  We study the impact of an Upwind scheme on the numerical convergence of
simulations of the Hall and Ohmic effect in neutron stars crusts. While
simulations of these effects have explored a variety of geometries and wide
ranges of physical parameters, they are limited to relatively low values of the
Hall parameter, playing the role of the magnetic Reynolds number, which should
be not exceed a few hundred for numerical convergence.
  We study the evolution of the magnetic field in a plane-parallel Cartesian
geometry. We discretise the induction equation using a finite difference scheme
and then integrate it via the Euler forward method. Two different approaches
are used for the integration of the advective terms appearing in the equation:
a Forward Time and Central in Space (FTCS) and an Upwind scheme. We compare
them in terms of accuracy and performance. We explore the impact of the Upwind
method on convergence according to the ratio of planar to vertical field and
the Hall parameter. In the limit of a low strength planar field the use of an
Upwind scheme provides a vast improvement leading to the convergence of
simulations where the Hall parameter is 2 orders of magnitude higher than that
of the FTCS. Upwind is still better if the planar field is stronger, yet, the
difference of the maximum value of the Hall parameter reached is within a
factor of 10 or a few. Moreover, we notice if the schemes diverge their
behaviour is very different, with FTCS producing infinite energy, while the
Upwind scheme only temporarily increasing the overall magnetic field energy.
  Overall, the Upwind scheme enhances the efficiency of the simulations
allowing the exploration of environments with higher value of electric
conductivity getting us closer than before to realistic environmental
conditions of magnetars.
