Before revision:  We then show that in the appropriate limit  this family of methods may be seen as a discretization of a family of second order ordinary differential equations , which properties can be also understood by a (continuous) Lyapunov function, which can also be obtained by studying the limiting behaviour of the discrete Lyapunov function .
After revision:  We then show that in the appropriate limit , this family of methods may be seen as a discretization of a family of second order ordinary differential equations , which properties can be also understood by a (continuous) Lyapunov function, which can also be obtained by studying the limiting behaviour of the discrete Lyapunov function .