# American Institute of Mathematical Sciences

September  2018, 23(7): 2641-2660. doi: 10.3934/dcdsb.2018081

## A comparison of boundary correction methods for Strang splitting

 Department of Mathematics, University of Innsbruck, A-6020 Innsbruck, Austria

* Corresponding author: Alexander Ostermann

Received  September 2016 Revised  August 2017 Published  March 2018

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.

Citation: Lukas Einkemmer, Alexander Ostermann. A comparison of boundary correction methods for Strang splitting. Discrete & Continuous Dynamical Systems - B, 2018, 23 (7) : 2641-2660. doi: 10.3934/dcdsb.2018081
##### References:

show all references

##### References:
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 $t = 0.19$
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
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
 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
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
 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
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
 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
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
 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
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
 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
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
 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
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
 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
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)
 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)
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
 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
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
 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
 [1] Joseph A. Connolly, Neville J. Ford. Comparison of numerical methods for fractional differential equations. Communications on Pure & Applied Analysis, 2006, 5 (2) : 289-307. doi: 10.3934/cpaa.2006.5.289 [2] Mohamed Badreddine, Thomas K. DeLillo, Saman Sahraei. A Comparison of some numerical conformal mapping methods for simply and multiply connected domains. Discrete & Continuous Dynamical Systems - B, 2019, 24 (1) : 55-82. doi: 10.3934/dcdsb.2018100 [3] R.G. Duran, J.I. Etcheverry, J.D. Rossi. Numerical approximation of a parabolic problem with a nonlinear boundary condition. Discrete & Continuous Dynamical Systems - A, 1998, 4 (3) : 497-506. doi: 10.3934/dcds.1998.4.497 [4] Jean-François Coulombel, Frédéric Lagoutière. The Neumann numerical boundary condition for transport equations. Kinetic & Related Models, 2020, 13 (1) : 1-32. doi: 10.3934/krm.2020001 [5] Dorina Mitrea, Marius Mitrea, Sylvie Monniaux. The Poisson problem for the exterior derivative operator with Dirichlet boundary condition in nonsmooth domains. Communications on Pure & Applied Analysis, 2008, 7 (6) : 1295-1333. doi: 10.3934/cpaa.2008.7.1295 [6] Yueding Yuan, Zhiming Guo, Moxun Tang. A nonlocal diffusion population model with age structure and Dirichlet boundary condition. Communications on Pure & Applied Analysis, 2015, 14 (5) : 2095-2115. doi: 10.3934/cpaa.2015.14.2095 [7] Khadijah Sharaf. A perturbation result for a critical elliptic equation with zero Dirichlet boundary condition. Discrete & Continuous Dynamical Systems - A, 2017, 37 (3) : 1691-1706. doi: 10.3934/dcds.2017070 [8] Qiang Long, Xue Wu, Changzhi Wu. Non-dominated sorting methods for multi-objective optimization: Review and numerical comparison. Journal of Industrial & Management Optimization, 2020  doi: 10.3934/jimo.2020009 [9] Pablo Amster, Colin Rogers. On a Ermakov-Painlevé II reduction in three-ion electrodiffusion. A Dirichlet boundary value problem. Discrete & Continuous Dynamical Systems - A, 2015, 35 (8) : 3277-3292. doi: 10.3934/dcds.2015.35.3277 [10] G. Acosta, Julián Fernández Bonder, P. Groisman, J.D. Rossi. Numerical approximation of a parabolic problem with a nonlinear boundary condition in several space dimensions. Discrete & Continuous Dynamical Systems - B, 2002, 2 (2) : 279-294. doi: 10.3934/dcdsb.2002.2.279 [11] Larissa V. Fardigola. Transformation operators in controllability problems for the wave equations with variable coefficients on a half-axis controlled by the Dirichlet boundary condition. Mathematical Control & Related Fields, 2015, 5 (1) : 31-53. doi: 10.3934/mcrf.2015.5.31 [12] Lan Zeng, Guoxi Ni, Yingying Li. Low Mach number limit of strong solutions for 3-D full compressible MHD equations with Dirichlet boundary condition. Discrete & Continuous Dynamical Systems - B, 2019, 24 (10) : 5503-5522. doi: 10.3934/dcdsb.2019068 [13] Jian Zhai, Zhihui Cai. $\Gamma$-convergence with Dirichlet boundary condition and Landau-Lifshitz functional for thin film. Discrete & Continuous Dynamical Systems - B, 2009, 11 (4) : 1071-1085. doi: 10.3934/dcdsb.2009.11.1071 [14] Zhiming Guo, Zhi-Chun Yang, Xingfu Zou. Existence and uniqueness of positive solution to a non-local differential equation with homogeneous Dirichlet boundary condition---A non-monotone case. Communications on Pure & Applied Analysis, 2012, 11 (5) : 1825-1838. doi: 10.3934/cpaa.2012.11.1825 [15] Tomáš Bodnár, Philippe Fraunié, Petr Knobloch, Hynek Řezníček. Numerical evaluation of artificial boundary condition for wall-bounded stably stratified flows. Discrete & Continuous Dynamical Systems - S, 2020  doi: 10.3934/dcdss.2020333 [16] Md. Masum Murshed, Kouta Futai, Masato Kimura, Hirofumi Notsu. Theoretical and numerical studies for energy estimates of the shallow water equations with a transmission boundary condition. Discrete & Continuous Dynamical Systems - S, 2019  doi: 10.3934/dcdss.2020230 [17] Inácio Andruski-Guimarães, Anselmo Chaves-Neto. Estimation in polytomous logistic model: Comparison of methods. Journal of Industrial & Management Optimization, 2009, 5 (2) : 239-252. doi: 10.3934/jimo.2009.5.239 [18] Emmanuel Frénod. Homogenization-based numerical methods. Discrete & Continuous Dynamical Systems - S, 2016, 9 (5) : i-ix. doi: 10.3934/dcdss.201605i [19] Sunghan Kim, Ki-Ahm Lee, Henrik Shahgholian. Homogenization of the boundary value for the Dirichlet problem. Discrete & Continuous Dynamical Systems - A, 2019, 39 (12) : 6843-6864. doi: 10.3934/dcds.2019234 [20] Xiaowei Tang, Xilin Fu. New comparison principle with Razumikhin condition for impulsive infinite delay differential systems. Conference Publications, 2009, 2009 (Special) : 739-743. doi: 10.3934/proc.2009.2009.739

2019 Impact Factor: 1.27