  The He-shell flash convection in AGB stars is the site for the
high-temperature component of the s-process in low- and intermediate mass
giants, driven by the Ne22 neutron source. [...] The upper convection boundary
plays a critical role during the H-ingestion episode that may lead to
neutron-bursts in the most metal-poor AGB stars. We address these problems
through global 3-dimensional hydrodynamic simulations including the entire
spherical He-shell flash convection zone (as oposed to the 3D box-in-a-star
simulations). An important aspect of our current effort is to establish the
feasibility of our appoach. We explain why we favour the explicit treatment
over the anelastic approximation for this problem. The simulations presented in
this paper use a Cartesian grid of 512^3 cells and have been run on four 8-core
workstations for four days to simulate ~5000s, which corresponds to almost ten
convective turn-over times. The convection layer extends radially at the
simulated point in the flash evolution over 7 H_p pressure scale-heights and
exceeds the size of the underlying core. Convection is dominated by large
convective cells that fill more than an entire octant. [...]
