Discrete transparent boundary conditions for the mixed KDV-BBM equation

Christophe Besse (1) , Pascal Noble (1) , David Sanchez (1)

1 IMT - Institut de Mathématiques de Toulouse UMR5219

In this paper, we consider artificial boundary conditions for the linearized mixed Korteweg-de Vries (KDV) Benjamin-Bona-Mahoney (BBM) equation which models water waves in the small amplitude, large wavelength regime. 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 Z-transform of an holomor-phic function). In this paper, we propose a new, stable and fairly general strategy to carry out this crucial step in the design of transparent boundary conditions. For large time simulations, we also introduce a methodology based on the asymptotic expansion of coefficients involved in exact direct transparent boundary conditions. We illustrate the accuracy of our methods for Gaussian and wave packets initial data.

Mots clés: stability analysis; artificial boundary conditions; Korteweg-de Vries and Benjamin-Bona-Mahoney equations; numerical simulation

Domaines: Equations aux dérivées partielles [math.AP]; Analyse numérique [math.NA]

Christophe Besse, Pascal Noble, David Sanchez. Discrete transparent boundary conditions for the mixed KDV-BBM equation. Journal of Computational Physics, 2017, 345, pp.484-509. ⟨10.1016/j.jcp.2017.05.031⟩. ⟨hal-01373388⟩