# Artificial boundary conditions for the linearized Benjamin–Bona–Mahony equation

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## Abstract

We consider various approximations of artificial boundary conditions for linearized Benjamin–Bona–Mahony BBM equation. Continuous (respectively discrete) artificial boundary conditions involve non local operators in time which in turn requires to compute time convolutions and invert the Laplace transform of an analytic function (respectively the \(\mathcal {Z}\)-transform of an holomorphic function). In this paper, we derive explicit transparent boundary conditions both continuous and discrete for the linearized BBM equation. The equation is discretized with the Crank Nicolson time discretization scheme and we focus on the difference between the upwind and the centered discretization of the convection term. We use these boundary conditions to compute solutions with compact support in the computational domain and also in the case of an incoming plane wave which is an exact solution of the linearized BBM equation. We focus on and prove consistency, stability and convergence of the numerical scheme and provide many numerical experiments to show the efficiency of our transparent boundary conditions.

## Mathematics Subject Classification

65M06 65M12 65M85 76M20## Notes

### Acknowledgements

Research of C. Besse and P. Noble was partially supported by the ANR project BoND ANR-13-BS01-0009-01. Research of B. Mésognon-Gireau was supported by the SHOM research contract 11CR0001.

## References

- 1.Abdallah, N.B., Méhats, F., Pinaud, O.: On an open transient Schrödinger–Poisson system. Math. Models Methods Appl. Sci.
**15**, 667 (2005)MathSciNetCrossRefzbMATHGoogle Scholar - 2.Achouri, T., Khiari, N., Omrani, K.: On the convergence of difference schemes for the Benjamin Bona Mahony BBM equation. Appl. Math. Comput.
**182**(2), 999–1005 (2006)MathSciNetzbMATHGoogle Scholar - 3.Alazman, A.A., Albert, J.P., Bona, J.L., Chen, M., Wu, J.: Comparisons between the BBM equation and a Boussinesq system. Adv. Differ. Equ.
**11**(2), 121–166 (2006)MathSciNetzbMATHGoogle Scholar - 4.Antoine, X., Arnold, A., Besse, C., Ehrhardt, M., Schädle, A.: A review of transparent and artificial boundary conditions techniques for linear and nonlinear Schrödinger equations. Commun. Comput. Phys.
**4**, 729–796 (2008)MathSciNetzbMATHGoogle Scholar - 5.Arnold, A.: Numerically absorbing boundary conditions for quantum evolution equations. VLSI Des.
**6**, 313–319 (1998)CrossRefGoogle Scholar - 6.Arnold, A., Ehrhardt, M., Sofronov, I.: Discrete transparent boundary conditions for the Schrödinger equation: fast calculation, approximation, and stability. Commun. Math. Sci.
**3**, 501–556 (2003)CrossRefzbMATHGoogle Scholar - 7.Besse, C., Ehrhardt, M., Lacroix-Violet, I.: Discrete Artificial Boundary Conditions for the Korteweg–de Vries Equation. Numer. Math. PDEs (2016). https://doi.org/10.1002/num.22058
- 8.Dutykh, D., Pelinovsky, E.: Numerical simulation of a solitonic gas in KdV and KdV-BBM equations. Phys. Lett. A
**378**(42), 3102–3110 (2004)MathSciNetCrossRefzbMATHGoogle Scholar - 9.Eilbeck, J.C., McGuire, G.R.: Numerical study of the regularized long-wave equation. II: interaction of solitary waves. J. Comput. Phys.
**23**(1), 63–73 (1977)MathSciNetCrossRefzbMATHGoogle Scholar - 10.El, G.A.: Resolution of a shock in hyperbolic systems modified by weak dispersion. Chaos Interdiscip. J. Nonlinear Sci.
**15**(3), 037103 (2005)MathSciNetCrossRefzbMATHGoogle Scholar - 11.Ehrhardt, M.: Discrete Artificial Boundary Conditions. Ph.D., Technische Universität Berlin (2001)Google Scholar
- 12.Ehrhardt, M., Arnold, A.: Discrete transparent boundary conditions for the Schrödinger equation. Riv. Math. Univ. Parma
**6**, 57–108 (2001)zbMATHGoogle Scholar - 13.Ehrhardt, M.: Discrete transparent boundary conditions for Schrödinger-type equations for non-compactly supported initial data. Appl. Numer. Math.
**58**, 660–673 (2008)MathSciNetCrossRefzbMATHGoogle Scholar - 14.El, G.A., Hoefer, M.A., Shearer, M.: Expansion shock solutions of the BBM and Boussinesq equations, arXiv preprint arXiv:1601.01071 (2016)
- 15.Grava, T., Klein, C.: Numerical study of a multi scale expansion of Korteweg–de Vries and Camass–Holm equation. Contemp. Math.
**458**, 81 (2008)CrossRefzbMATHGoogle Scholar - 16.Halpern, L.: Absorbing boundary conditions for the discretization schemes of the one-dimensional wave equation. Math. Comput.
**38**(158), 415–429 (1982)MathSciNetCrossRefzbMATHGoogle Scholar - 17.Jury, E.I.: Theory and Application of the z-Transform Method. Wiley, New York (1964)Google Scholar
- 18.Lannes, D.: The water waves problem: mathematical analysis and asymptotics. In: Mathematical Surveys and Monographs, vol 188. AMS (2013)Google Scholar
- 19.Lax, P.D., Levermore, D.: The small dispersion limit of the Korteweg-de Vries equation. I. Commun. Pure Appl. Math.
**36**(3), 253–290 (1983)MathSciNetCrossRefzbMATHGoogle Scholar - 20.Sofronov, I.L., Podgnornova, O.V.: A spectral approach for generating non-local boundary conditions for external wave problems in anisotropic media. J. Sci. Comput.
**27**(1–3), 419–430 (2006)MathSciNetCrossRefzbMATHGoogle Scholar - 21.Sofronov, I.L., Zaitsev, N.A.: Numerical generation of transparent boundary conditions on the side surface of a vertical transverse isotropic layer. J. Comput. Appl. Math.
**234**(6), 1732–1738 (2010)MathSciNetCrossRefzbMATHGoogle Scholar - 22.Whitham, G.B.: Linear and Nonlinear Waves, vol. 42. Wiley, Hoboken (2011)zbMATHGoogle Scholar
- 23.Zheng, C., Wen, X., Han, H.: Numerical solution to a linearized KdV equation on unbounded domain. Numer. Methods Partial Differ. Equ.
**24**(2), 383–399 (2008)MathSciNetCrossRefzbMATHGoogle Scholar