What determines the formation and characteristics of protoplanetary
  discs?
  Planets form in protoplanetary discs. Their masses, distribution, and orbits
sensitively depend on the structure of the protoplanetary discs. However, what
sets the initial structure of the discs in terms of mass, radius and accretion
rate is still unknown. We perform non-ideal MHD numerical simulations using the
adaptive mesh refinement code Ramses, of a collapsing, one solar mass,
molecular core to study the disc formation and early, up to 100 kyr, evolution,
paying great attention to the impact of numerical resolution and accretion
scheme. We found that while the mass of the central object is almost
independent of the numerical parameters such as the resolution and the
accretion scheme onto the sink particle, the disc mass, and to a lower extent
its size, heavily depend on the accretion scheme, which we found, is itself
resolution dependent. This implies that the accretion onto the star and through
the disc are largely decoupled. For a relatively large domain of initial
conditions (except at low magnetisation), we found that the properties of the
disc do not change too significantly. In particular both the level of initial
rotation and turbulence do not influence the disc properties provide the core
is sufficiently magnetized. After a short relaxation phase, the disc settles in
a stationary state. It then slowly grows in size but not in mass. The disc
itself is weakly magnetized but its immediate surrounding is on the contrary
highly magnetized. Our results show that the disc properties directly depend on
the inner boundary condition, i.e. the accretion scheme onto the central
object, suggesting that the disc mass is eventually controlled by the small
scale accretion process, possibly the star-disc interaction. Because of
ambipolar diffusion and its significant resistivity, the disc diversity remains
limited and except for low magnetisation, their properties are (abridged).
