  We provide rigorous and computable a-posteriori error estimates for first
order finite-volume approximations of nonlinear systems of hyperbolic
conservation laws in one spatial dimension. Our estimators rely on recent
stability results by Bressan, Chiri and Shen and a novel method to compute
negative order norms of residuals. Numerical experiments show that the error
estimator converges with the rate predicted by a-priori error estimates.
