  The Fourier analysis of the \emph{p}-multigrid acceleration technique is
considered for a dual-time scheme applied to the advection-diffusion equation
with various cycle configurations. It is found that improved convergence can be
achieved through \emph{V}-cycle asymmetry where additional prolongation
smoothing is applied. Experiments conducted on the artificial compressibility
formulation of the Navier--Stokes equations found that these analytic findings
could be observed numerically in the pressure residual, whereas velocity
terms---which are more hyperbolic in character---benefited primarily from
increased pseudo-time steps.
