Interface layers and coupling conditions on networks for linearized kinetic BGK equation

Ikrom Akramov; Raul Borsche; Nils Eckhard; Axel Klar

doi:10.3934/krm.2023029

Article Contents

2024, Volume 17, Issue 3: 468-488. Doi: 10.3934/krm.2023029

This issue Previous Article Low–dimensional reduction of the non-Abelian quantum synchronization models on the unitary group Next Article Well-posedness and singularity formation for Vlasov–Riesz system

Interface layers and coupling conditions on networks for linearized kinetic BGK equation

1.
TU Hamburg, Department of Mathematics, Germany
2.
RPTU Kaiserslautern, Department of Mathematics, Germany
3.
Fraunhofer ITWM Kaiserslautern, Germany

^*Corresponding author: Axel Klar
^*Corresponding author: Axel Klar

Received: January 2023

Revised: June 2023

Early access: July 24, 2023

Published online: July 24, 2023

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Abstract

We consider a linearized kinetic BGK equation and the associated acoustic system on a network. 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 solutions of kinetic half-space problems. This work extends the procedure developed in [15], where coupling conditions for a simplified BGK model have been derived. Numerical comparisons between different coupling conditions confirm the accuracy of the proposed approximation.

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Mathematics Subject Classification: Primary: 82C40, 65M08; Secondary: 35R02.

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Figure 1. Node connecting three edges and orientation of the edges

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Figure 2. Kinetic equation and half-moment approximation for different relations of $ \Delta = \frac{\epsilon}{T} $. $ \rho $ on edge $ 2 $ of the tripod. Upper row: $ \Delta = 0.001 $ (left) and $ \Delta = 0.01 $ (right). Lower row: $ \Delta = 0.1 $ (left) and $ \Delta = 1.0 $ (right)

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Figure 3. Kinetic equation and wave equation with coupling conditions given by the Maxwell/half-moment approach and by the equality of $ S $. $ \rho,q,S $ on edge $ 2 $ (left) and edge $ 3 $ (right) at time $ T = 1 $. The values at the nodes obtained from the coupling conditions are denoted by a cross

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Figure 4. Diamond network

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Figure 5. Kinetic equation and wave equation with coupling conditions given by the Maxwell/half-moment approach and by the equality of $ S $. $ \rho $ on edge $ 2 $ (left) and edge $ 4 $ (right) at time $ T = 1 $

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Table 1. Comparison of results with the different approximations

	Numerical solver	Maxwell	Half-moment	Maxw./Num.	Half/Num.
$\rho_{\infty}$	-2.41561	-1.59617	-2.23425	33.9%	7.5%
$S_{\infty}$	1.71757	1.59617	1.68775	7.1%	1.7%

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