  Over the past decade, the numerical modeling of the magnetic field evolution
in astrophysical scenarios has become an increasingly important field. In the
crystallized crust of neutron stars the evolution of the magnetic field is
governed by the Hall induction equation. In this equation the relative
contribution of the two terms (Hall term and Ohmic dissipation) varies
depending on the local conditions of temperature and magnetic field strength.
This results in the transition from the purely parabolic character of the
equations to the hyperbolic regime as the magnetic Reynolds number increases,
which presents severe numerical problems. Up to now, most attempts to study
this problem were based on spectral methods, but they failed in representing
the transition to large magnetic Reynolds numbers. We present a new code based
on upwind finite differences techniques that can handle situations with
arbitrary low magnetic diffusivity and it is suitable for studying the
formation of sharp current sheets during the evolution. The code is thoroughly
tested in different limits and used to illustrate the evolution of the crustal
magnetic field in a neutron star in some representative cases. Our code,
coupled to cooling codes, can be used to perform long-term simulations of the
magneto-thermal evolution of neutron stars.
