (circle) and exact solutions (solid line) for the case $ \gamma = 10 $

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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	e_2,p	e_2,q	e_2,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

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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

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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

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