Aims. We investigate the use of some high-resolution shock-capturing schemes on curvilinear grids in the context of general relativistic hydrodynamics (GRHD). We aim to demonstrate that these can be used to evolve accurately fluid flow onto a black hole. Methods. We describe a numerical scheme which applies high-resolution shock-capturing schemes and the curvilinear overlapping grids methodology to the evolution of the equations of GRHD. Results. We apply our scheme to the problem of Bondi-Hoyle-Lyttleton accretion onto a black hole.We validate our approach against an exact solution of the problem and against previous numerical results. Our approach allows for the incident wind to be at any angle to the spin-axis of the Kerr black hole, and also allows the flow density to be perturbed upstream. We give an illustration of the effects of these perturbations on the resulting flow-field.