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In this paper, we develop reliable a posteriori error estimates for numericalapproximations of scalar hyperbolic conservation laws in one space dimension.  Our methods have no inherent small-data limitations and are a step towardserror control of numerical schemes for systems. We are careful not to appeal tothe Kruzhkov theory for scalar conservation laws. Instead, we derive novelquantitative stability estimates that extend the theory of shifts, and inparticular, the framework for proving stability first developed by the secondauthor and Vasseur. This is the first time this methodology has been used forquantitative estimates.  We work entirely within the context of the theory of shifts and$a$-contraction, techniques which adapt well to systems. In fact, the stabilityframework by the second author and Vasseur has itself recently been pushed tosystems [Chen-Krupa-Vasseur. Uniqueness and weak-BV stability for $2\times 2$conservation laws. Arch. Ration. Mech. Anal., 246(1):299--332, 2022].  Our theoretical findings are complemented by a numerical implementation inMATLAB and numerical experiments.

Theory of shifts, shocks, and the intimate connections to $L^2$-type a posteriori error analysis of numerical schemes for hyperbolic problems