In this paper, we develop a posteriori error estimates for numerical approximations of scalar hyperbolic conservation laws in one space dimension. We develop novel quantitative partially $L^2$-type estimates by using the theory of shifts, and in particular, the framework for proving stability first developed in [Krupa-Vasseur. On uniqueness of solutions to conservation laws verifying a single entropy condition. J. Hyperbolic Differ. Equ., 2019]. In this paper, we solve two of the major obstacles to using the theory of shifts for quantitative estimates, including the change-of-variables problem and the loss of control on small shocks. Our methods have no inherit small-data limitations. Thus, our hope is to apply our techniques to the systems case to understand the numerical stability of large data. There is hope for our results to generalize to systems: the stability framework [Krupa-Vasseur. On uniqueness of solutions to conservation laws verifying a single entropy condition. J. Hyperbolic Differ. Equ., 2019] itself has been generalized 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]. Moreover, we are careful not to appeal to the Kruzhkov theory for scalar conservation laws. Instead, we work entirely within the context of the theory of shifts and $a$-contraction -- and these theories apply equally to systems. We present a MATLAB numerical implementation and numerical experiments. We also provide a brief introduction to the theory of shifts and $a$-contraction.

翻译：本文针对一维标量双曲守恒律的数值近似发展了后验误差估计。通过运用平移理论，特别是[Krupa-Vasseur. On uniqueness of solutions to conservation laws verifying a single entropy condition. J. Hyperbolic Differ. Equ., 2019]中首次建立的稳定性证明框架，我们提出了新颖的定量部分$L^2$型估计。本文解决了将平移理论应用于定量估计的两个主要障碍：变量替换问题和小激波控制失效问题。我们的方法不存在固有小数据限制，因此有望将技术应用于方程组情形以理解大数据的数值稳定性。该结果具有向方程组推广的潜力：稳定性框架[Krupa-Vasseur. On uniqueness of solutions to conservation laws verifying a single entropy condition. J. Hyperbolic Differ. Equ., 2019]本身已推广至方程组[Chen-Krupa-Vasseur. Uniqueness and weak-BV stability for $2\times 2$ conservation laws. Arch. Ration. Mech. Anal., 246(1):299--332, 2022]。此外，我们刻意避免调用标量守恒律的Kruzhkov理论，而是完全在平移理论和$a$-压缩的框架下开展工作——这些理论同样适用于方程组。我们给出了MATLAB数值实现与数值实验，并对平移理论和$a$-压缩进行了简要介绍。