Kinetic layers and coupling conditions for macroscopic equations on networks I: the wave equation

TL;DR

This paper derives and analyzes coupling conditions for macroscopic wave equations on networks from kinetic models, introducing a new approximation method and validating it through numerical comparisons.

Contribution

It introduces a novel approach to derive macroscopic coupling conditions from kinetic models on networks, including a new approximation method for kinetic half-space problems.

Findings

The new method accurately approximates kinetic half-space solutions.

Derived coupling conditions match kinetic solutions well in numerical tests.

Applicable to complex network structures like tripods.

Abstract

We consider kinetic and associated macroscopic equations on networks. The general approach will be explained in this paper for a linear kinetic BGK model and the corresponding limit for small Knudsen number, which is the wave equation. Coupling conditions for the macroscopic equations are derived from the kinetic conditions via an asymptotic analysis near the nodes of the network. This analysis leads to the consideration of a fixpoint problem involving the coupled solutions of kinetic half-space problems. A new approximate method for the solution of kinetic half-space problems is derived and used for the determination of the coupling conditions. Numerical comparisons between the solutions of the macroscopic equation with different coupling conditions and the kinetic solution are presented for the case of tripod and more complicated networks.