The work has two independent parts. The first part concerns the convergence toward equilibrium of discrete gradient flows or of some discretizations of autonomous systems which admit a Lyapunov function. The study is performed assuming sufficient conditions for the solutions of the continuous problem to converge toward a stationary state as time goes to infinity. Under mild hypotheses, the discrete system has the same property. This leads to new results on the large time asymptotic behavior of some known non-linear schemes. The second part concerns the numerical simulation of the motion...