and simulations have become the mainstream ([@chan86; @chan89; @chan96; @stein89; @malagoli90; @porter2000]). Numerical investigation of turbulent flows has two categories: direct numerical simulation (DNS) and large eddy simulation (LES). The problem being three-dimensional, multi-length-scale and multi-time-scale, a DNS of stellar convection demands enormous computer resources. In LES, the large eddies are calculated explicitly while the smaller eddies are to be handled by sub-grid scale (SGS) models. This is usually the approach adopted for stellar flow simulations. Being limited by the speed and memory of the computers, the progresses made by numerical simulations are usually slow. Even with today’s machines, we still cannot simulate directly the whole convection zone of a post-main-sequence star. A frequently used approach is to make high resolution local simulation.

Many problems of stellar structure are associated with convective overshooting. The overshooting distance beneath an outer convection zone may affect the surface chemical composition, e.g., the depletion of lithium in the Sun. The radiative-convective boundary is thought to be closely related to the solar dynamo and is the site for magnetic flux storage. The gravity waves generated by penetration may generate momentum exchange with the interior. However, calculation of overshooting is a challenge to all existing stellar 