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Non-standard boundary conditions for the linearized Korteweg-de Vries equation

  • * Corresponding author: sabah.kaouri@gmail.com

    * Corresponding author: sabah.kaouri@gmail.com
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  • This paper aims to solve numerically the linearized Korteweg-de Vries equation. We begin by deriving suitable boundary conditions then approximate them using finite difference method. The methodology of derivation, used in this paper, yields to Non-Standard Boundary Conditions (NSBC) that perfectly absorb wave reflections at the boundary. In addition, these NSBC are exact and local in time and space for non necessarily supported initial data and source terms. We finish with numerical examples that show the absorbing quality of these boundary conditions. Further comparisons are made using standard boundary conditions like, Dirichlet, Neumann and a variant of absorbing boundary conditions called discrete artificial ones.

    Mathematics Subject Classification: 93B30, 45Q05.

    Citation:

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  • Figure 1.  Instantaneous of $ u_{ref} $(red) and $ u_{nsbc} $(blue) for the first example with $ \tau = \frac{6}{64} $ and $ h = 10^{-1} $

    Figure 2.  Geophones of $ u_{ref} $(red) and $ u_{nsbc} $(blue) for the first example with $ \tau = \frac{6}{64} $ and $ h = 10^{-1} $ at different positions

    Figure 3.  Evolution of logarithm of error with respect to time and space for the first example with $ \tau = \frac{6}{64} $ and $ h = 10^{-1} $ for the first example

    Figure 4.  Energies $ E(t) $ and $ M(t) $ evolution in time of $ u_{nsbc} $ (left) and $ u_{ref} $ (right) for the first example with $ \tau = \frac{6}{64} $ and $ h = 10^{-1} $ for the first example

    Figure 5.  Instantaneous of $ u_{ref} $(red) and $ u_{nsbc} $(blue) for the second example with $ \tau = 3.2\times 10^{-6} $ and $ h = 2\times10^{-3} $

    Figure 6.  Geophones of $ u_{ref} $(red) and $ u_{nsbc} $(blue) for the second example with $ \tau = 3.2\times 10^{-6} $ and $ h = 2\times10^{-3} $ at different positions

    Figure 7.  Evolution of logarithm of error with respect to time and space for the second example with $ \tau = 3.2\times 10^{-6} $ and $ h = 2\times10^{-3} $

    Figure 8.  Energies $ E(t) $ and $ M(t) $ evolution in time of $ u_{nsbc} $ (left) and $ u_{ref} $ (right) for the second example with $ \tau = 3.2\times 10^{-6} $ and $ h = 2\times10^{-3} $

    Figure 9.  Instantaneous of $ u_{ref} $(red) and $ u_{nsbc} $(blue) for the third example using (43) for $ c = 1 $ with $ \tau = 1.56\times 10^{-2} $ and $ h = 10^{-1} $

    Figure 10.  Geophones of $ u_{ref} $(red) and $ u_{nsbc} $(blue) for the third example using (43) for $ c = 1 $ with $ \tau = 1.56\times 10^{-2} $ and $ h = 10^{-1} $ at different positions

    Figure 11.  Evolution of logarithm of error with respect to time and space for the third example using (43) for $ c = 1 $ with $ \tau = 1.56\times 10^{-2} $ and $ h = 10^{-1} $

    Figure 12.  Energies $ E(t) $ and $ M(t) $ evolution in time of $ u_{nsbc} $ (left) and $ u_{ref} $ (right) third example using (43) for $ c = 1 $ with $ \tau = 1.56\times 10^{-2} $ and $ h = 10^{-1} $

    Figure 13.  Geophones of $ u_{ref} $(red) and $ u_{nsbc} $(blue) for the third example using (44) for $ \omega = 30 $ with $ \tau = 1.56\times 10^{-2} $ and $ h = 10^{-1} $ at different positions

    Figure 14.  Instantaneous of $ u_{ref} $(red) and $ u_{nsbc} $(blue) for the fifth example using (43) for $ c = 2 $ and (44) for $ \omega = 40 $ with $ \tau = 3.2\times 10^{-6} $ and $ h = 4\times10^{-4} $

    Figure 15.  Geophones of $ u_{ref} $(red) and $ u_{nsbc} $(blue) for the fifth example using (43) for $ c = 2 $ and (44) for $ \omega = 40 $ with $ \tau = 1.56\times 10^{-2} $ and $ h = 10^{-1} $ at different positions

    Figure 16.  Evolution of logarithm of error with respect to time and space for the fifth example using using (43) for $ c = 2 $ and (44) for $ \omega = 40 $ with $ \tau = 1.56\times 10^{-2} $ and $ h = 10^{-1} $

    Figure 17.  Energies $ E(t) $ and $ M(t) $ evolution in time of $ u_{nsbc} $ (left) and $ u_{ref} $ (right) fifth example using using (43) for $ c = 2 $ and (44) for $ \omega = 40 $ with $ \tau = 1.56\times 10^{-2} $ and $ h = 10^{-1} $

    Figure 18.  Domain of existence (white)

    Table 1.  Comparison of Errors for the first example using various BC, $ \tau $ and $ h $

    u $ \tau $ $ h $ $ err(u) $ $ e_2(u) $ $ e_\infty(u) $
    $ u_{nsbc} $ $ 7.8\times 10^{-3} $ $ 10^{-1} $ $ 1.54\times 10^{-2} $ $ 1.62\times 10^{-2} $ $ 1.7\times 10^{-2} $
    $ u_{dabc}(rcn) $ $ 7.8\times 10^{-3} $ $ 10^{-1} $ $ - $ $ \approx 5\times10^{-2} $ $ \approx 10^{-1} $
    $ u_{dabc}(ccn) $ $ 7.8\times 10^{-3} $ $ 10^{-1} $ $ - $ $ \approx 10^{-1} $ $ \approx 10^{-1} $
    $ u_{dbc} $ $ 7.8\times 10^{-3} $ $ 10^{-1} $ $ 5.52\times 10^{-1} $ $ 5.17\times 10^{-1} $ $ 4.05\times 10^{-1} $
    $ u_{nbc} $ $ 7.8\times 10^{-3} $ $ 10^{-1} $ $ NaN $ $ NaN $ $ NaN $
    $ u_{nsbc} $ $ 3.9\times 10^{-3} $ $ 10^{-1} $ $ 2.44\times 10^{-2} $ $ 1.74\times 10^{-2} $ $ 2.59\times 10^{-2} $
    $ u_{dabc}(rcn) $ $ 3.9\times 10^{-3} $ $ 10^{-1} $ $ - $ $ \approx 5\times10^{-2} $ $ \approx 10^{-1} $
    $ u_{dabc}(ccn) $ $ 3.9\times 10^{-3} $ $ 10^{-1} $ $ - $ $ \approx 10^{-2} $ $ \approx 5\times10^{-2} $
    $ u_{nsbc} $ $ 3.9\times 10^{-3} $ $ 10^{-2} $ $ 6.5\times 10^{-3} $ $ 9.4\times 10^{-3} $ $ 9.7\times 10^{-3} $
    $ u_{dabc}(rcn) $ $ 3.9\times 10^{-3} $ $ 10^{-2} $ $ - $ $ \approx 10^{-4} $ $ \approx 10^{-3} $
    $ u_{dabc}(ccn) $ $ 3.9\times 10^{-3} $ $ 10^{-2} $ $ - $ $ \approx 10^{-3} $ $ \approx 5\times10^{-3} $
    $ u_{nsbc} $ $ 4.88\times 10^{-4} $ $ 10^{-1} $ $ 1.32\times 10^{-2} $ $ 1.52\times 10^{-2} $ $ 1.63\times 10^{-2} $
    $ u_{dabc}(rcn) $ $ 4.88\times 10^{-4} $ $ 10^{-1} $ $ - $ $ \approx 10^{-2} $ $ \approx 10^{-1} $
    $ u_{dabc}(ccn) $ $ 4.88\times 10^{-4} $ $ 10^{-1} $ $ - $ $ \approx 10^{-3} $ $ \approx 10^{-2} $
    $ u_{nsbc} $ $ 2.44\times 10^{-4} $ $ 10^{-1} $ $ 1.32\times 10^{-2} $ $ 1.52\times 10^{-2} $ $ 1.63\times 10^{-2} $
    $ u_{dabc}(rcn) $ $ 2.44\times 10^{-4} $ $ 10^{-1} $ $ - $ $ \approx 10^{-2} $ $ \approx 10^{-1} $
    $ u_{dabc}(ccn) $ $ 2.44\times 10^{-4} $ $ 10^{-1} $ $ - $ $ \approx 10^{-3} $ $ \approx 10^{-2} $
     | Show Table
    DownLoad: CSV

    Table 2.  Errors of the first example for homogeneous LKdV equation with NSBC for long time simulation and various values of $ \alpha $ and $ \beta $

    $ T $ $ \alpha $ $ \beta $ $ \tau $ $ h $ $ err $ $ e_2 $ $ e_\infty $
    $ 1 $ $ 1 $ $ 0 $ $ \frac{1}{32} $ $ 10^{-1} $ $ 6.8\times 10^{-3} $ $ 3.3\times 10^{-3} $ $ 4.6\times 10^{-3} $
    $ 1 $ $ 1 $ $ 0 $ $ \frac{1}{64} $ $ 10^{-1} $ $ 6.5\times 10^{-3} $ $ 3.2\times 10^{-3} $ $ 4.3\times 10^{-3} $
    $ 1 $ $ 1 $ $ 0 $ $ \frac{1}{128} $ $ 10^{-1} $ $ 6.4\times 10^{-3} $ $ 3\times 10^{-3} $ $ 4.2\times 10^{-3} $
    $ 1 $ $ 1 $ $ 0 $ $ \frac{1}{128} $ $ 5\times 10^{-2} $ $ 7.4\times 10^{-3} $ $ 3.6\times 10^{-3} $ $ 5.1\times 10^{-3} $
    $ 2 $ $ 1 $ $ 0 $ $ \frac{2}{32} $ $ 10^{-1} $ $ 8.1\times 10^{-3} $ $ 6.2\times 10^{-3} $ $ 6.3\times 10^{-3} $
    $ 2 $ $ 1 $ $ 0 $ $ \frac{2}{64} $ $ 10^{-1} $ $ 6.9\times 10^{-3} $ $ 5.8\times 10^{-3} $ $ 5.6\times 10^{-3} $
    $ 2 $ $ 1 $ $ 0 $ $ \frac{2}{64} $ $ 5\times10^{-2} $ $ 6.4\times 10^{-3} $ $ 3\times 10^{-3} $ $ 4.2\times 10^{-3} $
    $ 3 $ $ 1 $ $ 0 $ $ \frac{3}{32} $ $ 10^{-1} $ $ 8.2\times 10^{-3} $ $ 1.02\times 10^{-2} $ $ 10^{-2} $
    $ 3 $ $ 1 $ $ 0 $ $ \frac{3}{64} $ $ 5\times10^{-2} $ $ -6.4\times 10^{-3} $ $ -3\times 10^{-3} $ $ -4.2\times 10^{-3} $
    $ 4 $ $ 1 $ $ 0 $ $ \frac{4}{32} $ $ 10^{-1} $ $ 1.55\times 10^{-2} $ $ 1.82\times 10^{-2} $ $ 1.92\times 10^{-2} $
    $ 4 $ $ 1 $ $ 0 $ $ \frac{4}{64} $ $ 10^{-1} $ $ 1.33\times 10^{-2} $ $ 1.43\times 10^{-2} $ $ 1.72\times 10^{-2} $
    $ 5 $ $ 1 $ $ 0 $ $ \frac{5}{32} $ $ 10^{-1} $ $ 1.82\times 10^{-2} $ $ 3.04\times 10^{-2} $ $ 2.5\times 10^{-2} $
    $ 5 $ $ 1 $ $ 0 $ $ \frac{5}{64} $ $ 10^{-1} $ $ 1.75\times 10^{-2} $ $ 2.94\times 10^{-2} $ $ 2.4\times 10^{-2} $
    $ 6 $ $ 1 $ $ 0 $ $ \frac{6}{32} $ $ 10^{-1} $ $ 1.84\times 10^{-2} $ $ 3.76\times 10^{-2} $ $ 2.61\times 10^{-2} $
    $ 6 $ $ 1 $ $ 0 $ $ \frac{6}{64} $ $ 10^{-1} $ $ 1.77\times 10^{-2} $ $ 3.66\times 10^{-2} $ $ 2.59\times 10^{-2} $
    $ 4 $ $ 2 $ $ 2 $ $ \frac{4}{32} $ $ 10^{-1} $ $ 1.52\times 10^{-2} $ $ 7.6\times 10^{-3} $ $ 1.67\times 10^{-2} $
    $ 4 $ $ 2 $ $ -2 $ $ \frac{4}{32} $ $ 10^{-1} $ $ 8.8\times 10^{-3} $ $ 4.5\times 10^{-3} $ $ 5.9\times 10^{-3} $
    $ 4 $ $ 0.5 $ $ -0.5 $ $ \frac{4}{32} $ $ 10^{-1} $ $ 8.04\times 10^{-4} $ $ 7.56\times 10^{-4} $ $ 7.89\times 10^{-4} $
    $ 4 $ $ 1 $ $ 4 $ $ \frac{4}{32} $ $ 10^{-1} $ $ 1.59\times 10^{-2} $ $ 1.05\times 10^{-2} $ $ 1.78\times 10^{-2} $
    $ 4 $ $ 1 $ $ -4 $ $ \frac{4}{32} $ $ 10^{-1} $ $ 9\times 10^{-3} $ $ 3.8\times 10^{-3} $ $ 5.9\times 10^{-3} $
    $ 4 $ $ 8 $ $ 2 $ $ \frac{4}{32} $ $ 10^{-1} $ $ 5.25\times 10^{-2} $ $ 4.18\times 10^{-2} $ $ 7.14\times 10^{-2} $
    $ 4 $ $ 8 $ $ -2 $ $ \frac{4}{32} $ $ 10^{-1} $ $ 2.31\times 10^{-2} $ $ 1.82\times 10^{-2} $ $ 3.29\times 10^{-2} $
    $ 4 $ $ 0 $ $ 1 $ $ \frac{4}{32} $ $ 10^{-1} $ $ 4.82\times 10^{-4} $ $ 6.86\times 10^{-5} $ $ 1.76\times 10^{-4} $
    $ 4 $ $ 0 $ $ 1 $ $ \frac{4}{32} $ $ 5\times10^{-2} $ $ 1.38\times 10^{-4} $ $ 1.87\times 10^{-5} $ $ 4.9\times 10^{-5} $
     | Show Table
    DownLoad: CSV

    Table 3.  Comparison of Errors for the second example using various BC, $ \tau $ and $ h $

    u $ \tau $ $ h $ $ err(u) $ $ e_2(u) $ $ e_\infty(u) $
    $ u_{nsbc} $ $ 3.2\times 10^{-6} $ $ 2\times10^{-2} $ $ 5.17\times 10^{-2} $ $ 1.22\times 10^{-3} $ $ 7.08\times 10^{-2} $
    $ u_{dabc}(ccn) $ $ 3.2\times 10^{-6} $ $ 2\times10^{-2} $ $ - $ $ - $ $ \approx 5\times10^{-2} $
    $ u_{dbc} $ $ 3.2\times 10^{-6} $ $ 2\times10^{-2} $ $ 7.6\times 10^{-2} $ $ 5.4\times 10^{-3} $ $ 4.65\times 10^{-1} $
    $ u_{nbc} $ $ 3.2\times 10^{-6} $ $ 2\times10^{-2} $ $ 9.4\times10^{-2} $ $ 5.7\times10^{-3} $ $ 5.72\times10^{-1} $
    $ u_{nsbc} $ $ 3.2\times 10^{-6} $ $ 2\times10^{-3} $ $ 6.3\times 10^{-3} $ $ 2.18\times 10^{-4} $ $ 2.09\times 10^{-2} $
    $ u_{dabc}(ccn) $ $ 3.2\times 10^{-6} $ $ 2\times10^{-3} $ $ - $ $ - $ $ \approx 5\times10^{-3} $
    $ u_{nsbc} $ $ 3.2\times 10^{-6} $ $ 4\times10^{-4} $ $ 6.49\times 10^{-4} $ $ 5.97\times 10^{-6} $ $ 3.8\times 10^{-4} $
    $ u_{dabc}(ccn) $ $ 3.2\times 10^{-6} $ $ 4\times10^{-4} $ $ - $ $ - $ $ \approx 5\times10^{-3} $
     | Show Table
    DownLoad: CSV

    Table 4.  Errors of the third example using (43) for various values of $ c $ in (43)

    $ c $ $ \tau $ $ err $ $ e_2 $ $ e_\infty $
    $ 0.5 $ $ \frac{1}{32} $ $ 8.5\times 10^{-3} $ $ 1.77\times 10^{-3} $ $ 2.33\times 10^{-3} $
    $ 0.5 $ $ \frac{1}{64} $ $ 8.3\times 10^{-3} $ $ 1.7\times 10^{-3} $ $ 2.3\times 10^{-3} $
    $ 1 $ $ \frac{1}{64} $ $ 8.5\times 10^{-3} $ $ 10^{-3} $ $ 1.4\times 10^{-3} $
    $ 2 $ $ \frac{1}{64} $ $ 8.5\times 10^{-3} $ $ 5.56\times 10^{-4} $ $ 7.6\times 10^{-4} $
     | Show Table
    DownLoad: CSV

    Table 5.  Errors of the third example using (44) for various values of $ \omega $

    $ \omega $ $ \tau $ $ err $ $ e_2 $ $ e_\infty $
    $ 5 $ $ \frac{1}{32} $ $ 6.22\times 10^{-2} $ $ 6.02\times 10^{-2} $ $ 8.48\times 10^{-2} $
    $ 10 $ $ \frac{1}{32} $ $ 3\times 10^{-2} $ $ 3.89\times 10^{-2} $ $ 5.63\times 10^{-2} $
    $ 20 $ $ \frac{1}{32} $ $ 2.23\times 10^{-2} $ $ 1.67\times 10^{-2} $ $ 2.63\times 10^{-2} $
    $ 30 $ $ \frac{1}{32} $ $ 5.2\times 10^{-3} $ $ 1.02\times 10^{-2} $ $ 1.36\times 10^{-2} $
    $ 40 $ $ \frac{1}{32} $ $ 9.6\times 10^{-3} $ $ 5.7\times 10^{-3} $ $ 8.7\times 10^{-3} $
     | Show Table
    DownLoad: CSV

    Table 6.  Errors of the third example using (44) and (43)

    $ \omega $ $ c $ $ err $ $ e_2 $ $ e_\infty $
    $ 40 $ $ 2 $ $ 8.5\times 10^{-3} $ $ 8.89\times 10^{-4} $ $ 1.1\times 10^{-3} $
     | Show Table
    DownLoad: CSV
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