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Fully decoupled schemes for the coupled Schrödinger-KdV system

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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  • 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.

    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

    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
     | Show Table
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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
     | Show Table
    DownLoad: CSV

    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
     | Show Table
    DownLoad: CSV
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