
-
Previous Article
Global attractor for a one dimensional weakly damped half-wave equation
- DCDS-S Home
- This Issue
-
Next Article
Single point blow-up and final profile for a perturbed nonlinear heat equation with a gradient and a non-local term
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 |
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.
References:
[1] |
M. Abounouh, H. 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. |
[2] |
C. Besse, M. 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. |
[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. |
[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. |
[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.
|
[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. |
[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. |
[8] |
B. Klaus, The Korteweg-de Vries Equation: History, Exact Solutions, and Graphical Representation, University of Osnabrück/Germany, 2000. |
[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. |
[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. |
[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. |
[12] |
L. A. Ostrovsky and Y. A. Stepanyants,
Do internal solitons exist in the ocean?, Rev. Geophys., 27 (1989), 293-310.
|
[13] |
G. M. Phillips, Interpolation and Approximation by Polynomials, Canadian Mathematical Society, Springer, 2003.
doi: 10.1007/b97417. |
[14] |
J. S. Russell, Report of the committee on waves, Rep. Meet. Brit. Assoc. Adv. Sci., 7th Liverpool, 417, John Murray, London, 1837. |
[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. |
[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. |
[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. |
[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.
|
[19] |
C. Zheng, W. 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. |
show all references
References:
[1] |
M. Abounouh, H. 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. |
[2] |
C. Besse, M. 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. |
[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. |
[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. |
[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.
|
[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. |
[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. |
[8] |
B. Klaus, The Korteweg-de Vries Equation: History, Exact Solutions, and Graphical Representation, University of Osnabrück/Germany, 2000. |
[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. |
[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. |
[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. |
[12] |
L. A. Ostrovsky and Y. A. Stepanyants,
Do internal solitons exist in the ocean?, Rev. Geophys., 27 (1989), 293-310.
|
[13] |
G. M. Phillips, Interpolation and Approximation by Polynomials, Canadian Mathematical Society, Springer, 2003.
doi: 10.1007/b97417. |
[14] |
J. S. Russell, Report of the committee on waves, Rep. Meet. Brit. Assoc. Adv. Sci., 7th Liverpool, 417, John Murray, London, 1837. |
[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. |
[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. |
[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. |
[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.
|
[19] |
C. Zheng, W. 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. |


















u | |||||
u | |||||
u | |||||
u | |||||
[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 and 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 and 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, 2021, 29 (5) : 3361-3382. 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 and 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 and 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 and 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 and 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 and 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 and 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 and 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 and 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 and 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 and Imaging, 2016, 10 (4) : 1111-1139. doi: 10.3934/ipi.2016034 |
[15] |
Gabriella Di Blasio. Ultraparabolic equations with nonlocal delayed boundary conditions. Discrete and 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 and 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 and 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 and 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 and Applied Analysis, 2019, 18 (1) : 107-128. doi: 10.3934/cpaa.2019007 |
2021 Impact Factor: 1.865
Tools
Metrics
Other articles
by authors
[Back to Top]