Non-ideal magnetohydrodynamics on a moving mesh I: Ohmic and ambipolar diffusion

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.