  In real-time lattice simulations of cosmic strings in the Abelian Higgs
model, the broken translational invariance introduces lattice artefacts;
relativistic strings therefore decelerate and radiate. We introduce two
different methods to construct a moving string on the lattice, and study in
detail the lattice effects on moving strings. We find that there are two types
of lattice artefact: there is an effective maximum speed with which a moving
string can be placed on the lattice, and a moving string also slows down, with
the deceleration approximately proportional to the exponential of the velocity.
To mitigate this, we introduce and study an improved discretisation, based on
the tree-level L\"{u}scher-Weisz action, which is found to reduce the
deceleration by an order of magnitude, and to increase the string speed limit
by an amount equivalent to halving the lattice spacing. The improved algorithm
is expected to be very useful for 3D simulations of cosmic strings in the early
universe, where one wishes to simulate as large a volume as possible.
