  We describe our modelling of the radiatively cooling shocks and their thin
shells with various numerical tools in different physical and calculational
setups. We inspect structure of the dense shell, its formation and evolution,
pointing out physical and numerical factors that sustain its shape and also may
lead to instabilities. We have found that under certain physical conditions,
the circular shaped shells show a strong bending instability and successive
fragmentation on Cartesian grids soon after their formation, while remain
almost unperturbed when simulated on polar meshes. We explain this by physical
Rayleigh--Taylor like instabilities triggered by corrugation of the dense shell
surfaces by numerical noise. Conditions for these instabilities follow from
both the shell structure itself and from episodes of transient acceleration
during re-establishing of dynamical pressure balance after sudden radiative
cooling onset. They are also easily excited by physical perturbations of the
ambient medium. The widely mentioned Non-linear Thin Shell Instability, in
contrast, in tests with physical perturbations is shown to have only limited
chances to develop in real radiative shocks, as it seems to require a special
spatial arrangement of fluctuations to be excited efficiently. The described
phenomena also set new requirements on further simulations of the radiatively
cooling shocks in order to be physically correct and free of numerical
artefacts.
