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

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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… Click to show full 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 $$\mathcal {Z}$$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.

Keywords: boundary conditions; linearized benjamin; benjamin bona; artificial boundary; equation; conditions linearized

Journal Title: Numerische Mathematik
Year Published: 2018

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