  We study the Richtmyer--Meshkov (RM) instability of a relativistic perfect
fluid by means of high order numerical simulations with adaptive mesh
refinement (AMR). The numerical scheme adopts a finite volume Weighted
Essentially Non-Oscillatory (WENO) reconstruction to increase accuracy in
space, a local space-time discontinuous Galerkin predictor method to obtain
high order of accuracy in time and a high order one-step time update scheme
together with a "cell-by-cell" space-time AMR strategy with time-accurate local
time stepping. In this way, third order accurate (both in space and in time)
numerical simulations of the RM instability are performed, spanning a wide
parameter space. We present results both for the case in which a light fluid
penetrates into a higher density one (Atwood number $A>0$), and for the case in
which a heavy fluid penetrates into a lower density one (Atwood number $A<0$).
We find that, for large Lorentz factors \gamma_s of the incident shock wave,
the relativistic RM instability is substantially weakened and ultimately
suppressed. More specifically, the growth rate of the RM instability in the
linear phase has a local maximum which occurs at a critical value of \gamma_s ~
[1.2,2]. Moreover, we have also revealed a genuine relativistic effect, absent
in Newtonian hydrodynamics, which arises in three dimensional configurations
with a non-zero velocity component tangent to the incident shock front. In this
case, the RM instability is strongly affected, typically resulting in less
efficient mixing of the fluid.
