Kinetic layers and coupling conditions for nonlinear scalar equations on networks

TL;DR

This paper derives and compares kinetic and macroscopic coupling conditions for nonlinear scalar equations on networks, using asymptotic analysis and numerical validation to ensure consistency across models.

Contribution

It introduces a method to derive macroscopic coupling conditions from kinetic models on networks, combining kinetic half-space and Riemann problems.

Findings

Coupling conditions derived from kinetic models match macroscopic models in tripod networks.

Asymptotic analysis effectively links kinetic and macroscopic coupling conditions.

Numerical comparisons confirm the validity of the derived coupling conditions.

Abstract

We consider a kinetic relaxation model and an associated macroscopic scalar nonlinear hyperbolic equation on a network. Coupling conditions for the macroscopic equations are derived from the kinetic coupling conditions via an asymptotic analysis near the nodes of the network. This analysis leads to the combination of kinetic half-space problems with Riemann problems at the junction. Detailed numerical comparisons between the different models show the agreement of the coupling conditions for the case of tripod networks.