We discuss the general design of the ANTARES code which is intended for simulations in stellar hydrodynamics with radiative transfer and realistic microphysics in 1D, 2D and 3D. We then compare the quality of various numerical methods. We have applied ANTARES in order to obtain high resolution simulations of solar granulation which we describe and analyze. In order to obtain high resolution, we apply grid refinement to a region predominantly occupied by an exploding granule. Strong, rapidly rotating vortex tubes of small diameter (∼ 100 km) generated by the downdrafts and ascending into the photosphere near the granule boundaries evolve, often entering the photosphere from below in an arclike fashion. They essentially contribute to the turbulent velocity field near the granule boundaries.

The review considers the modelling process for stellar convection rather than specific astrophysical results. For achieving reasonable depth and length we deal with hydrodynamics only, omitting MHD. A historically oriented introduction offers first glimpses on the physics of stellar convection. Examination of its basic properties shows that two very different kinds of modelling keep being needed: low dimensional models (mixing length, Reynolds stress, etc.) and "full" 3D simulations. A list of affordable and not affordable tasks for the latter is given. Various low dimensional modelling approaches are put in a hierarchy and basic principles which they should respect are formulated. In 3D simulations of low Mach number convection the inclusion of then unimportant sound waves with their rapid time variation is numerically impossible. We describe a number of approaches where the Navier-Stokes equations are modified for their elimination (anelastic approximation, etc.). We then turn to working with the full Navier-Stokes equations and deal with numerical principles for faithful and efficient numerics. Spatial differentiation as well as time marching aspects are considered. A list of codes allows assessing the state of the art. An important recent development is the treatment of even the low Mach number problem without prior modification of the basic equation (obviating side effects) by specifically designed

Context. Magnetic bright points (MBPs) are small-scale magnetic features in the solar photosphere. They may be a possible source of coronal heating by rapid footpoint motions that cause magnetohydrodynamical waves. The number and size distribution are of vital importance in estimating the small scale-magnetic-field energy. Aims. The size distribution of MBPs is derived for G-band images acquired by the Hinode/SOT instrument. Methods. For identification purposes, a new automated segmentation and identification algorithm was developed. Results. For a sampling of 0.108 arcsec/pixel, we derived a mean diameter of (218 ± 48) km for the MBPs. For the full resolved data set with a sampling of 0.054 arcsec/pixel, the size distribution shifted to a mean diameter of (166 ± 31) km. The determined diameters are consistent with earlier published values. The shift is most probably due to the different spatial sampling. Conclusions. We conclude that the smallest magnetic elements in the solar photosphere cannot yet be resolved by G-band observations. The influence of discretisation effects (sampling) has also not yet been investigated sufficiently.

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We have extended the ANTARES code to simulate the coupling of pulsation with convection in Cepheid-like variables in an increasingly realistic way, in particular in multidimensions, 2D at this stage. Present days models of radially pulsating stars assume radial symmetry and have the pulsation-convection interaction included via model equations containing ad hoc closures and moreover parameters whose values are barely known. We intend to construct ever more realistic multidimensional models of Cepheids. In the present paper, the first of a series, we describe the basic numerical approach and how it is motivated by physical properties of these objects which are sometimes more, sometimes less obvious. -For the construction of appropriate models a polar grid co-moving with the mean radial velocity has been introduced to optimize radial resolution throughout the different pulsation phases. The grid is radially stretched to account for the change of spatial scales due to vertical stratification and a new grid refinement scheme is introduced to resolve the upper, hydrogen ionisation zone where the gradient of temperature is steepest. We demonstrate that the simulations are not conservative when the original weighted essentially non-oscillatory method implemented in ANTARES is used and derive a new scheme which allows a conservative time evolution. The numerical approximation of diffusion follows the same principles. Moreover, the radiative transfer solver has been modified to improve the efficiency of calculations on parallel computers. We show that with these improvements the ANTARES code can be used for realistic simulations of the convection-pulsation interaction in Cepheids. We discuss the properties of several numerical models of this kind which include the upper 42% of a Cepheid along its radial coordinate and assume different opening angles. The models are suitable for an in-depth study of convection and pulsation in these objects.