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:
[1]

M. AbounouhH. Al-Moatassime and S. Kaouri, Novel transparent boundary conditions for the regularized long wave equation, Int. J. Comput. Methods, 16 (2019), 1950021-1950047.  doi: 10.1142/S021987621950021X.  Google Scholar

[2]

C. BesseM. Ehrhardt and I. Lacroix-Violet, Discrete artificial boundary conditions for the linearized Korteweg-de Vries equation, Num. Meth. For. Partial Differential Equations, 32 (2016), 1455-1484.  doi: 10.1002/num.22058.  Google Scholar

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J. Boussinesq, Essai sur la théorie des eaux courantes, l'Acad. des Sci. Inst. Nat. France, 23 (1877). Available from: https://gallica.bnf.fr/ark:/12148/bpt6k56673076.texteImage. Google Scholar

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D. S. Clark, Short proof of a discrete Gronwall inequality, Discrete Applied Mathematics, 16 (1987), 279-281.  doi: 10.1016/0166-218X(87)90064-3.  Google Scholar

[5]

O. Goubet and J. Shen, On the dual Petrov-Galerkin formulation of the KdV equation in a finite interval, Adv. Differ. Equat., 12 (2007), 221-239.   Google Scholar

[6]

R. Hirota, Exact solution of the Korteweg–de Vries equation for multiple collisions of solitons, Phys. Rev. Lett., 27 (1971), 1192-1196.  doi: 10.1103/PhysRevLett.27.1192.  Google Scholar

[7]

W. Hereman, Shallow water waves and solitary waves, Encyclopedia of Complexity and Applied System Science, 480 (2009), 8112-8125.  doi: 10.1007/978-1-4614-1806-1_96.  Google Scholar

[8]

B. Klaus, The Korteweg-de Vries Equation: History, Exact Solutions, and Graphical Representation, University of Osnabrück/Germany, 2000. Google Scholar

[9]

D. J. Korteweg and G. de Vries, On the change of form of long waves advancing in a rectangular canal and on a new type of long stationary waves, Philos Mag., 39 (1895), 422-443.  doi: 10.1080/14786449508620739.  Google Scholar

[10]

N. A. Kudryashov and I. L. Chernyavskii, Nonlinear waves in fluid flow through a viscoelastic tube, Fluid. Dyn., 41 (2006), 49-62.  doi: 10.1007/s10697-006-0021-3.  Google Scholar

[11]

H. G. Lee and J. Kim, A simple and robust boundary treatment for the forced Korteweg–de Vries equation, Commun. Nonlinear Sci. Numer. Simulat., 19 (2014), 2262-2271.  doi: 10.1016/j.cnsns.2013.12.019.  Google Scholar

[12]

L. A. Ostrovsky and Y. A. Stepanyants, Do internal solitons exist in the ocean?, Rev. Geophys., 27 (1989), 293-310.   Google Scholar

[13]

G. M. Phillips, Interpolation and Approximation by Polynomials, Canadian Mathematical Society, Springer, 2003. doi: 10.1007/b97417.  Google Scholar

[14]

J. S. Russell, Report of the committee on waves, Rep. Meet. Brit. Assoc. Adv. Sci., 7th Liverpool, 417, John Murray, London, 1837. Google Scholar

[15]

J. Shen, A new dual-Petrov-Galerkin method for third and higher odd-order differential equations: Application to the KdV equation, SIAM J. Numer. Anal., 41 (2003), 1595-1619.  doi: 10.1137/S0036142902410271.  Google Scholar

[16]

R. S. Varga, Matrix iterative analysis, Springer Series in Computational Mathematics, SSCM, 27, Springer, Berlin, Heidelberg, 1962. doi: 10.1007/978-3-642-05156-2.  Google Scholar

[17]

H. Washimi and T. Taniuti, Propagation of ion-acoustic solitary waves of small amplitude, Phys. Rev. Lett., 966 (1966). doi: 10.1103/PhysRevLett.17.996.  Google Scholar

[18]

N. J. Zabusky and M. D. Kruskal, Interactions of solitons in a collisionless plasma and the recurrence of initial states, Phys. Rev. Lett., 15 (1965), 240-43.   Google Scholar

[19]

C. ZhengW. X. Wen and H. Han, Numerical solution to a linearized KdV equation on unbounded domain, Numer. Methods Partial Differ. Equations, 24 (2007), 383-399.  doi: 10.1002/num.20267.  Google Scholar

show all references

References:
[1]

M. AbounouhH. Al-Moatassime and S. Kaouri, Novel transparent boundary conditions for the regularized long wave equation, Int. J. Comput. Methods, 16 (2019), 1950021-1950047.  doi: 10.1142/S021987621950021X.  Google Scholar

[2]

C. BesseM. Ehrhardt and I. Lacroix-Violet, Discrete artificial boundary conditions for the linearized Korteweg-de Vries equation, Num. Meth. For. Partial Differential Equations, 32 (2016), 1455-1484.  doi: 10.1002/num.22058.  Google Scholar

[3]

J. Boussinesq, Essai sur la théorie des eaux courantes, l'Acad. des Sci. Inst. Nat. France, 23 (1877). Available from: https://gallica.bnf.fr/ark:/12148/bpt6k56673076.texteImage. Google Scholar

[4]

D. S. Clark, Short proof of a discrete Gronwall inequality, Discrete Applied Mathematics, 16 (1987), 279-281.  doi: 10.1016/0166-218X(87)90064-3.  Google Scholar

[5]

O. Goubet and J. Shen, On the dual Petrov-Galerkin formulation of the KdV equation in a finite interval, Adv. Differ. Equat., 12 (2007), 221-239.   Google Scholar

[6]

R. Hirota, Exact solution of the Korteweg–de Vries equation for multiple collisions of solitons, Phys. Rev. Lett., 27 (1971), 1192-1196.  doi: 10.1103/PhysRevLett.27.1192.  Google Scholar

[7]

W. Hereman, Shallow water waves and solitary waves, Encyclopedia of Complexity and Applied System Science, 480 (2009), 8112-8125.  doi: 10.1007/978-1-4614-1806-1_96.  Google Scholar

[8]

B. Klaus, The Korteweg-de Vries Equation: History, Exact Solutions, and Graphical Representation, University of Osnabrück/Germany, 2000. Google Scholar

[9]

D. J. Korteweg and G. de Vries, On the change of form of long waves advancing in a rectangular canal and on a new type of long stationary waves, Philos Mag., 39 (1895), 422-443.  doi: 10.1080/14786449508620739.  Google Scholar

[10]

N. A. Kudryashov and I. L. Chernyavskii, Nonlinear waves in fluid flow through a viscoelastic tube, Fluid. Dyn., 41 (2006), 49-62.  doi: 10.1007/s10697-006-0021-3.  Google Scholar

[11]

H. G. Lee and J. Kim, A simple and robust boundary treatment for the forced Korteweg–de Vries equation, Commun. Nonlinear Sci. Numer. Simulat., 19 (2014), 2262-2271.  doi: 10.1016/j.cnsns.2013.12.019.  Google Scholar

[12]

L. A. Ostrovsky and Y. A. Stepanyants, Do internal solitons exist in the ocean?, Rev. Geophys., 27 (1989), 293-310.   Google Scholar

[13]

G. M. Phillips, Interpolation and Approximation by Polynomials, Canadian Mathematical Society, Springer, 2003. doi: 10.1007/b97417.  Google Scholar

[14]

J. S. Russell, Report of the committee on waves, Rep. Meet. Brit. Assoc. Adv. Sci., 7th Liverpool, 417, John Murray, London, 1837. Google Scholar

[15]

J. Shen, A new dual-Petrov-Galerkin method for third and higher odd-order differential equations: Application to the KdV equation, SIAM J. Numer. Anal., 41 (2003), 1595-1619.  doi: 10.1137/S0036142902410271.  Google Scholar

[16]

R. S. Varga, Matrix iterative analysis, Springer Series in Computational Mathematics, SSCM, 27, Springer, Berlin, Heidelberg, 1962. doi: 10.1007/978-3-642-05156-2.  Google Scholar

[17]

H. Washimi and T. Taniuti, Propagation of ion-acoustic solitary waves of small amplitude, Phys. Rev. Lett., 966 (1966). doi: 10.1103/PhysRevLett.17.996.  Google Scholar

[18]

N. J. Zabusky and M. D. Kruskal, Interactions of solitons in a collisionless plasma and the recurrence of initial states, Phys. Rev. Lett., 15 (1965), 240-43.   Google Scholar

[19]

C. ZhengW. X. Wen and H. Han, Numerical solution to a linearized KdV equation on unbounded domain, Numer. Methods Partial Differ. Equations, 24 (2007), 383-399.  doi: 10.1002/num.20267.  Google Scholar

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} $
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} $
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} $
$ 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} $
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} $
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} $
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} $
$ 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} $
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} $
$ \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} $
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} $
$ \omega $ $ c $ $ err $ $ e_2 $ $ e_\infty $
$ 40 $ $ 2 $ $ 8.5\times 10^{-3} $ $ 8.89\times 10^{-4} $ $ 1.1\times 10^{-3} $
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