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

Raul Borsche, Tobias Damm, Axel Klar, Yizhou Zhou

TL;DR

The paper develops a rigorous framework to derive macroscopic coupling conditions for wave-type equations on networks from underlying kinetic BGK models. It introduces a discrete velocity/discrete-Maxwellian formulation and a spectral method to solve coupled kinetic half-space problems at network nodes, enabling accurate computation of extrapolation-length-type coupling coefficients. A central result is the well-posedness of the coupled half-space problem for symmetric, multi-edge nodes and the numerical extraction of coupling coefficients that align kinetic-network solutions with macroscopic network behavior. The approach is extended to unbounded velocity spaces using Hermite discretization, with consistent invariants and numerical evidence demonstrating fast convergence and high fidelity to kinetic dynamics. The work lays groundwork for extending to full BGK/Euler systems and viscous-layer analysis, with open avenues for higher-order asymptotics and convergence proofs, and it provides reproducible code and data for computing the coupling coefficients at nodes.

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.

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

TL;DR

The paper develops a rigorous framework to derive macroscopic coupling conditions for wave-type equations on networks from underlying kinetic BGK models. It introduces a discrete velocity/discrete-Maxwellian formulation and a spectral method to solve coupled kinetic half-space problems at network nodes, enabling accurate computation of extrapolation-length-type coupling coefficients. A central result is the well-posedness of the coupled half-space problem for symmetric, multi-edge nodes and the numerical extraction of coupling coefficients that align kinetic-network solutions with macroscopic network behavior. The approach is extended to unbounded velocity spaces using Hermite discretization, with consistent invariants and numerical evidence demonstrating fast convergence and high fidelity to kinetic dynamics. The work lays groundwork for extending to full BGK/Euler systems and viscous-layer analysis, with open avenues for higher-order asymptotics and convergence proofs, and it provides reproducible code and data for computing the coupling coefficients at nodes.

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.

Paper Structure

This paper contains 19 sections, 5 theorems, 148 equations, 6 figures.

Table of Contents

  1. Introduction
  2. A kinetic model equation and coupling conditions
  3. Kinetic layers at the nodes and coupling conditions for macroscopic equations
  4. The discrete velocity model
  5. The discrete layer problem
  6. The coupled half-space problems
  7. Well-posedness of the coupled half-space problem
  8. Numerical solution of the coupling problem
  9. The kinetic solution at the node
  10. Approximate coupling conditions
  11. Numerical results
  12. A kinetic model with unbounded velocity space
  13. Equations and coupling conditions
  14. The discrete velocity model
  15. The discrete layer problem and coupling conditions
  16. ...and 4 more sections

Key Result

Lemma 1

\newlabellemma10 $A_{22}$ is strictly hyperbolic, that means it is diagonalizable with real and distinct eigenvalues. Moreover, $N-1$ eigenvalues of $A_{22}$ are strictly positive. The remaining $N-1$ eigenvalues have the corresponding negative values. We denote the eigenvectors associated to posit

Figures (6)

  • Figure 1: Node connecting three edges and orientation of the edges.
  • Figure 1: Coefficient $\delta$ depending on $N$ for $n=3$ (left) and $n=\infty$ (right) using Gauss-Legendre polynomials and points. Associated increment depending on $N$. The black line denotes the limit value $\delta (\infty)$ of $\delta (N)$.
  • Figure 1: Coefficient $\delta$ depending on $N$ for $n=3$ (left) and $n=\infty$ (right) using Hermite polynomials and points. Associated increment depending on $N$. The black line denotes the limit value $\delta (\infty)$ of $\delta (N)$.
  • Figure 1: $\rho$ for all edges, kinetic solution for $\epsilon= 10^{-1}, \epsilon= 10^{-2}$ and $\epsilon= 5 \cdot 10^{-3}$ at time $t=0.1$ (left). Zoom to solution on edge 2 (right).
  • Figure 2: $\rho$ for edge $2$, kinetic solution for $\epsilon= 10^{-2}$ and $\epsilon= 5 \cdot 10^{-3}$ at time $t=0.1$, vertical zoom(left). Horizontal zoom to the kinetic layer near the node (right).
  • ...and 1 more figures

Theorems & Definitions (13)

  • Remark 1
  • Remark 2
  • Lemma 1
  • Remark 3
  • Theorem 1
  • Lemma 2
  • Lemma 3
  • Proof 1
  • Lemma 4
  • Proof 2
  • ...and 3 more