  In the fields of statistics, machine learning, image science, and related
areas, there is an increasing demand for decentralized collection or storage of
large-scale datasets, as well as distributed solution methods. To tackle this
challenge, the alternating direction method of multipliers (ADMM) has emerged
as a widely used approach, particularly well-suited to distributed convex
optimization. However, the iterative behavior of ADMM has not been well
understood. In this paper, we employ dimensional analysis to derive a system of
high-resolution ordinary differential equations (ODEs) for ADMM. This system
captures an important characteristic of ADMM, called the $\lambda$-correction,
which causes the trajectory of ADMM to deviate from the constrained hyperplane.
To explore the convergence behavior of the system of high-resolution ODEs, we
utilize Lyapunov analysis and extend our findings to the discrete ADMM
algorithm. Through this analysis, we identify that the numerical error
resulting from the implicit scheme is a crucial factor that affects the
convergence rate and monotonicity in the discrete ADMM algorithm. In addition,
we further discover that if one component of the objective function is assumed
to be strongly convex, the iterative average of ADMM converges strongly with a
rate $O(1/N)$, where $N$ is the number of iterations.
