AbstractIt is commonly, but erroneously, assumed that the best way to treat upwind (closed) boundaries in numerical approximations of hyperbolic equations consists in a literal transcription, letting the numerical value be equal to the prescribed value. This results in a total reflection of spurious solutions that may arrive at the boundary from the computing domain. Those reflected solutions cannot be distinguished from consistent solutions, and they may seriously degrade the overall accuracy. We show that modifications of this treatment of the boundary may result in the absorption of spurious solutions. The effect of the absorbing properties of these boundary schemes is analyzed in Fourier space. We also analyze their numerical stability properties, and their effect on the accuracy of solutions generated in response to a time dependent boundary condition.