Numerische Mathematik

, Volume 139, Issue 2, pp 281–314 | Cite as

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

  • Christophe BesseEmail author
  • Benoît Mésognon-Gireau
  • Pascal Noble


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 



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.


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Copyright information

© Springer-Verlag GmbH Germany, part of Springer Nature 2018

Authors and Affiliations

  • Christophe Besse
    • 1
    Email author
  • Benoît Mésognon-Gireau
    • 2
  • Pascal Noble
    • 2
  1. 1.Institut de Mathématiques de Toulouse, UMR5219, CNRS, UPS IMTUniversité de ToulouseToulouse Cedex 9France
  2. 2.Institut de Mathématiques de Toulouse, UMR5219, CNRS, INSAUniversité de ToulouseToulouseFrance

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