# American Institute of Mathematical Sciences

August  2021, 14(8): 2625-2654. doi: 10.3934/dcdss.2021066

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

 Cadi Ayyad University, Faculty of Sciences et Techniques, Departement of Mathematics, Team: Optimisation, Systèmes d'Évolution et Réseaux (OSER), 549, Av.Abdelkarim Elkhattabi, Marrakesh-Morocco

* Corresponding author: sabah.kaouri@gmail.com

Received  November 2019 Revised  May 2021 Published  August 2021 Early access  June 2021

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.

Citation: Mostafa Abounouh, Hassan Al-Moatassime, Sabah Kaouri. Non-standard boundary conditions for the linearized Korteweg-de Vries equation. Discrete & Continuous Dynamical Systems - S, 2021, 14 (8) : 2625-2654. doi: 10.3934/dcdss.2021066
##### References:

show all references

##### References:
Instantaneous of $u_{ref}$(red) and $u_{nsbc}$(blue) for the first example with $\tau = \frac{6}{64}$ and $h = 10^{-1}$
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
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
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
Instantaneous of $u_{ref}$(red) and $u_{nsbc}$(blue) for the second example with $\tau = 3.2\times 10^{-6}$ and $h = 2\times10^{-3}$
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
Evolution of logarithm of error with respect to time and space for the second example with $\tau = 3.2\times 10^{-6}$ and $h = 2\times10^{-3}$
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 $h = 2\times10^{-3}$
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 $h = 10^{-1}$
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
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 $h = 10^{-1}$
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 $h = 10^{-1}$
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
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 $h = 4\times10^{-4}$
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
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 $h = 10^{-1}$
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 $h = 10^{-1}$
Domain of existence (white)
Comparison of Errors for the first example using various BC, $\tau$ and $h$
 u $\tau$ $h$ $err(u)$ $e_2(u)$ $e_\infty(u)$ $u_{nsbc}$ $7.8\times 10^{-3}$ $10^{-1}$ $1.54\times 10^{-2}$ $1.62\times 10^{-2}$ $1.7\times 10^{-2}$ $u_{dabc}(rcn)$ $7.8\times 10^{-3}$ $10^{-1}$ $-$ $\approx 5\times10^{-2}$ $\approx 10^{-1}$ $u_{dabc}(ccn)$ $7.8\times 10^{-3}$ $10^{-1}$ $-$ $\approx 10^{-1}$ $\approx 10^{-1}$ $u_{dbc}$ $7.8\times 10^{-3}$ $10^{-1}$ $5.52\times 10^{-1}$ $5.17\times 10^{-1}$ $4.05\times 10^{-1}$ $u_{nbc}$ $7.8\times 10^{-3}$ $10^{-1}$ $NaN$ $NaN$ $NaN$ $u_{nsbc}$ $3.9\times 10^{-3}$ $10^{-1}$ $2.44\times 10^{-2}$ $1.74\times 10^{-2}$ $2.59\times 10^{-2}$ $u_{dabc}(rcn)$ $3.9\times 10^{-3}$ $10^{-1}$ $-$ $\approx 5\times10^{-2}$ $\approx 10^{-1}$ $u_{dabc}(ccn)$ $3.9\times 10^{-3}$ $10^{-1}$ $-$ $\approx 10^{-2}$ $\approx 5\times10^{-2}$ $u_{nsbc}$ $3.9\times 10^{-3}$ $10^{-2}$ $6.5\times 10^{-3}$ $9.4\times 10^{-3}$ $9.7\times 10^{-3}$ $u_{dabc}(rcn)$ $3.9\times 10^{-3}$ $10^{-2}$ $-$ $\approx 10^{-4}$ $\approx 10^{-3}$ $u_{dabc}(ccn)$ $3.9\times 10^{-3}$ $10^{-2}$ $-$ $\approx 10^{-3}$ $\approx 5\times10^{-3}$ $u_{nsbc}$ $4.88\times 10^{-4}$ $10^{-1}$ $1.32\times 10^{-2}$ $1.52\times 10^{-2}$ $1.63\times 10^{-2}$ $u_{dabc}(rcn)$ $4.88\times 10^{-4}$ $10^{-1}$ $-$ $\approx 10^{-2}$ $\approx 10^{-1}$ $u_{dabc}(ccn)$ $4.88\times 10^{-4}$ $10^{-1}$ $-$ $\approx 10^{-3}$ $\approx 10^{-2}$ $u_{nsbc}$ $2.44\times 10^{-4}$ $10^{-1}$ $1.32\times 10^{-2}$ $1.52\times 10^{-2}$ $1.63\times 10^{-2}$ $u_{dabc}(rcn)$ $2.44\times 10^{-4}$ $10^{-1}$ $-$ $\approx 10^{-2}$ $\approx 10^{-1}$ $u_{dabc}(ccn)$ $2.44\times 10^{-4}$ $10^{-1}$ $-$ $\approx 10^{-3}$ $\approx 10^{-2}$
 u $\tau$ $h$ $err(u)$ $e_2(u)$ $e_\infty(u)$ $u_{nsbc}$ $7.8\times 10^{-3}$ $10^{-1}$ $1.54\times 10^{-2}$ $1.62\times 10^{-2}$ $1.7\times 10^{-2}$ $u_{dabc}(rcn)$ $7.8\times 10^{-3}$ $10^{-1}$ $-$ $\approx 5\times10^{-2}$ $\approx 10^{-1}$ $u_{dabc}(ccn)$ $7.8\times 10^{-3}$ $10^{-1}$ $-$ $\approx 10^{-1}$ $\approx 10^{-1}$ $u_{dbc}$ $7.8\times 10^{-3}$ $10^{-1}$ $5.52\times 10^{-1}$ $5.17\times 10^{-1}$ $4.05\times 10^{-1}$ $u_{nbc}$ $7.8\times 10^{-3}$ $10^{-1}$ $NaN$ $NaN$ $NaN$ $u_{nsbc}$ $3.9\times 10^{-3}$ $10^{-1}$ $2.44\times 10^{-2}$ $1.74\times 10^{-2}$ $2.59\times 10^{-2}$ $u_{dabc}(rcn)$ $3.9\times 10^{-3}$ $10^{-1}$ $-$ $\approx 5\times10^{-2}$ $\approx 10^{-1}$ $u_{dabc}(ccn)$ $3.9\times 10^{-3}$ $10^{-1}$ $-$ $\approx 10^{-2}$ $\approx 5\times10^{-2}$ $u_{nsbc}$ $3.9\times 10^{-3}$ $10^{-2}$ $6.5\times 10^{-3}$ $9.4\times 10^{-3}$ $9.7\times 10^{-3}$ $u_{dabc}(rcn)$ $3.9\times 10^{-3}$ $10^{-2}$ $-$ $\approx 10^{-4}$ $\approx 10^{-3}$ $u_{dabc}(ccn)$ $3.9\times 10^{-3}$ $10^{-2}$ $-$ $\approx 10^{-3}$ $\approx 5\times10^{-3}$ $u_{nsbc}$ $4.88\times 10^{-4}$ $10^{-1}$ $1.32\times 10^{-2}$ $1.52\times 10^{-2}$ $1.63\times 10^{-2}$ $u_{dabc}(rcn)$ $4.88\times 10^{-4}$ $10^{-1}$ $-$ $\approx 10^{-2}$ $\approx 10^{-1}$ $u_{dabc}(ccn)$ $4.88\times 10^{-4}$ $10^{-1}$ $-$ $\approx 10^{-3}$ $\approx 10^{-2}$ $u_{nsbc}$ $2.44\times 10^{-4}$ $10^{-1}$ $1.32\times 10^{-2}$ $1.52\times 10^{-2}$ $1.63\times 10^{-2}$ $u_{dabc}(rcn)$ $2.44\times 10^{-4}$ $10^{-1}$ $-$ $\approx 10^{-2}$ $\approx 10^{-1}$ $u_{dabc}(ccn)$ $2.44\times 10^{-4}$ $10^{-1}$ $-$ $\approx 10^{-3}$ $\approx 10^{-2}$
Errors of the first example for homogeneous LKdV equation with NSBC for long time simulation and various values of $\alpha$ and $\beta$
 $T$ $\alpha$ $\beta$ $\tau$ $h$ $err$ $e_2$ $e_\infty$ $1$ $1$ $0$ $\frac{1}{32}$ $10^{-1}$ $6.8\times 10^{-3}$ $3.3\times 10^{-3}$ $4.6\times 10^{-3}$ $1$ $1$ $0$ $\frac{1}{64}$ $10^{-1}$ $6.5\times 10^{-3}$ $3.2\times 10^{-3}$ $4.3\times 10^{-3}$ $1$ $1$ $0$ $\frac{1}{128}$ $10^{-1}$ $6.4\times 10^{-3}$ $3\times 10^{-3}$ $4.2\times 10^{-3}$ $1$ $1$ $0$ $\frac{1}{128}$ $5\times 10^{-2}$ $7.4\times 10^{-3}$ $3.6\times 10^{-3}$ $5.1\times 10^{-3}$ $2$ $1$ $0$ $\frac{2}{32}$ $10^{-1}$ $8.1\times 10^{-3}$ $6.2\times 10^{-3}$ $6.3\times 10^{-3}$ $2$ $1$ $0$ $\frac{2}{64}$ $10^{-1}$ $6.9\times 10^{-3}$ $5.8\times 10^{-3}$ $5.6\times 10^{-3}$ $2$ $1$ $0$ $\frac{2}{64}$ $5\times10^{-2}$ $6.4\times 10^{-3}$ $3\times 10^{-3}$ $4.2\times 10^{-3}$ $3$ $1$ $0$ $\frac{3}{32}$ $10^{-1}$ $8.2\times 10^{-3}$ $1.02\times 10^{-2}$ $10^{-2}$ $3$ $1$ $0$ $\frac{3}{64}$ $5\times10^{-2}$ $-6.4\times 10^{-3}$ $-3\times 10^{-3}$ $-4.2\times 10^{-3}$ $4$ $1$ $0$ $\frac{4}{32}$ $10^{-1}$ $1.55\times 10^{-2}$ $1.82\times 10^{-2}$ $1.92\times 10^{-2}$ $4$ $1$ $0$ $\frac{4}{64}$ $10^{-1}$ $1.33\times 10^{-2}$ $1.43\times 10^{-2}$ $1.72\times 10^{-2}$ $5$ $1$ $0$ $\frac{5}{32}$ $10^{-1}$ $1.82\times 10^{-2}$ $3.04\times 10^{-2}$ $2.5\times 10^{-2}$ $5$ $1$ $0$ $\frac{5}{64}$ $10^{-1}$ $1.75\times 10^{-2}$ $2.94\times 10^{-2}$ $2.4\times 10^{-2}$ $6$ $1$ $0$ $\frac{6}{32}$ $10^{-1}$ $1.84\times 10^{-2}$ $3.76\times 10^{-2}$ $2.61\times 10^{-2}$ $6$ $1$ $0$ $\frac{6}{64}$ $10^{-1}$ $1.77\times 10^{-2}$ $3.66\times 10^{-2}$ $2.59\times 10^{-2}$ $4$ $2$ $2$ $\frac{4}{32}$ $10^{-1}$ $1.52\times 10^{-2}$ $7.6\times 10^{-3}$ $1.67\times 10^{-2}$ $4$ $2$ $-2$ $\frac{4}{32}$ $10^{-1}$ $8.8\times 10^{-3}$ $4.5\times 10^{-3}$ $5.9\times 10^{-3}$ $4$ $0.5$ $-0.5$ $\frac{4}{32}$ $10^{-1}$ $8.04\times 10^{-4}$ $7.56\times 10^{-4}$ $7.89\times 10^{-4}$ $4$ $1$ $4$ $\frac{4}{32}$ $10^{-1}$ $1.59\times 10^{-2}$ $1.05\times 10^{-2}$ $1.78\times 10^{-2}$ $4$ $1$ $-4$ $\frac{4}{32}$ $10^{-1}$ $9\times 10^{-3}$ $3.8\times 10^{-3}$ $5.9\times 10^{-3}$ $4$ $8$ $2$ $\frac{4}{32}$ $10^{-1}$ $5.25\times 10^{-2}$ $4.18\times 10^{-2}$ $7.14\times 10^{-2}$ $4$ $8$ $-2$ $\frac{4}{32}$ $10^{-1}$ $2.31\times 10^{-2}$ $1.82\times 10^{-2}$ $3.29\times 10^{-2}$ $4$ $0$ $1$ $\frac{4}{32}$ $10^{-1}$ $4.82\times 10^{-4}$ $6.86\times 10^{-5}$ $1.76\times 10^{-4}$ $4$ $0$ $1$ $\frac{4}{32}$ $5\times10^{-2}$ $1.38\times 10^{-4}$ $1.87\times 10^{-5}$ $4.9\times 10^{-5}$
 $T$ $\alpha$ $\beta$ $\tau$ $h$ $err$ $e_2$ $e_\infty$ $1$ $1$ $0$ $\frac{1}{32}$ $10^{-1}$ $6.8\times 10^{-3}$ $3.3\times 10^{-3}$ $4.6\times 10^{-3}$ $1$ $1$ $0$ $\frac{1}{64}$ $10^{-1}$ $6.5\times 10^{-3}$ $3.2\times 10^{-3}$ $4.3\times 10^{-3}$ $1$ $1$ $0$ $\frac{1}{128}$ $10^{-1}$ $6.4\times 10^{-3}$ $3\times 10^{-3}$ $4.2\times 10^{-3}$ $1$ $1$ $0$ $\frac{1}{128}$ $5\times 10^{-2}$ $7.4\times 10^{-3}$ $3.6\times 10^{-3}$ $5.1\times 10^{-3}$ $2$ $1$ $0$ $\frac{2}{32}$ $10^{-1}$ $8.1\times 10^{-3}$ $6.2\times 10^{-3}$ $6.3\times 10^{-3}$ $2$ $1$ $0$ $\frac{2}{64}$ $10^{-1}$ $6.9\times 10^{-3}$ $5.8\times 10^{-3}$ $5.6\times 10^{-3}$ $2$ $1$ $0$ $\frac{2}{64}$ $5\times10^{-2}$ $6.4\times 10^{-3}$ $3\times 10^{-3}$ $4.2\times 10^{-3}$ $3$ $1$ $0$ $\frac{3}{32}$ $10^{-1}$ $8.2\times 10^{-3}$ $1.02\times 10^{-2}$ $10^{-2}$ $3$ $1$ $0$ $\frac{3}{64}$ $5\times10^{-2}$ $-6.4\times 10^{-3}$ $-3\times 10^{-3}$ $-4.2\times 10^{-3}$ $4$ $1$ $0$ $\frac{4}{32}$ $10^{-1}$ $1.55\times 10^{-2}$ $1.82\times 10^{-2}$ $1.92\times 10^{-2}$ $4$ $1$ $0$ $\frac{4}{64}$ $10^{-1}$ $1.33\times 10^{-2}$ $1.43\times 10^{-2}$ $1.72\times 10^{-2}$ $5$ $1$ $0$ $\frac{5}{32}$ $10^{-1}$ $1.82\times 10^{-2}$ $3.04\times 10^{-2}$ $2.5\times 10^{-2}$ $5$ $1$ $0$ $\frac{5}{64}$ $10^{-1}$ $1.75\times 10^{-2}$ $2.94\times 10^{-2}$ $2.4\times 10^{-2}$ $6$ $1$ $0$ $\frac{6}{32}$ $10^{-1}$ $1.84\times 10^{-2}$ $3.76\times 10^{-2}$ $2.61\times 10^{-2}$ $6$ $1$ $0$ $\frac{6}{64}$ $10^{-1}$ $1.77\times 10^{-2}$ $3.66\times 10^{-2}$ $2.59\times 10^{-2}$ $4$ $2$ $2$ $\frac{4}{32}$ $10^{-1}$ $1.52\times 10^{-2}$ $7.6\times 10^{-3}$ $1.67\times 10^{-2}$ $4$ $2$ $-2$ $\frac{4}{32}$ $10^{-1}$ $8.8\times 10^{-3}$ $4.5\times 10^{-3}$ $5.9\times 10^{-3}$ $4$ $0.5$ $-0.5$ $\frac{4}{32}$ $10^{-1}$ $8.04\times 10^{-4}$ $7.56\times 10^{-4}$ $7.89\times 10^{-4}$ $4$ $1$ $4$ $\frac{4}{32}$ $10^{-1}$ $1.59\times 10^{-2}$ $1.05\times 10^{-2}$ $1.78\times 10^{-2}$ $4$ $1$ $-4$ $\frac{4}{32}$ $10^{-1}$ $9\times 10^{-3}$ $3.8\times 10^{-3}$ $5.9\times 10^{-3}$ $4$ $8$ $2$ $\frac{4}{32}$ $10^{-1}$ $5.25\times 10^{-2}$ $4.18\times 10^{-2}$ $7.14\times 10^{-2}$ $4$ $8$ $-2$ $\frac{4}{32}$ $10^{-1}$ $2.31\times 10^{-2}$ $1.82\times 10^{-2}$ $3.29\times 10^{-2}$ $4$ $0$ $1$ $\frac{4}{32}$ $10^{-1}$ $4.82\times 10^{-4}$ $6.86\times 10^{-5}$ $1.76\times 10^{-4}$ $4$ $0$ $1$ $\frac{4}{32}$ $5\times10^{-2}$ $1.38\times 10^{-4}$ $1.87\times 10^{-5}$ $4.9\times 10^{-5}$
Comparison of Errors for the second example using various BC, $\tau$ and $h$
 u $\tau$ $h$ $err(u)$ $e_2(u)$ $e_\infty(u)$ $u_{nsbc}$ $3.2\times 10^{-6}$ $2\times10^{-2}$ $5.17\times 10^{-2}$ $1.22\times 10^{-3}$ $7.08\times 10^{-2}$ $u_{dabc}(ccn)$ $3.2\times 10^{-6}$ $2\times10^{-2}$ $-$ $-$ $\approx 5\times10^{-2}$ $u_{dbc}$ $3.2\times 10^{-6}$ $2\times10^{-2}$ $7.6\times 10^{-2}$ $5.4\times 10^{-3}$ $4.65\times 10^{-1}$ $u_{nbc}$ $3.2\times 10^{-6}$ $2\times10^{-2}$ $9.4\times10^{-2}$ $5.7\times10^{-3}$ $5.72\times10^{-1}$ $u_{nsbc}$ $3.2\times 10^{-6}$ $2\times10^{-3}$ $6.3\times 10^{-3}$ $2.18\times 10^{-4}$ $2.09\times 10^{-2}$ $u_{dabc}(ccn)$ $3.2\times 10^{-6}$ $2\times10^{-3}$ $-$ $-$ $\approx 5\times10^{-3}$ $u_{nsbc}$ $3.2\times 10^{-6}$ $4\times10^{-4}$ $6.49\times 10^{-4}$ $5.97\times 10^{-6}$ $3.8\times 10^{-4}$ $u_{dabc}(ccn)$ $3.2\times 10^{-6}$ $4\times10^{-4}$ $-$ $-$ $\approx 5\times10^{-3}$
 u $\tau$ $h$ $err(u)$ $e_2(u)$ $e_\infty(u)$ $u_{nsbc}$ $3.2\times 10^{-6}$ $2\times10^{-2}$ $5.17\times 10^{-2}$ $1.22\times 10^{-3}$ $7.08\times 10^{-2}$ $u_{dabc}(ccn)$ $3.2\times 10^{-6}$ $2\times10^{-2}$ $-$ $-$ $\approx 5\times10^{-2}$ $u_{dbc}$ $3.2\times 10^{-6}$ $2\times10^{-2}$ $7.6\times 10^{-2}$ $5.4\times 10^{-3}$ $4.65\times 10^{-1}$ $u_{nbc}$ $3.2\times 10^{-6}$ $2\times10^{-2}$ $9.4\times10^{-2}$ $5.7\times10^{-3}$ $5.72\times10^{-1}$ $u_{nsbc}$ $3.2\times 10^{-6}$ $2\times10^{-3}$ $6.3\times 10^{-3}$ $2.18\times 10^{-4}$ $2.09\times 10^{-2}$ $u_{dabc}(ccn)$ $3.2\times 10^{-6}$ $2\times10^{-3}$ $-$ $-$ $\approx 5\times10^{-3}$ $u_{nsbc}$ $3.2\times 10^{-6}$ $4\times10^{-4}$ $6.49\times 10^{-4}$ $5.97\times 10^{-6}$ $3.8\times 10^{-4}$ $u_{dabc}(ccn)$ $3.2\times 10^{-6}$ $4\times10^{-4}$ $-$ $-$ $\approx 5\times10^{-3}$
Errors of the third example using (43) for various values of $c$ in (43)
 $c$ $\tau$ $err$ $e_2$ $e_\infty$ $0.5$ $\frac{1}{32}$ $8.5\times 10^{-3}$ $1.77\times 10^{-3}$ $2.33\times 10^{-3}$ $0.5$ $\frac{1}{64}$ $8.3\times 10^{-3}$ $1.7\times 10^{-3}$ $2.3\times 10^{-3}$ $1$ $\frac{1}{64}$ $8.5\times 10^{-3}$ $10^{-3}$ $1.4\times 10^{-3}$ $2$ $\frac{1}{64}$ $8.5\times 10^{-3}$ $5.56\times 10^{-4}$ $7.6\times 10^{-4}$
 $c$ $\tau$ $err$ $e_2$ $e_\infty$ $0.5$ $\frac{1}{32}$ $8.5\times 10^{-3}$ $1.77\times 10^{-3}$ $2.33\times 10^{-3}$ $0.5$ $\frac{1}{64}$ $8.3\times 10^{-3}$ $1.7\times 10^{-3}$ $2.3\times 10^{-3}$ $1$ $\frac{1}{64}$ $8.5\times 10^{-3}$ $10^{-3}$ $1.4\times 10^{-3}$ $2$ $\frac{1}{64}$ $8.5\times 10^{-3}$ $5.56\times 10^{-4}$ $7.6\times 10^{-4}$
Errors of the third example using (44) for various values of $\omega$
 $\omega$ $\tau$ $err$ $e_2$ $e_\infty$ $5$ $\frac{1}{32}$ $6.22\times 10^{-2}$ $6.02\times 10^{-2}$ $8.48\times 10^{-2}$ $10$ $\frac{1}{32}$ $3\times 10^{-2}$ $3.89\times 10^{-2}$ $5.63\times 10^{-2}$ $20$ $\frac{1}{32}$ $2.23\times 10^{-2}$ $1.67\times 10^{-2}$ $2.63\times 10^{-2}$ $30$ $\frac{1}{32}$ $5.2\times 10^{-3}$ $1.02\times 10^{-2}$ $1.36\times 10^{-2}$ $40$ $\frac{1}{32}$ $9.6\times 10^{-3}$ $5.7\times 10^{-3}$ $8.7\times 10^{-3}$
 $\omega$ $\tau$ $err$ $e_2$ $e_\infty$ $5$ $\frac{1}{32}$ $6.22\times 10^{-2}$ $6.02\times 10^{-2}$ $8.48\times 10^{-2}$ $10$ $\frac{1}{32}$ $3\times 10^{-2}$ $3.89\times 10^{-2}$ $5.63\times 10^{-2}$ $20$ $\frac{1}{32}$ $2.23\times 10^{-2}$ $1.67\times 10^{-2}$ $2.63\times 10^{-2}$ $30$ $\frac{1}{32}$ $5.2\times 10^{-3}$ $1.02\times 10^{-2}$ $1.36\times 10^{-2}$ $40$ $\frac{1}{32}$ $9.6\times 10^{-3}$ $5.7\times 10^{-3}$ $8.7\times 10^{-3}$
Errors of the third example using (44) and (43)
 $\omega$ $c$ $err$ $e_2$ $e_\infty$ $40$ $2$ $8.5\times 10^{-3}$ $8.89\times 10^{-4}$ $1.1\times 10^{-3}$
 $\omega$ $c$ $err$ $e_2$ $e_\infty$ $40$ $2$ $8.5\times 10^{-3}$ $8.89\times 10^{-4}$ $1.1\times 10^{-3}$
 [1] Igor Shevchenko, Barbara Kaltenbacher. Absorbing boundary conditions for the Westervelt equation. Conference Publications, 2015, 2015 (special) : 1000-1008. doi: 10.3934/proc.2015.1000 [2] Michael Renardy. A backward uniqueness result for the wave equation with absorbing boundary conditions. Evolution Equations & Control Theory, 2015, 4 (3) : 347-353. doi: 10.3934/eect.2015.4.347 [3] Roland Schnaubelt, Martin Spitz. Local wellposedness of quasilinear Maxwell equations with absorbing boundary conditions. Evolution Equations & Control Theory, 2021, 10 (1) : 155-198. doi: 10.3934/eect.2020061 [4] Youngmok Jeon, Dongwook Shin. Immersed hybrid difference methods for elliptic boundary value problems by artificial interface conditions. Electronic Research Archive, , () : -. doi: 10.3934/era.2021043 [5] Bernard Ducomet, Alexander Zlotnik, Ilya Zlotnik. On a family of finite-difference schemes with approximate transparent boundary conditions for a generalized 1D Schrödinger equation. Kinetic & Related Models, 2009, 2 (1) : 151-179. doi: 10.3934/krm.2009.2.151 [6] Alexander Zlotnik, Ilya Zlotnik. Finite element method with discrete transparent boundary conditions for the time-dependent 1D Schrödinger equation. Kinetic & Related Models, 2012, 5 (3) : 639-667. doi: 10.3934/krm.2012.5.639 [7] Takeshi Fukao, Shuji Yoshikawa, Saori Wada. Structure-preserving finite difference schemes for the Cahn-Hilliard equation with dynamic boundary conditions in the one-dimensional case. Communications on Pure & Applied Analysis, 2017, 16 (5) : 1915-1938. doi: 10.3934/cpaa.2017093 [8] Elena Kosygina. Brownian flow on a finite interval with jump boundary conditions. Discrete & Continuous Dynamical Systems - B, 2006, 6 (4) : 867-880. doi: 10.3934/dcdsb.2006.6.867 [9] Daniela Calvetti, Paul J. Hadwin, Janne M. J. Huttunen, Jari P. Kaipio, Erkki Somersalo. Artificial boundary conditions and domain truncation in electrical impedance tomography. Part II: Stochastic extension of the boundary map. Inverse Problems & Imaging, 2015, 9 (3) : 767-789. doi: 10.3934/ipi.2015.9.767 [10] Yosra Boukari, Houssem Haddar. The factorization method applied to cracks with impedance boundary conditions. Inverse Problems & Imaging, 2013, 7 (4) : 1123-1138. doi: 10.3934/ipi.2013.7.1123 [11] Daniela Calvetti, Paul J. Hadwin, Janne M. J. Huttunen, David Isaacson, Jari P. Kaipio, Debra McGivney, Erkki Somersalo, Joseph Volzer. Artificial boundary conditions and domain truncation in electrical impedance tomography. Part I: Theory and preliminary results. Inverse Problems & Imaging, 2015, 9 (3) : 749-766. doi: 10.3934/ipi.2015.9.749 [12] Antonio Suárez. A logistic equation with degenerate diffusion and Robin boundary conditions. Communications on Pure & Applied Analysis, 2008, 7 (5) : 1255-1267. doi: 10.3934/cpaa.2008.7.1255 [13] Eugenio Montefusco, Benedetta Pellacci, Gianmaria Verzini. Fractional diffusion with Neumann boundary conditions: The logistic equation. Discrete & Continuous Dynamical Systems - B, 2013, 18 (8) : 2175-2202. doi: 10.3934/dcdsb.2013.18.2175 [14] Vesselin Petkov. Location of eigenvalues for the wave equation with dissipative boundary conditions. Inverse Problems & Imaging, 2016, 10 (4) : 1111-1139. doi: 10.3934/ipi.2016034 [15] Gabriella Di Blasio. Ultraparabolic equations with nonlocal delayed boundary conditions. Discrete & Continuous Dynamical Systems, 2013, 33 (11&12) : 4945-4965. doi: 10.3934/dcds.2013.33.4945 [16] Lahcen Maniar, Martin Meyries, Roland Schnaubelt. Null controllability for parabolic equations with dynamic boundary conditions. Evolution Equations & Control Theory, 2017, 6 (3) : 381-407. doi: 10.3934/eect.2017020 [17] Xiaoyu Fu. Stabilization of hyperbolic equations with mixed boundary conditions. Mathematical Control & Related Fields, 2015, 5 (4) : 761-780. doi: 10.3934/mcrf.2015.5.761 [18] Gennaro Infante. Positive solutions of differential equations with nonlinear boundary conditions. Conference Publications, 2003, 2003 (Special) : 432-438. doi: 10.3934/proc.2003.2003.432 [19] V. Casarino, K.-J. Engel, G. Nickel, S. Piazzera. Decoupling techniques for wave equations with dynamic boundary conditions. Discrete & Continuous Dynamical Systems, 2005, 12 (4) : 761-772. doi: 10.3934/dcds.2005.12.761 [20] Alassane Niang. Boundary regularity for a degenerate elliptic equation with mixed boundary conditions. Communications on Pure & Applied Analysis, 2019, 18 (1) : 107-128. doi: 10.3934/cpaa.2019007

2020 Impact Factor: 2.425