Two conservative difference schemes for a model of nonlinear dispersive equations

Two conservative differences schemes for the nonlinear dispersive Benjamin-Bona-Mahony-KdV (BBM-KdV) equation are proposed. The first scheme is two-level and nonlinear-implicit. The second scheme is three-level and linear implicit. Existence of its difference solutions has been shown. It is proved by the discrete energy method that the two schemes are uniquely solvable, unconditionally stable and the convergence is of second-order in the maximum norm. An iterative algorithm is proposed for solving the nonlinear scheme. The particular case known as the RLW equation is also discussed numerically in detail. Furthermore, three invariants of motion are evaluated to determine the conservation properties of the problem. Interaction of solitary waves with different amplitudes are shown. The three invariants of the motion are evaluated to determine the conservation proprieties of the system. The temporal evaluation of a Maxwellian initial pulse is then studied. Some numerical examples are given in order to validate the theoretical results.