In this paper, we develop reliable a posteriori error estimates for numerical approximations of scalar hyperbolic conservation laws in one space dimension. Our methods have no inherent small-data limitations and are a step towards error control of numerical schemes for systems. We are careful not to appeal to the Kruzhkov theory for scalar conservation laws. Instead, we derive novel quantitative stability estimates that extend the theory of shifts, and in particular, the framework for proving stability first developed by the second author and Vasseur. This is the first time this methodology has been used for quantitative estimates. We work entirely within the context of the theory of shifts and $a$-contraction, techniques which adapt well to systems. In fact, the stability framework by the second author and Vasseur has itself recently been pushed to systems [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 in MATLAB and numerical experiments.

翻译：本文针对一维标量双曲守恒律的数值近似，发展了可靠的后验误差估计。我们的方法不存在固有的小数据限制，是向系统数值格式误差控制迈出的一步。我们审慎避免诉诸标量守恒律的Kruzhkov理论，而是推导出新的定量稳定性估计，扩展了移位理论，特别是第二作者与Vasseur首次建立的稳定性框架。这是该方法首次用于定量估计。我们完全在移位理论与$a$-压缩框架内开展工作，该技术对系统具有良好的适应性。事实上，第二作者与Vasseur的稳定性框架近期已推广至系统[Chen-Krupa-Vasseur. $2\times 2$守恒律的唯一性与弱BV稳定性. Arch. Ration. Mech. Anal., 246(1):299--332, 2022]。我们的理论发现通过MATLAB数值实现与数值实验得到了补充。