  We investigate the effect of numerical magnetic diffusion in
magnetohydrodynamic simulations of magnetically supported molecular clouds. To
this end, we have performed numerical studies on adaptive mesh isothermal
simulations of marginally sub-critical molecular clouds. We find that
simulations with low and intermediate resolutions collapse, contrary to what is
theoretically expected. However, the simulation with the highest numerical
resolution oscillates around an equilibrium state without collapsing. In order
to quantify the numerical diffusion of the magnetic field, we ran a second suit
of current-sheet simulations in which the numerical magnetic diffusion
coefficient can be directly measured, and computed the corresponding diffusion
times at various numerical resolutions. On this basis, we propose a criterion
for the resolution of magnetic fields in MHD simulations based on requiring
that the diffusion time to be larger than the characteristic timescale of the
physical process responsible for the dynamic evolution of the structure.
