  An efficient multigrid framework is developed for the time marching of
steady-state compressible flows with a spatially high-order ($p$-order
polynomial) modal discontinuous Galerkin method. The core algorithm that based
on a global coupling, exponential time integration scheme provides strong
damping effects to accelerate the convergence towards the steady state, while
high-frequency, high-order spatial error modes are smoothed out with a
$s$-stage preconditioned Runge-Kutta method. Numerical studies show that the
exponential time integration substantially improves the damping and propagative
efficiency of Runge-Kutta time-stepping for use with the $p$-multigrid method,
yielding rapid and $p$-independent convergences to steady flows in both two and
three dimensions.
