  We present in this paper both a linear study and numerical relativistic MHD
simulations of the non-resonant streaming instability occurring in the
precursor of relativistic shocks. In the shock front restframe, we perform a
linear analysis of this instability in a likely configuration for
ultra-relativistic shock precursors. This considers magneto-acoustic waves
having a wave vector perpendicular to the shock front and the large scale
magnetic field. Our linear analysis is achieved without any assumption on the
shock velocity and is thus valid for all velocity regimes. In order to check
our calculation, we also perform relativistic MHD simulations describing the
propagation of the aforementioned magneto-acoustic waves through the shock
precursor. The numerical calculations confirm our linear analysis, which
predicts that the growth rate of the instability is maximal for
ultra-relativistic shocks and exhibits a wavenumber dependence $\propto
k_x^{1/2}$. Our numerical simulations also depict the saturation regime of the
instability where we show that the magnetic amplification is moderate but
nevertheless significant ($\delta B/B\leq 10$). This latter fact may explain
the presence of strong turbulence in the vicinity of relativistic magnetized
shocks. Our numerical approach also introduces a convenient means to handle
isothermal (ultra-)relativistic MHD conditions.
