A spectral approach to interface layers on networks for the linearized BGK equation and its acoustic limit

TL;DR

This paper develops a spectral method to analyze interface layers on networks for the linearized BGK equation, deriving coupling conditions for macroscopic limits and demonstrating the approach's accuracy and efficiency.

Contribution

It introduces a spectral approach to study interface layers and coupling conditions for kinetic and macroscopic equations on networks, including viscous layers.

Findings

Spectral method effectively solves coupled kinetic half-space problems.

Derived coupling conditions accurately connect kinetic and macroscopic models.

Numerical results confirm the approach's accuracy and computational efficiency.

Abstract

We consider in this paper a velocity discretized version of the full linear kinetic BGK model and the corresponding limit for small Knudsen number, the linearised Euler or acoustic system. Considering these equations on networks, coupling conditions for the macroscopic equations are derived from the kinetic conditions via an asymptotic analysis near the nodes of the network. Here, a degeneracy in the limit equations requires not only the investigation of kinetic layers, but also the discussion of viscous layers. Using the kinetic coupling conditions at the junction and coupling kinetic and viscous layers to the outer problems on the edges one obtains a coupled kinetic half-space problem at each node. A spectral method is developed to solve this coupled kinetic half-space problems. This allows to obtain a detailed picture of the various interface layers near the nodes and to determine the relevant coefficients in the kinetic derived coupling conditions for the macroscopic equations.Numerical results show the accuracy and efficiency of the approach.