Before revision:  To facilitate our analysis, we introduce a second-order ordinary differential equation (ODE) and demonstrate that this ODE is the limiting case of PAGD as the step size tends to zero. Using a simple energy argument , we show an exponential convergence of the ODE solution to its steady state .
After revision:  To facilitate our analysis, we introduce a second-order ordinary differential equation (ODE) with a preconditioner built-in and show that PAGD is an explicit time-discretization of this ODE, which requires a natural time step restriction for energy stability. At the continuous time level , we show an exponential convergence of the ODE solution to its steady state .