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

Abstract: 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.