  This paper provides a self-contained ordinary differential equation solver
approach for separable convex optimization problems. A novel primal-dual
dynamical system with built-in time rescaling factors is introduced, and the
exponential decay of a tailored Lyapunov function is established. Then several
time discretizations of the continuous model are considered and analyzed via a
unified discrete Lyapunov function. Moreover, two families of accelerated
proximal alternating direction methods of multipliers are obtained, and
nonergodic optimal mixed-type convergence rates shall be proved for the primal
objective residual, the feasibility violation and the Lagrangian gap. Finally,
numerical experiments are provided to validate the practical performances.
