Systematic construction of upwind constrained transport schemes for MHD
The constrained transport (CT) method reflects the state of the art numerical technique for preserving the divergence-free condition of magnetic field to machine accuracy in multi-dimensional MHD simulations performed with Godunov-type, or upwind, conservative codes. The evolution of the different magnetic field components, located at zone interfaces using a staggered representation, is achieved by calculating the electric field components at cell edges, in a way that has to be consistent with the Riemann solver used for the update of cell-centered fluid quantities at interfaces. Albeit several approaches have been undertaken, the purpose of this work is, on the one hand, to compare existing methods in terms of robustness and accuracy and, on the other, to extend the \emph{upwind contrained transport} (UCT) method by Londrillo \& Del Zanna (2004) and Del Zanna et al. (2007) for the systematic construction of new averaging schemes. In particular, we propose a general formula for the upwind fluxes of the induction equation which simply involves the information available from the base Riemann solver employed for the fluid part, provided it does not require full spectral decomposition, and 1D reconstructions of velocity and magnetic field components from nearby intercell faces to cell edges. Our results are presented here in the context of second-order schemes for classical MHD, but they can be easily generalized to higher than second order schemes, either based on finite volumes or finite differences, and to other physical systems retaining the same structure of the equations, such as that of relativistic or general relativistic MHD.