Numerical simulation of Kadomtsev-Petviashvili-Benjamin-Bona-Mahony equations using finite difference method

Keywords: KP–BBM-II equations Crank–Nicholson method Finite difference scheme Convergence Stability a b s t r a c t In this paper, the finite difference method is employed to solve Kadomtsev–Petviashvili– Benjamin–Bona–Mahony II (KP–BBM-II) partial differential equations. The time and space variable are discretized by the Crank–Nicholson method and the central-difference scheme, respectively. The consistence and stability are also proved. Some examples are investigated to verify the efficiency of the present method. Water wave problems are of great interest to engineers working in the fields of naval architecture and ship design, offshore structures, physical oceanography and marine hydrodynamics. In the past few decades, research in water waves has been very active, driven by the increasing demand of sea transport and offshore oil exploration [17]. Various offshore structures currently in use; including both mobile and fixed drilling platforms, such as: suspended bridges, Tension leg platform (TLP), the more traditional Jacket and Jack-up barge.. . pose their own peculiar demands in terms of hydrodynamic loading effects [20] (see Fig. 1). In this context, we are interested in evaluating forces applied by traveling waves on an offshore structure. More precise, we deal with determining the pressure field exerted to the structure. To overcome that difficulty, Korteweg–de Vries and Benjamin–Bona–Mahony started from the general equations governing fluids motion: Navier–Stokes equations, to develop their unidirectional propagate water waves models known as KdV equations and BBM equations, where they took into account of the dispersive character of the wave. Note that the Navier–Stokes equations are derived from the general principle of mass and momentum conservation and this means, respectively, that the velocity of the fluid at the surface matches the velocity of the surface (kinematic boundary condition) and the pressure jump across the free surface is proportional to the curvature of the surface (dynamic boundary condition). So, the first step in the problem of determining pressure applied to an offshore structure is to solve a water wave model which is, generally, in the form of partial differential equations [1,6–8,18]. Then, using obtained results to find out pressure forces. In the case of an offshore structure, the fluid flow is considered, relatively, unbounded. So, it will exert pressure forces around this structure. For this reason we are interested to one of the bidirectional traveling waves models, such as the KP–BBM-II equations [9]. And, in this work, we will just take care to solve numerically this type of equations …