  Non-local, time-dependent convection models have been used to explain the
location of double-mode pulsations in Cepheids in the HR diagram as well as the
existence and location of the red edge of the instability strip. These
properties are highly sensitive to model parameters. We use 2D radiation
hydrodynamical simulations with realistic microphysics and grey
radiative-transfer to model a short period Cepheid. The simulations show that
the strength of the convection zone varies significantly over the pulsation
period and exhibits a phase shift relative to the variations in radius. We
evaluate the convective flux and the work integral as predicted by the most
common convection models. It turns out that over one pulsation cycle the model
parameter $\alpha_{\rm c}$, has to be varied by up to a factor of beyond 2 to
match the convective flux obtained from the simulations. To bring convective
fluxes integrated over the He II convection zone and the overshoot zone below
into agreement, this parameter has to be varied by a factor of up to $\sim 7.5$
(Kuhfu{\ss}). We then present results on the energetics of the convection and
overshoot zone by radially symmetric and fluctuating quantities. To
successfully model this scenario by a static, one dimensional or even by a
simple time-dependent model appears extremely challenging. We conclude that
significant improvements are needed to make predictions based on 1D models more
robust and to improve the reliability of conclusions on the
convection-pulsation coupling drawn from them. Multidimensional simulations can
provide guidelines for developing descriptions of convection then applied in
traditional 1D modelling.
