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A comparison of boundary correction methods for Strang splitting

  • * Corresponding author: Alexander Ostermann

    * Corresponding author: Alexander Ostermann
Abstract / Introduction Full Text(HTML) Figure(2) / Table(10) Related Papers Cited by
  • In this paper we investigate splitting methods in the presence of non-homogeneous boundary conditions. In particular, we consider the corrections that have been described and analyzed in Einkemmer, Ostermann 2015 and Alonso-Mallo, Cano, Reguera 2016. The latter method is extended to the non-linear case, and a rigorous convergence analysis is provided. We perform numerical simulations for diffusion-reaction, advection-reaction, and dispersion-reaction equations in order to evaluate the relative performance of these two corrections. Furthermore, we introduce an extension of both methods to obtain order three locally and evaluate under what circumstances this is beneficial.

    Mathematics Subject Classification: Primary: 65M12, 65J08.

    Citation:

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  • Figure 1.  The global error in the infinity norm as a function of the time step size is shown. The error for TDBC2 and CEC2 is almost identical and therefore only the (erratic) error for TDBC2 is shown in the plot. In addition, the dashed lines are of slope $1$ and $2$, respectively. In all simulations equation (22) with $f(u) = {\rm{e}}^{u-1}$ is employed and the initial value $u(0, x) = 1+\sin \pi x + i \sin 2 \pi x$ is imposed. The space discretization is conducted by using the standard centered finite difference stencil with $200$ grid points and all simulations are conducted until $t = 0.19$

    Figure 2.  The local (full lines) and global errors (dashed lines) in the infinity norm are shown as a function of time for the second order CEC (top) and the third order CEC (bottom) corrections. The following step sizes are used (from top to bottom in this order in both cases): $1.5\cdot10^{-3}$ (yellow), $7.5\cdot10^{-4}$ (magenta), $3.75\cdot10^{-4}$ (cyan), $1.88\cdot10^{-4}$ (blue), $9.38\cdot10^{-5}$ (green), $4.69\cdot10^{-5}$ (red). In all simulations equation (22) with $f(u) = {\rm{e}}^{u-1}$ is employed and the initial value $u(0, x) = 1+\sin \pi x + i \sin 2 \pi x$ is imposed. The space discretization is conducted by using the standard centered finite difference stencil with $200$ grid points

    Table 1.  The local (at $t = 0$) and global errors using the unmodified Strang splitting applied to equation (19) are shown. The three different reaction terms indicated in the text are used. The space discretization is conducted by using the standard centered finite difference stencil with $200$ grid points. All problems are integrated until $t = 0.25$ and use the initial value $u(0, x) = 0$

    Local error
    $f(u, x)=u+1$ $f(u, x)=u+p(x)$ $f(u, x)=u+q(x)$
    step size $l^{\infty}$ error order $l^{\infty}$ error order $l^{\infty}$ error order
    6.40e-02 3.14e-02 - 4.08e-04 - 4.54e-04 -
    3.20e-02 1.54e-02 1.03 9.93e-05 2.04 6.13e-05 2.89
    1.60e-02 7.51e-03 1.03 2.48e-05 2.00 7.72e-06 2.99
    8.00e-03 3.64e-03 1.04 6.21e-06 2.00 9.69e-07 2.99
    4.00e-03 1.75e-03 1.06 1.55e-06 2.00 1.22e-07 2.99
    2.00e-03 8.24e-04 1.08 3.88e-07 2.00 1.54e-08 2.99
    Global error
    $f(u, x)=u+1$ $f(u, x)=u+p(x)$ $f(u, x)=u+q(x)$
    step size $l^{\infty}$ error order $l^{\infty}$ error order $l^{\infty}$ error order
    6.40e-02 3.15e-02 - 6.75e-04 - 9.66e-04 -
    3.20e-02 1.54e-02 1.03 1.71e-04 1.98 2.41e-04 2.00
    1.60e-02 7.52e-03 1.03 4.34e-05 1.98 6.01e-05 2.00
    8.00e-03 3.65e-03 1.04 1.09e-05 1.99 1.50e-05 2.00
    4.00e-03 1.75e-03 1.06 2.75e-06 1.99 3.76e-06 2.00
    2.00e-03 8.29e-04 1.08 6.91e-07 1.99 9.40e-07 2.00
     | Show Table
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    Table 2.  The local (at $t = 0$) and global errors for the unmodified Strang splitting as well as the second order TDBC and CEC corrected Strang splitting applied to equation (19) with $f(u) = {\rm{e}}^{u-1}$ are shown. The space discretization is conducted by using the standard centered finite difference stencil with $200$ grid points. All problems are integrated until $t = 0.25$ and use the initial value $u(0, x) = \sin \pi x$

    Local error
    unmodified TDBC CEC
    step size $l^{\infty}$ error order $l^{\infty}$ error order $l^{\infty}$ error order
    1.60e-02 7.49e-03 - 1.25e-04 - 1.06e-04 -
    8.00e-03 3.64e-03 1.04 3.25e-05 1.94 2.76e-05 1.94
    4.00e-03 1.75e-03 1.06 8.17e-06 1.99 6.91e-06 2.00
    2.00e-03 8.24e-04 1.08 2.04e-06 2.00 1.73e-06 2.00
    1.00e-03 3.79e-04 1.12 5.13e-07 2.00 4.31e-07 2.00
    5.00e-04 1.68e-04 1.18 1.27e-07 2.01 1.07e-07 2.00
    Global error
    unmodified TDBC CEC
    step size $l^{\infty}$ error order $l^{\infty}$ error order $l^{\infty}$ error order
    1.60e-02 7.52e-03 - 3.13e-05 - 4.15e-05 -
    8.00e-03 3.65e-03 1.04 7.72e-06 2.02 1.04e-05 2.00
    4.00e-03 1.75e-03 1.06 1.91e-06 2.02 2.60e-06 2.00
    2.00e-03 8.29e-04 1.08 4.69e-07 2.02 6.49e-07 2.00
    1.00e-03 3.82e-04 1.12 1.15e-07 2.03 1.62e-07 2.00
    5.00e-04 1.70e-04 1.17 2.81e-08 2.03 4.06e-08 2.00
     | Show Table
    DownLoad: CSV

    Table 3.  The local (at $t = 0$) and global errors for the unmodified Strang splitting as well as the third order TDBC and CEC corrected Strang splitting applied to equation (19) with $f(u) = {\rm{e}}^{u-1}$ are shown. The space discretization is conducted by using the standard centered finite difference stencil with $200$ grid points. All problems are integrated until $t = 0.25$ and use the initial value $u(0, x) = \sin \pi x$

    Local error
    unmodified TDBC CEC
    step size $l^{\infty}$ error order $l^{\infty}$ error order $l^{\infty}$ error order
    1.60e-02 7.49e-03 - 1.71e-04 - 8.81e-05 -
    8.00e-03 3.64e-03 1.04 1.86e-05 3.20 1.44e-05 2.61
    4.00e-03 1.75e-03 1.06 2.29e-06 3.02 2.11e-06 2.77
    2.00e-03 8.24e-04 1.08 3.11e-07 2.88 2.87e-07 2.88
    1.00e-03 3.79e-04 1.12 4.06e-08 2.94 3.75e-08 2.94
    5.00e-04 1.68e-04 1.18 5.18e-09 2.97 4.80e-09 2.97
    Global error
    unmodified TDBC CEC
    step size $l^{\infty}$ error order $l^{\infty}$ error order $l^{\infty}$ error order
    1.60e-02 7.52e-03 - 2.32e-04 - 6.85e-05 -
    8.00e-03 3.65e-03 1.04 3.30e-05 2.81 1.67e-05 2.04
    4.00e-03 1.75e-03 1.06 5.89e-06 2.49 4.11e-06 2.02
    2.00e-03 8.29e-04 1.08 1.22e-06 2.27 1.02e-06 2.01
    1.00e-03 3.82e-04 1.12 2.77e-07 2.14 2.54e-07 2.00
    5.00e-04 1.70e-04 1.17 6.59e-08 2.07 6.34e-08 2.00
     | Show Table
    DownLoad: CSV

    Table 4.  The local (at $t = 0$) and global errors using the unmodified Strang splitting applied to equation (20) are shown for the three different reaction terms indicated in the table. The space discretization is conducted by using a second order upwind finite difference stencil with $10^3$ grid points. All problems are integrated until $t = 1.9$ and use the initial value $u(0, x) = 0$

    Local error
    $f(u, x)=u+1$ $f(u, x)=u+x$ $f(u, x)=u+x^2$
    step size $l^{\infty}$ error order $l^{\infty}$ error order $l^{\infty}$ error order
    2.40e-01 1.26e-01 - 7.70e-03 - 2.41e-03 -
    1.20e-01 6.08e-02 1.05 1.84e-03 2.07 2.94e-04 3.04
    6.00e-02 2.94e-02 1.05 4.44e-04 2.05 3.63e-05 3.02
    3.00e-02 1.41e-02 1.06 1.06e-04 2.06 4.51e-06 3.01
    1.50e-02 6.53e-03 1.11 2.47e-05 2.10 5.62e-07 3.00
    7.50e-03 2.76e-03 1.24 5.45e-06 2.18 6.98e-08 3.01
    Global error
    $f(u, x)=u+1$ $f(u, x)=u+x$ $f(u, x)=u+x^2$
    step size $l^{\infty}$ error order $l^{\infty}$ error order $l^{\infty}$ error order
    2.40e-01 1.25e-01 - 1.11e-02 - 1.14e-02 -
    1.20e-01 5.98e-02 1.07 2.16e-03 2.36 2.92e-03 1.96
    6.00e-02 2.85e-02 1.07 4.44e-04 2.28 7.32e-04 1.99
    3.00e-02 1.31e-02 1.12 1.06e-04 2.06 1.83e-04 2.00
    1.50e-02 5.54e-03 1.24 2.55e-05 2.06 4.56e-05 2.00
    7.50e-03 1.94e-03 1.51 6.37e-06 2.00 1.14e-05 2.00
     | Show Table
    DownLoad: CSV

    Table 5.  The local (at $t = 0$) and global errors for the unmodified Strang splitting as well as the second order TDBC and CEC corrected Strang splitting applied to equation (21) with $f(u) = {\rm{e}}^{u-1}$ are shown. The space discretization is conducted by using a second order upwind finite difference stencil with $500$ grid points. All problems are integrated until $t = 1.9$ and use the initial value $u(0, x) = 1+x$

    Local error
    unmodified TDBC CEC
    step size $l^{\infty}$ error order $l^{\infty}$ error order $l^{\infty}$ error order
    2.40e-01 1.25e-01 - 1.51e-02 - 8.80e-03 -
    1.20e-01 5.98e-02 1.07 2.14e-03 2.82 1.93e-03 2.19
    6.00e-02 2.84e-02 1.07 4.73e-04 2.18 4.42e-04 2.13
    3.00e-02 1.31e-02 1.12 1.15e-04 2.04 1.01e-04 2.13
    1.50e-02 5.53e-03 1.24 2.84e-05 2.02 2.20e-05 2.19
    7.50e-03 1.89e-03 1.55 6.91e-06 2.04 4.68e-06 2.24
    Global error
    unmodified TDBC CEC
    step size $l^{\infty}$ error order $l^{\infty}$ error order $l^{\infty}$ error order
    2.40e-01 3.73e-01 - 1.09e-01 - 2.59e-02 -
    1.20e-01 9.07e-02 2.04 3.56e-02 1.62 4.72e-03 2.46
    6.00e-02 2.84e-02 1.67 1.02e-02 1.80 1.30e-03 1.86
    3.00e-02 1.31e-02 1.12 2.74e-03 1.90 4.15e-04 1.65
    1.50e-02 5.54e-03 1.24 7.07e-04 1.95 1.16e-04 1.83
    7.50e-03 1.94e-03 1.51 1.80e-04 1.98 3.08e-05 1.92
     | Show Table
    DownLoad: CSV

    Table 6.  The local (at $t = 0$) and global errors for the unmodified Strang splitting as well as the third order TDBC and CEC corrected Strang splitting applied to equation (21) with $f(u) = {\rm{e}}^{u-1}$ are shown. The space discretization is conducted by using a second order upwind finite difference stencil with $500$ grid points. All problems are integrated until $t = 1.9$ and use the initial value $u(0, x) = 1+x$

    Local error
    unmodified TDBC CEC
    step size $l^{\infty}$ error order $l^{\infty}$ error order $l^{\infty}$ error order
    2.40e-01 1.25e-01 - 1.51e-02 - 1.39e-02 -
    1.20e-01 5.98e-02 1.07 2.14e-03 2.82 1.61e-03 3.11
    6.00e-02 2.84e-02 1.07 2.87e-04 2.90 1.90e-04 3.09
    3.00e-02 1.31e-02 1.12 3.72e-05 2.95 2.28e-05 3.06
    1.50e-02 5.53e-03 1.24 4.73e-06 2.97 2.79e-06 3.03
    7.50e-03 1.89e-03 1.55 5.98e-07 2.99 3.44e-07 3.02
    Global error
    unmodified TDBC CEC
    step size $l^{\infty}$ error order $l^{\infty}$ error order $l^{\infty}$ error order
    2.40e-01 3.73e-01 - 5.46e-02 - 4.65e-02 -
    1.20e-01 9.07e-02 2.04 2.04e-02 1.42 1.08e-02 2.10
    6.00e-02 2.84e-02 1.67 6.25e-03 1.70 2.57e-03 2.08
    3.00e-02 1.31e-02 1.12 1.73e-03 1.85 6.24e-04 2.05
    1.50e-02 5.54e-03 1.24 4.55e-04 1.93 1.53e-04 2.03
    7.50e-03 1.94e-03 1.51 1.17e-04 1.96 3.79e-05 2.01
     | Show Table
    DownLoad: CSV

    Table 7.  The local (at $t = 0$) and global errors computed in $[\tfrac{1}{2}, 1]$ for the unmodified Strang splitting as well as the second order TDBC and CEC corrected Strang splitting applied to equation (21) with $f(u) = {\rm{e}}^{u-1}$ are shown. The space discretization is conducted by using a second order upwind finite difference stencil with $500$ grid points. All problems are integrated until $t = 1.9$ and use the initial value $u(0, x) = 1+x$

    Local error
    unmodified TDBC CEC
    step size $l^{\infty}$ error order $l^{\infty}$ error order $l^{\infty}$ error order
    2.40e-01 1.44e-02 - 1.44e-02 - 7.19e-03 -
    1.20e-01 2.05e-03 2.81 2.05e-03 2.81 9.33e-04 2.95
    6.00e-02 2.75e-04 2.90 2.75e-04 2.90 1.24e-04 2.91
    3.00e-02 3.58e-05 2.95 3.58e-05 2.95 1.62e-05 2.94
    1.50e-02 4.56e-06 2.97 4.56e-06 2.97 2.08e-06 2.96
    7.50e-03 5.76e-07 2.99 5.76e-07 2.99 2.63e-07 2.98
    Global error
    unmodified TDBC CEC
    step size $l^{\infty}$ error order $l^{\infty}$ error order $l^{\infty}$ error order
    2.40e-01 3.73e-01 - 1.02e-01 - 2.59e-02 -
    1.20e-01 9.07e-02 2.04 3.31e-02 1.63 4.72e-03 2.46
    6.00e-02 1.26e-02 2.85 9.46e-03 1.81 9.71e-04 2.28
    3.00e-02 1.87e-03 2.75 2.52e-03 1.91 3.16e-04 1.62
    1.50e-02 4.88e-04 1.94 6.51e-04 1.95 8.99e-05 1.81
    7.50e-03 1.25e-04 1.97 1.65e-04 1.98 2.39e-05 1.91
     | Show Table
    DownLoad: CSV

    Table 8.  The accuracy (at times $t = 0.5$ and $t = 2$) of the best TDBC approach (this can be the second or third order correction) divided by the accuracy of the best CEC approach is shown for five different reactions $f_1 = \sqrt{u+1}$, $f_2 = {\rm{e}}^{u/5}$, $f_3 = \log(2+u)$, $f_4 = 1/2+\text{arsinhpt}{u}$, $f_5 = \cos u$ and five different advection coefficients $a_1 = 1+\sin x$, $a_2 = \sin(\pi x/2)+2/5$, $a_3 = 3/2-x$, $a_4 = 1/5+{\rm{e}}^{-50 (x-1/2)^2}$, $a_5 = 1 + \sin(2\pi x)/5$. The number in parentheses shows the gain in accuracy achieved by going from CEC2 to CEC3 and from TDBC2 to TDBC3, respectively (values larger than one indicate a gain in accuracy, while values smaller than one indicate a loss in accuracy). The space discretization is conducted by using a second order upwind finite difference stencil with $500$ grid points

    t=0.5
    $a_1$ $a_2$ $a_3$ $a_4$ $a_5$
    $f_1$ 27.6(6.9, 0.8) 4.5(1.2, 0.9) 27.3(23, 1.5) 14.1(7.2, 2.2) 4.0(1.0, 0.7)
    $f_2$ 18.2(1.5, 0.9) 15.4(1.3, 1.0) 14.1(7.2, 2.2) 9.4(0.2, 1.0) 8.2(0.9, 0.7)
    $f_3$ 22.7(5.7, 0.8) 4.6(1.3, 0.9) 15.8(13.7, 1.5) 4.0(0.3, 1.0) 4.2(1.1, 0.7)
    $f_4$ 5.7(3.1, 0.6) 2.2(1.4, 0.7) 1.5(3.3, 0.6) 1.7(0.4, 1.0) 2.7(1.4, 0.6)
    $f_5$ 2.4(1.0, 0.7) 2.4(1.0, 0.9) 2.5(1.0, 1.7) 3.7(1.0, 1.0) 3.4(1.0, 0.7)
    $t=2$
    $a_1$ $a_2$ $a_3$ $a_4$ $a_5$
    $f_1$ 21.6(2.9, 0.6) 5.7(0.8, 0.4) 35.3(24.7, 1.3) 2.6(0.8, 0.5) 3.9(1.9, 0.4)
    $f_2$ 11.1(1.4, 0.5) 19.3(3.8, 0.3) 15.5(7.1, 2.0) 0.9(0.9, 0.2) 6.0(1.6, 0.5)
    $f_3$ 18.6(2.6, 0.6) 5.7(0.8, 0.4) 19.3(13.9, 1.3) 2.5(1.1, 0.4) 3.8(1.9, 0.4)
    $f_4$ 6.8(1.4, 0.6) 8.8(1.8, 0.5) 1.2(29, 0.4) 4.0(2.7, 0.2) 2.1(1.5, 0.4)
    $f_5$ 1.7(1.0, 0.4) 1.0(1.0, 0.2) 2.6(1.0, 1.6) 0.8(1.0, 0.5) 2.1(1.0, 0.5)
     | Show Table
    DownLoad: CSV

    Table 9.  The local (at $t = 0$) and global errors for the unmodified Strang splitting as well as the second order TDBC and CEC corrected Strang splitting applied to equation (22) with $f(u) = {\rm{e}}^{u-1}$ are shown. The space discretization is conducted by using the standard centered finite difference stencil with $200$ grid points. All problems are integrated until $t = 0.19$ and use the initial value $u(0, x) = 1+\sin \pi x + i \sin 2 \pi x$

    Local error
    unmodified TDBC CEC
    step size $l^{\infty}$ error order $l^{\infty}$ error order $l^{\infty}$ error order
    1.20e-02 5.84e-03 - 1.48e-03 - 1.50e-03 -
    6.00e-03 2.79e-03 1.07 2.72e-04 2.45 2.70e-04 2.47
    3.00e-03 1.23e-03 1.19 3.49e-05 2.96 3.47e-05 2.96
    1.50e-03 6.38e-04 0.94 8.77e-06 1.99 8.65e-06 2.00
    7.50e-04 2.95e-04 1.11 2.11e-06 2.05 2.08e-06 2.05
    3.75e-04 1.30e-04 1.18 5.15e-07 2.04 5.07e-07 2.04
    Global error
    unmodified TDBC CEC
    step size $l^{\infty}$ error order $l^{\infty}$ error order $l^{\infty}$ error order
    1.20e-02 2.02e-02 - 2.35e-03 1.89 2.33e-03 -
    6.00e-03 1.18e-02 0.78 5.32e-04 2.14 5.25e-04 2.15
    3.00e-03 4.77e-03 1.30 1.09e-04 2.28 1.08e-04 2.29
    1.50e-03 1.04e-03 2.20 4.85e-05 1.17 4.82e-05 1.16
    7.50e-04 6.09e-04 0.77 1.82e-05 1.41 1.82e-05 1.41
    3.75e-04 2.20e-04 1.47 1.44e-05 0.34 1.44e-05 0.34
    1.88e-04 9.37e-05 1.23 4.70e-07 4.94 4.66e-07 4.95
     | Show Table
    DownLoad: CSV

    Table 10.  The local (at $t = 0$) and global errors for the unmodified Strang splitting as well as the third order TDBC and CEC corrected Strang splitting applied to equation (22) with $f(u) = {\rm{e}}^{u-1}$ are shown. The space discretization is conducted by using the standard centered finite difference stencil with $200$ grid points. All problems are integrated until $t = 0.19$ and use the initial value $u(0, x) = 1+\sin \pi x + i \sin 2 \pi x$

    Local error
    unmodified TDBC CEC
    step size $l^{\infty}$ error order $l^{\infty}$ error order $l^{\infty}$ error order
    1.20e-02 5.84e-03 - 1.51e-03 - 1.53e-03 -
    6.00e-03 2.79e-03 1.07 2.48e-04 2.61 2.38e-04 2.69
    3.00e-03 1.23e-03 1.19 2.92e-05 3.09 2.81e-05 3.08
    1.50e-03 6.38e-04 0.94 3.21e-06 3.18 3.14e-06 3.16
    7.50e-04 2.95e-04 1.11 3.82e-07 3.07 3.72e-07 3.08
    3.75e-04 1.30e-04 1.18 4.65e-08 3.04 4.64e-08 3.00
    Global error
    unmodified TDBC CEC
    step size $l^{\infty}$ error order $l^{\infty}$ error order $l^{\infty}$ error order
    1.20e-02 2.02e-02 - 9.02e-03 2.09 7.91e-03 -
    6.00e-03 1.18e-02 0.78 1.84e-03 2.29 1.73e-03 2.19
    3.00e-03 4.77e-03 1.30 4.16e-04 2.15 4.12e-04 2.07
    1.50e-03 1.04e-03 2.20 9.85e-05 2.08 9.97e-05 2.05
    7.50e-04 6.09e-04 0.77 2.42e-05 2.03 2.44e-05 2.03
    3.75e-04 2.20e-04 1.47 6.01e-06 2.01 6.01e-06 2.02
    1.88e-04 9.37e-05 1.23 1.45e-06 2.06 1.50e-06 2.00
     | Show Table
    DownLoad: CSV
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