Interface layers and coupling conditions for discrete kinetic models on networks: a spectral approac

TL;DR

This paper develops a spectral method to derive and analyze coupling conditions for macroscopic wave equations from kinetic models on networks, demonstrating accuracy and efficiency through numerical validation.

Contribution

It introduces a spectral approach to determine coupling coefficients for macroscopic equations from kinetic network models, bridging kinetic and macroscopic descriptions.

Findings

Spectral method accurately computes coupling coefficients.

Fast convergence of the numerical approach.

Good agreement between kinetic and macroscopic solutions.

Abstract

We consider kinetic and related macroscopic equations on networks. A class of linear kinetic BGK models is considered, where the limit equation for small Knudsen numbers is given by the wave equation. Coupling conditions for the macroscopic equations are obtained from the kinetic coupling conditions via an asymptotic analysis near the nodes of the network and the consideration of coupled solutions of kinetic half-space problems. Analytical results are obtained for a discrete velocity version of the coupled half-space problems. Moreover, an efficient spectral method is developed to solve the coupled discrete velocity half-space problems. In particular, this allows to determine the relevant coefficients in the coupling conditions for the macroscopic equations from the underlying kinetic network problem. These coefficients correspond to the so-called extrapolation length for kinetic boundary value problems. Numerical results show the accuracy and fast convergence of the approach. Moreover, a comparison of the kinetic solution on the network with the macroscopic solution is presented.