  We study the amplification of magnetic fields in the collapse and the
post-bounce evolution of the core of a non-rotating star of 15 solar masses in
axisymmetry. To this end, we solve the coupled equations of
magnetohydrodynamics and neutrino transport in the two-moment approximation.
The pre-collapse magnetic field is strongly amplified by compression in the
infall. Initial fields of the order of 1010 G translate into proto-neutron star
fields similar to the ones observed in pulsars, while stronger initial fields
yield magnetar-like final field strengths. After core bounce, the field is
advected through the hydrodynamically unstable neutrino-heating layer, where
non-radial flows due to convection and the standing accretion shock instability
amplify the field further. Consequently, the resulting amplification factor of
order five is the result of the number of small-eddy turnovers taking place
within the time scale of advection through the post-shock layer. Due to this
limit, most of our models do not reach equipartition between kinetic and
magnetic energy and, consequently, evolve similarly to the non-magnetic case,
exploding after about 800 ms when a single or few high-entropy bubbles persist
over several dynamical time scales. In the model with the strongest initial
field we studied, \$10^{12}\$ G, for which equipartition between flow and field
is achieved, the magnetic tension favours a much earlier development of such
long-lived high-entropy bubbles and enforces a fairly ordered large-scale flow
pattern. Consequently, this model, after exhibiting very regular shock
oscillations, explodes much earlier than non-magnetic ones.
