In this paper, a finite difference algorithm using a three-layer approximation for the vertical flow region to solve the 2D Euler equations is considered. In this algorithm, the pressure is split into hydrostatic and hydrodynamic parts, and the predictor-corrector procedure is applied. In the predictor step, the momentum hydrostatic model is formulated. In the corrector step, the hydrodynamic pressure is accommodated after solving the Laplace equation using the Successive Over Relaxation (SOR) iteration method. The resulting algorithm is first tested to simulate a standing wave over an intermediate constant depth. Dispersion relation of the scheme is derived, and it is shown to agree with the analytical dispersion relation for kd < π with 94% accuracy. The second test case is a solitary wave simulation. Our computed solitary wave propagates with constant velocity, undisturbed in shape, and confirm the analytical solitary wave. Finally, the scheme is tested to simulate the appearance of the undular bore. The result shows a good agreement with the result from the finite volume scheme for the Boussinesq-type model by Soares-Frazão and Guinot (2008).

It is well known that transcritical flow past an obstacle may generate undular bores propagating away from the obstacle. This flow has been successfully modelled in the framework of the forced Korteweg–de Vries equation, where numerical simulations and asymptotic analyses have shown that the unsteady undular bores are connected by a locally steady solution over the obstacle. In this paper we present an overview of the underlying theory, together with some recent work on the case where the obstacle has a large width.

Quintic B-spline collocation schemes for numerical solution of the regularized long wave (RLW) equation have been proposed. The schemes are based on the Crank–Nicolson formulation for time integration and quintic B-spline functions for space integration. The quintic B-spline collocation method over finite intervals is also applied to the time-split RLW equation and space-split RLW equation. After stability analysis is applied to all the schemes, the results of the three algorithms are compared by studying the propagation of the solitary wave, interaction of two solitary waves and wave undulation.