Before revision:  Finally, we show that the majority of typical discretizations of this ODE , such as the Heavy ball method, do not possess suitable discrete Lyapunov functions , and hence fail to reproduce the desired limiting behaviour of this ODE, which in turn implies that their converge rates when seen as optimization methods cannot behave in an "accerelated" manner .
After revision:  Finally, we show that the majority of typical discretizations of the family of ODEs , such as the Heavy ball method, do not possess suitable discrete Lyapunov functions , and hence fail to reproduce the desired limiting behaviour of this ODE, which in turn implies that their converge rates when seen as optimization methods cannot behave in an "accerelated" manner .