The authors describe a Petrov-Galerkin finite element method with third-order accuracy for the numerical solution of the 1-term Beji-Nadaoka model for weakly nonlinear dispersive water waves in one dimensional horizontal domains. Finite elements are used in both space and time domains. Dispersion correction and a highly selective dissipation mechanism are introduced through additional streamline upwind in the weighting functions. Linear interpolation in space is retained and coupled with an implicit one-level time integration scheme. An accuracy and stability analysis is presented. The numerical results are compared to experimental data of waves propagating over a bar. It is concluded that the proposed finite element method possesses very good features.
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