  Especially in cold and high-density regions, the assumptions of ideal
magnetohydrodynamics (MHD) can break down, making first order non-ideal terms
such as Ohmic and ambipolar diffusion as well as the Hall effect important. In
this study we present a new numerical scheme for the first two resistive terms,
which we implement in the moving-mesh code AREPO using the single-fluid
approximation combined with a new gradient estimation technique based on a
least-squares fit per interface. Through various test calculations including
the diffusion of a magnetic peak, the structure of a magnetic C-shock, and the
damping of an Alfv\'en wave, we show that we can achieve an accuracy comparable
to the state-of-the-art code ATHENA++. We apply the scheme to the linear growth
of the magnetorotational instability and find good agreement with the
analytical growth rates. By simulating the collapse of a magnetised cloud with
constant magnetic diffusion, we show that the new scheme is stable even for
large density contrasts. Thanks to the Lagrangian nature of the moving mesh
method the new scheme is thus well suited for intended future applications
where a high resolution in the dense cores of collapsing protostellar clouds
needs to be achieved. In a forthcoming work we will extend the scheme to the
Hall effect.

Earth and Planetary Astrophysics;Instrumentation and Methods for Astrophysics