  Convergence analysis of accelerated first-order methods for convex
optimization problems are presented from the point of view of ordinary
differential equation solvers. A new dynamical system, called Nesterov
accelerated gradient flow, has been derived from the connection between
acceleration mechanism and $A$-stability of ODE solvers, and the exponential
decay of a tailored Lyapunov function along with the solution trajectory is
proved. Numerical discretizations are then considered and convergence rates are
established via a unified discrete Lyapunov function. The proposed differential
equation solver approach can not only cover existing accelerated methods, such
as FISTA, G\"{u}ler's proximal algorithm and Nesterov's accelerated gradient
method, but also produce new algorithms for composite convex optimization that
possess accelerated convergence rates.
