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Locally conservative finite difference schemes for the modified KdV equation

  • * Corresponding author: Peter E. Hydon

    * Corresponding author: Peter E. Hydon
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  • Finite difference schemes that preserve two conservation laws of a given partial differential equation can be found directly by a recently-developed symbolic approach. Until now, this has been used only for equations with quadratic nonlinearity.

    In principle, a simplified version of the direct approach also works for equations with polynomial nonlinearity of higher degree. For the modified Korteweg-de Vries equation, whose nonlinear term is cubic, this approach yields several new families of second-order accurate schemes that preserve mass and either energy or momentum. Two of these families contain Average Vector Field schemes of the type developed by Quispel and co-workers. Numerical tests show that each family includes schemes that are highly accurate compared to other mass-preserving methods that can be found in the literature.

    Mathematics Subject Classification: Primary: 65M06, 37K05; Secondary: 39A14.

    Citation:

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  • Figure 1.  Example of a rectangular stencil for mKdV. PDEs and conservation laws are preserved to second order at the central point $ (x, t) $; densities and fluxes are second-order at $ (x, t-\Delta t/2) $ and $ (x-\Delta x/2, t) $, respectively

    Figure 2.  Two-soliton problem for the mKdV equation. Top: Initial condition (dashed line) and solution of $ \mbox{EC}_{10}(0.04) $ with $ \Delta x = 0.1 $, $ \Delta t = 0.025 $ at time $ T = 10 $ (solid line). Bottom: Top of the faster soliton; exact profile (solid line) and solutions of $ \mbox{EC}_{10}(0.04) $ (circles), narrow box (squares), multisymplectic (diamonds) and $ \mbox{EC}_{10}(0) $ (crosses)

    Figure 3.  Two-soliton problem for the mKdV equation with $ \Delta x = 0.2 $, $ \Delta t = 0.05 $ at time $ T = 10 $. Top of the soliton: exact profile (solid line) and solutions of $ \mbox{EC}_{10}(0.05) $ (circles), narrow box scheme (squares), multisymplectic scheme (diamonds) and $ \mbox{EC}_{10}(0) $ (crosses)

    Figure 4.  Breather problem. Top: Numerical solution given by method $ \mbox{EC}_{8}(2.22) $ with stepsizes $ \Delta x = 0.02 $, $ \Delta t = 0.002 $ at time $ T = 0.4 $. Bottom: exact profile (solid line) and solutions of methods $ \mbox{EC}_{8}(2.22) $ (circles), multisymplectic (diamonds) and $ \mbox{EC}_{10}(0) $ (crosses) at the final time (markers at every tenth point)

    Table 1.  Errors in conservation laws and solutions for the two-soliton problem for the mKdV equation, with $ \Delta x\! = \!0.1 $, $ \Delta t\! = \!0.025 $. An asterisk denotes the error that is minimized

    Method $ \text{Err}_1 $ $ \text{Err}_2 $ $ \text{Err}_3 $ Sol. Err. $ \text{Err}_{\phi_1} $ $ \text{Err}_{\phi_2} $ $ \text{Err}_{\phi} $
    $ \mbox{EC}_{8}(0) $ 1.74e-13 0.0036 5.13e-13 0.3701 -0.51 -0.06 -0.45
    $ \mbox{EC}_{8}(1) $ 1.33e-13 0.0732 8.01e-13$ ^* $ 0.0085$ ^* $ 0 -0.01 0.01
    $ \mbox{EC}_{8}(-0.05) $ 6.22e-14$ ^* $ 1.81e-04$ ^* $ 4.65e-13 0.3857 -0.53 -0.07 -0.46
    $ \mbox{MC}_{8}(0) $ 2.13e-13 3.69e-13 0.0632 0.2396 -0.32 -0.04 -0.28
    $ \mbox{MC}_{8}(-0.077) $ 1.21e-13 3.32e-13 0.0032$ ^* $ 0.0051$ ^* $ 0 0.01 -0.01
    $ \mbox{MC}_{8}(-0.073) $ 6.93e-14 1.46e-13$ ^* $ 5.55e-04$ ^* $ 0.0139 -0.02 0.01 -0.03
    $ \mbox{AVF}_\text{EC} $; $ \mbox{EC}_{10}(0) $ 3.91e-14 0.0142 4.80e-14 0.0167 -0.02 0.01 -0.03
    $ \mbox{EC}_{10}(0.04) $ 3.73e-14 0.0114 9.41e-14$ ^* $ 0.0030$ ^* $ 0 0.01 -0.01
    $ \mbox{EC}_{10}(0.20) $ 5.15e-14$ ^* $ 1.82e-04$ ^* $ 5.51e-14 0.0627 0.08 0.02 0.06
    $ \mbox{AVF}_\text{MC} $; $ \mbox{MC}_{10}(0) $ 4.62e-14 5.68e-14 0.0358 0.0756 -0.10 0 -0.10
    $ \mbox{MC}_{10}(0.19) $ 4.26e-14 5.33e-14 0.0359$ ^* $ 0.0051$ ^* $ 0 0.01 -0.01
    Narrow box 1.28e-13 0.0117 7.0014 0.0742 0.10 0.02 0.08
    Multisymplectic 6.04e-14 0.0058 6.8991 0.2279 -0.31 -0.04 -0.27
     | Show Table
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    Table 2.  Errors in conservation laws and solutions for the two-soliton problem for the mKdV equation, with $ \Delta x = 0.2 $, $ \Delta t = 0.05 $. An asterisk denotes the error that is minimized

    Method $ \text{Err}_1 $ $ \text{Err}_2 $ $ \text{Err}_3 $ Sol. Err. $ \text{Err}_{\phi_1} $ $ \text{Err}_{\phi_2} $ $ \text{Err}_{\phi} $
    $ \mbox{EC}_{8}(0) $ 4.62e-14 0.0155 6.93e-14 0.9599 -1.84 -0.26 -1.58
    $ \mbox{EC}_{8}(0.97) $ 3.55e-14 0.2754 1.14e-13$ ^* $ 0.0358$ ^* $ 0 -0.03 0.03
    $ \mbox{EC}_{8}(-0.06) $ 4.26e-14$ ^* $ 5.19e-04$ ^* $ 1.15e-13 0.9798 -1.93 -0.26 -1.67
    $ \mbox{MC}_{8}(0) $ 4.44e-14 9.41e-14 0.2363 0.7553 -1.21 -0.15 -1.06
    $ \mbox{MC}_{8}(-0.079) $ 6.57e-14 1.42e-13 0.0138$ ^* $ 0.0215$ ^* $ 0 0.05 -0.05
    $ \mbox{MC}_{8}(-0.075) $ 4.09e-14 7.11e-14$ ^* $ 0.0021$ ^* $ 0.0567 -0.06 0.04 -0.10
    $ \mbox{AVF}_\text{EC} $; $ \mbox{EC}_{10}(0) $ 2.13e-14 0.0574 3.73e-14 0.0725 -0.1 0.02 -0.12
    $ \mbox{EC}_{10}(0.05) $ 2.66e-14 0.0438 4.09e-14$ ^* $ 0.0116$ ^* $ 0 0.03 -0.03
    $ \mbox{EC}_{10}(0.21) $ 2.13e-14$ ^* $ 6.74e-04$ ^* $ 4.97e-14 0.2571 0.35 0.08 0.27
    $ \mbox{AVF}_\text{MC} $; $ \mbox{MC}_{10}(0) $ 1.95e-14 4.80e-14 0.1461 0.2959 -0.40 -0.01 -0.39
    $ \mbox{MC}_{10}(0.19) $ 2.31e-14 2.49e-14 0.1477$ ^* $ 0.0205$ ^* $ 0 0.08 -0.08
    Narrow box 4.97e-14 0.0459 6.8421 0.3054 0.40 0.09 0.31
    Multisymplectic 2.66e-14 0.0228 6.4635 0.7278 -1.15 -0.17 -0.98
     | Show Table
    DownLoad: CSV

    Table 3.  Errors in conservation laws and solution for the breather problem, setting $ \Delta x = 0.02 $, $ \Delta t = 0.002 $. An asterisk denotes the error that is minimized

    Method $ \text{Err}_1 $ $ \text{Err}_2 $ $ \text{Err}_3 $ Solution error
    $ \mbox{EC}_{8}(0) $ 6.76e-13 0.1091 1.33e-10 0.9099
    $ \mbox{EC}_{8}(2.22) $ 2.97e-13 0.3979 1.55e-10$ ^* $ 0.0144$ ^* $
    $ \mbox{EC}_{8}(0.49) $ 9.59e-13$ ^* $ 0.0079$ ^* $ 1.05e-10 0.7442
    $ \mbox{MC}_{8}(0) $ 6.25e-13 5.31e-12 7.534 0.7666
    $ \mbox{MC}_{8}(-0.165) $ 3.03e-13 2.74e-12 2.3599$ ^* $ 0.0497$ ^* $
    $ \mbox{MC}_{8}(-0.128) $ 7.24e-13 7.60e-12$ ^* $ 0.1728$ ^* $ 0.1931
    $ \mbox{AVF}_\text{EC} $; $ \mbox{EC}_{10}(0) $ 3.28e-14 0.1765 1.24e-11 0.4042
    $ \mbox{EC}_{10}(0.92) $ 3.53e-14 0.0296 2.63e-11$ ^* $ 0.0295$ ^* $
    $ \mbox{EC}_{10}(0.78) $ 5.96e-14$ ^* $ 0.0095$ ^* $ 1.35e-11 0.0708
    $ \mbox{AVF}_\text{MC} $; $ \mbox{MC}_{10}(0) $ 1.14e-13 9.24e-13 4.3586 0.5040
    $ \mbox{MC}_{10}(1.15) $ 5.34e-14 7.03e-13 4.8298$ ^* $ 0.0219$ ^* $
    Narrow box 1.00e-12 0.0382 566.37 0.3477
    Multisymplectic 2.60e-13 0.0184 539.400.7994
     | Show Table
    DownLoad: CSV
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