g. [@BDKMc] and its references. We describe in detail the implementation of this fully discrete hybrid method and make a computational study of its accuracy and stability properties when it is applied to the Benjamin and Benjamin-Ono (i. e. when $\delta$ is set to zero) equations. In addition, we validate the hybrid scheme by making a detailed comparison of the solutions that it produces with those of a standard fully discrete pseudospectral scheme in the case of three numerical experiments involving the propagation of solitary waves of the Benjamin and Benjamin-Ono equations.

In Section \[sec3\] we review the continuation-conjugate gradient-Newton technique of [@DDM1] for generating single and multi-pulse solitary-wave solutions (i. e. solutions of (\[E15\])) of the Benjamin equation for various values of $\gamma$ with particular attention to values close to $1$. We use these numerical profiles as initial conditions in numerical evolution experiments with the hybrid scheme and investigate with various metrics their accuracy as travelling wave solutions of the Benjamin equation. Our conclusion from the numerical experiments of Sections \[sec2\] and \[sec3\] is that the hybrid scheme yields very accurate and stable approximations of solutions of the Benjamin equation, and in particular of the solitary waves for values 