  The main method for calibrating the luminosity at Large Hadron Collider (LHC)
is van der Meer scan where the beams are swept transversely across each other.
This beautiful method was invented in 1968. Despite the honourable age, it
remains the preferable tool at hadron colliders. It delivers the lowest
calibration systematics, which still often dominates the overall luminosity
uncertainty at LHC experiments. Various details of the method are discussed in
the paper. One of the main factors limiting proton-proton van der Meer scan
accuracy is the beam-beam electromagnetic interaction. It modifies the shapes
of the colliding bunches and biases the measured luminosity. In the first years
of operation, four main LHC experiments did not attempt to correct the bias
because of its complexity. In 2012 a correction method was proposed and then
subsequently used by all experiments. It was based, however, on a simplified
linear approximation of the beam-beam force and, therefore, had limited
accuracy. In this paper, a new simulation is presented, which takes into
account the exact non-linear force. Depending on the beam parameters, the
results of the new and old methods differ by $\sim1\%$. This needs to be
propagated to all LHC cross-section measurements after 2012. The new simulation
is going to be used at LHC in future luminosity calibrations.
