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

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.

Figures (8)

• Figure 1: Node connecting three edges and orientation of the edges.
• Figure 1: Coefficients $\delta_1$ and $\delta_2$ depending on $N$ for $n=3$. Associated error depending on $N$.
• Figure 1: Testcase 1: $\rho$ for all edges, kinetic solution for $\epsilon= 5 \cdot 10^{-4}$ at time $t=0.1$ (left). Zoom to solution on edge 2 (right).
• Figure 2: Coefficients $\delta_1$ and $\delta_2$ depending on $N$ for $n=\infty$. Associated error depending on $N$.
• Figure 2: Testcase2: $\rho$ for all edges, kinetic solution for $\epsilon= \cdot 10^{-4}$ at time $t=0.1$ (left). Zoom to solution on edge 2 (right).

Theorems & Definitions (4)

• Remark 1
• Remark 2
• Lemma 1
• Remark 3