We present two-dimensional, numerical simulations of relativistic accretion onto a Schwarzschild black hole, including a density gradient in the initial conditions of the flow. Varying flow velocities, sound speeds, adiabatic indices and density gradients, we find that not only the morphology but also the stability of the flow is strongly dependent on the above parameters. More specifically, we find significant differences regarding the onset of the so-called flip-flop instability (which is associated with the shock cone in the rear part of the accretor) with respect to previous Newtonian studies.