The Follow-the-Leader scheme with non-monotone velocity

Marco Di Francesco

Abstract

This paper addresses the approximation of entropy solutions to a one-dimensional scalar conservation law via the so-called Follow-the-Leader particle scheme in case the velocity map is \emph{non monotone}. The result is based on a discrete maximum principle, on $BV$ estimates of the approximated density, and on the characterization of the limit as entropy solution in the sense of Kru\v zkov.

Disclosure

“8W4, Nonlinear Evolutions PDEs, fluid dynamics and transport equations: theoretical foundations and applica- tions. The author is also partially supported by the InterMaths Network, www.intermaths.eu. The author acknowledges the use of the AI assistant Claude (Anthropic) during the early stage of this project, in identifying the connection between the follow-the-leader scheme and Godunov-type numerical Hamiltonians for Hamilton–Jacobi equations, and in locating the relevant references [3”
PDF page 15

Classification: Brainstorming or outlining
Multiplier: 2
Verified: 2026-08-23T19:13:42.374087Z

Structural counts

Pages 16 pdf

Theorems 4 source

Lemmas 2 source

Propositions 1 source

Corollaries 0 source

Definitions 1 source

Displayed equations 86 source

Bibliography entries 29 source

Appendix pages 2 estimated

Count notes

Source counts use the expanded primary TeX file ftl_non_monotone_MDF.tex.
Appendix pages include the first PDF page with an explicit Appendix heading through the final page.