Kolmogorov $\varepsilon$-entropy of numerical solutions for scalar conservation laws with convex flux

TL;DR

This paper establishes quantitative compactness estimates for numerical solutions of scalar conservation laws with convex flux, demonstrating that certain finite-difference schemes preserve the Kolmogorov $\varepsilon$-entropy scaling of exact solutions.

Contribution

It proves that conservative, monotone finite-difference schemes satisfying a discrete OSLC preserve the $1/ ext{ extsterling}\varepsilon$ Kolmogorov entropy scaling, matching known bounds.

Findings

Numerical solutions preserve the entropy scaling under specific grid constraints.

Discrete OSLC ensures the upper bound of the Kolmogorov entropy.

A uniform approximation argument establishes the lower bound.

Abstract

Building on the information-theoretic perspective of P.~D.~Lax [\textit{Proc.\ Sympos., Math.\ Res.\ Center, Univ.\ Wisconsin}, 1978], we establish a two-sided quantitative compactness estimate for numerical solutions of scalar conservation laws with a uniformly convex flux, expressed in terms of Kolmogorov $\varepsilon$-entropy. We prove that, under specific grid constraints, conservative, monotone finite-difference schemes satisfying a discrete one-sided Lipschitz condition (OSLC) preserve the $1/\varepsilon$ Kolmogorov entropy scaling of the corresponding exact entropy solution set, matching the bounds obtained by De~Lellis and Golse [\textit{Comm.\ Pure Appl.\ Math.}\ \textbf{58} (2005)] and by Ancona, Glass, and Nguyen [\textit{Comm.\ Pure Appl.\ Math.}\ \textbf{65} (2012)]. Specifically, the upper bound follows from the discrete OSLC, while the lower bound relies on a uniform approximation argument on a bounded-variation precursor class. Our results show that prototypical first-order methods are high-resolution in Lax's sense. Finally, we abstract the lower bound mechanism into a general transfer principle, discuss implications for information recovery via post-processing, and indicate directions for future work.