  There has been a long history of using ordinary differential equations (ODEs)
to understand the dynamics of discrete-time algorithms (DTAs). Surprisingly,
there are still two fundamental and unanswered questions: (i) it is unclear how
to obtain a \emph{suitable} ODE from a given DTA, and (ii) it is unclear the
connection between the convergence of a DTA and its corresponding ODEs. In this
paper, we propose a new machinery -- an $O(s^r)$-resolution ODE framework --
for analyzing the behavior of a generic DTA, which (partially) answers the
above two questions. The framework contains three steps: 1. To obtain a
suitable ODE from a given DTA, we define a hierarchy of $O(s^r)$-resolution
ODEs of a DTA parameterized by the degree $r$, where $s$ is the step-size of
the DTA. We present a principal approach to construct the unique
$O(s^r)$-resolution ODEs from a DTA; 2. To analyze the resulting ODE, we
propose the $O(s^r)$-linear-convergence condition of a DTA with respect to an
energy function, under which the $O(s^r)$-resolution ODE converges linearly to
an optimal solution; 3. To bridge the convergence properties of a DTA and its
corresponding ODEs, we define the properness of an energy function and show
that the linear convergence of the $O(s^r)$-resolution ODE with respect to a
proper energy function can automatically guarantee the linear convergence of
the DTA. To better illustrate this machinery, we utilize it to study three
classic algorithms -- gradient descent ascent (GDA), proximal point method
(PPM) and extra-gradient method (EGM) -- for solving the unconstrained minimax
problem $\min_{x\in\RR^n} \max_{y\in \RR^m} L(x,y)$.
