A class of fully discrete schemes for the numerical simulation of solutions of the periodic initial-value problem for a class of generalized Korteweg-de Vries equations is analysed, implemented and tested. These schemes may have arbitrarily high order in both the spatial and the temporal variable, but at the same time they feature weak theoretical stability limitations. The spatial discretization is effected using smooth splines of quadratic or higher degree, while the temporal discretization is a multi-stage, implicit, Runge-Kutta method. A proof is presented showing convergence of the numerical approximations to the true solution of the initial-value problem in the limit of vanishing spatial and temporal discretization. In addition, a careful analysis of the efficiency of particular versions of our schemes is given. The information thus gleaned is used in the investigation of the instability of the solitary-wave solutions of a certain class of these equations.