  We present numerical simulations of cold, axisymmetric, magnetically driven
relativistic outflows. The outflows are initially sub-Alfv\'enic and Poynting
flux-dominated, with total--to--rest-mass energy flux ratio up to $\mu \sim
620$. To study the magnetic acceleration of jets we simulate flows confined
within a funnel with rigid wall of prescribed shape, which we take to be
$z\propto r^a$ (in cylindrical coordinates, with $a$ ranging from 1 to 2). This
allows us to eliminate the numerical dissipative effects induced by a free
boundary with an ambient medium. We find that in all cases they converge to a
steady state characterized by a spatially extended acceleration region. For the
jet solutions the acceleration process is very efficient - on the outermost
scale of the simulation more than half of the Poynting flux has been converted
into kinetic energy flux, and the terminal Lorentz factor approached its
maximum possible value ($\Gamma_\infty \simeq \mu$). The acceleration is
accompanied by the collimation of magnetic field lines in excess of that
dictated by the funnel shape. The numerical solutions are generally consistent
with the semi-analytic self-similar jets solutions and the spatially extended
acceleration observed in some astrophysical relativistic jets. In agreement
with previous studies we also find that the acceleration is significantly less
effective for wind solutions suggesting that pulsar winds may remain Poynting
dominated when they reach the termination shock.
