  We describe a novel Godunov-type numerical method for solving the equations
of resistive relativistic magnetohydrodynamics. In the proposed approach, the
spatial components of both magnetic and electric fields are located at zone
interfaces and are evolved using the constrained transport formalism. Direct
application of Stokes' theorem to Faraday's and Ampere's laws ensures that the
resulting discretization is divergence-free for the magnetic field and
charge-conserving for the electric field. Hydrodynamic variables retain,
instead, the usual zone-centred representation commonly adopted in
finite-volume schemes. Temporal discretization is based on Runge-Kutta
implicit-explicit (IMEX) schemes in order to resolve the temporal scale
disparity introduced by the stiff source term in Ampere's law. The implicit
step is accomplished by means of an improved and more efficient Newton-Broyden
multidimensional root-finding algorithm. The explicit step relies on a
multidimensional Riemann solver to compute the line-averaged electric and
magnetic fields at zone edges and it employs a one-dimensional Riemann solver
at zone interfaces to update zone-centred hydrodynamic quantities. For the
latter, we introduce a five-wave solver based on the frozen limit of the
relaxation system whereby the solution to the Riemann problem can be decomposed
into an outer Maxwell solver and an inner hydrodynamic solver. A number of
numerical benchmarks demonstrate that our method is superior in stability and
robustness to the more popular charge-conserving divergence cleaning approach
where both primary electric and magnetic fields are zone-centered. In addition,
the employment of a less diffusive Riemann solver noticeably improves the
accuracy of the computations.
