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.

Figures (14)

• Figure 1: The function $\bar{u}^0$ is divided into rapidly decreasing parts and nearly non-decreasing parts. In $\hat{u}(\cdot,0)=\hat{\psi}(\cdot,0)$, we differentiate between three different types of shocks: large shocks, front tracking shocks, and boundary shocks.
• Figure 2: The philosophy of the present paper is to control $\hat{\psi}-\bar{u}$ via the $L^2$ stability we gain via shifting, while on the other hand measuring $\hat{\psi}-\hat{u}$ via the control on the shifts we get due to dissipation.
• Figure 3: To ensure that our analysis is applicable to a function $\hat{u}$ that is defined using $\hat{v}_i$ globally constant (as opposed to the global extensions from \ref{['ext_section']}), we need to ensure the level set of $\hat{\Lambda}$ is not too far away from the corresponding characteristic of $v_i$ (and similarly with $\hat{P}$).
• Figure 4: An illustration showing $h_i,h_j,\hat{h}_i,\hat{h}_j,\mathfrak{h}_{\tilde{L}_i(t),\hat{R}_i(t-)}$, and $\mathfrak{h}_{\hat{L}_j(t-),\tilde{R}_j(t)}$. We also show the error interval around $\hat{h}_i$ and $\mathfrak{h}_{\tilde{L}_i(t),\hat{R}_i(t-)}$ which contains $h_i$, as well as the error interval around $\hat{h}_j$ and $\mathfrak{h}_{\hat{L}_j(t-),\tilde{R}_j(t)}$ which contains $h_j$. Note the green X marks where the two error intervals stop overlapping, and thus the first time we can conclude that the $h_i$ and $h_j$ must have merged.
• Figure 5: The two "boundary" shocks, here denoted by $\hat{h}_{*}$ and $\hat{h}_{**}$, are separated by a Lipschitz-continuous piece (originating from a nearly non-decreasing part of $\bar{u}^0$), and are about to collide in $\hat{u}$.

Theorems & Definitions (30)

• Definition 2.1: Strong Trace Property
• Theorem 2.2: Main Theorem
• Remark 1
• Remark 2
• Remark 3
• Remark 4
• Remark 5
• Remark 6
• Remark 7
• Remark 8