A lot of effort has been devoted to the hydrodynamical modelling of Cepheids in one dimension. While the recovery of the most basic properties such as the pulsational instability itself has been achieved already a long time ago, properties such as the observed double-mode pulsation of some objects and the red-edge of the classical instability strip and their dependence on metallicity have remained a delicate issue. The uncertainty introduced by adjustable parameters and further physical approximations introduced in one-dimensional model equations motivate an investigation based on numerical simulations which use the full hydrodynamical equations. In this talk, results from such two-dimensional numerical simulations of a short period Cepheid are presented. The importance of a carefully designed numerical setup, in particular of sufficient resolution and domain extent, is discussed. The problematic issue of how to reliably choose fixed parameters for the one-dimensional model is illustrated. Results from an analysis of the interaction of pulsation with convection are shown concerning the large-scale structure of the He ii ionization zone. We also address the influence of convection on the atmospheric structure. Considering the potential of hydrodynamical simulations and the wealth of ever improving observational data an outlook on possible future work in this field of research is given.

We discuss how 3D numerical simulations can be used to analyse the different contributions within dynamical equations of non-local Reynolds stress models of convection.

We introduce an extended 3D numerical simulation study of Reynolds stress models of stellar convection and probe fluxes as well as mean temperature gradient profiles.

We provide results from an extended numerical simulation study of Reynolds stress models of stellar convection and probe the modelling of third order moments and non-locality.

We present the essential features of the ANTARES code. ANTARES has been developed to perform the simulation of astrophysical flows in one, two, or three dimensions. Using, in particular, the option of grid refinement, we present results for solar granulation achieved at very high spatial resolution.

We provide results from an extended 3D numerical simulation study of Reynolds stress models of stellar convection and probe the modelling of compressibility, pressure fluctuations, and dissipation of turbulent kinetic energy.