The BBM-KdV equation plays an important role because it can describe a large number of physical phenomena

such as shallow water waves and ion waves. It is an important model for long-wave unidirectional propagation in weakly nonlinear dispersive media

but its numerical investigations are rarely made. For the initial-boundary value problem of the generalized BBM-KdV equations with homogeneous boundary conditions

a two-level nonlinear finite difference scheme with the second-order theoretical accuracy is proposed

which reasonably simulates the two conserved quantities of the problem. With a priori estimation

the existence and uniqueness of the difference solutions are dicussed. By the discrete functional analysis method the convergence and unconditional stability of the scheme are also proved. Finally

some numerical experiments verify the robustness of the proposed scheme.