Magnetic reconnection is simulated with the help of covariant transport equations for the electro-magnetic field. They include an induction equation for the magnetic as well as the electric field. The transporting flow is a four-vector field (W4). The space components of this four-vector field are of stagnation flow type and the time-component vanishes at the reconnection site. The numerical method and necessary special numerical treatment at the line along which reconnection occurs is described in detail. The way in which the time-component (W0) of the transporting flow approaches zero at the reconnection line controls the type of non-ideal dynamics. Different profiles resulting in various current density distributions are studied. In reality the non-idealness necessary for reconnection may result from current-dependent micro-instabilities and therefore a current-dependent profile of W0 is included.