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Error Estimates for Hyperbolic Scaling Limits of Linear Kinetic Models on Networks

Axel Klar, Yizhou Zhou

Abstract

This paper studies linear discrete kinetic models on networks and their asymptotic behavior in the small Knudsen number limit. For coupling conditions at an n-edge junction under a symmetric formulation, we introduce a change of variables that reformulates the system into n independent initial-boundary value problems. The asymptotic expansions are then constructed and rigorously justified by deriving an error estimate based on the energy method.

Error Estimates for Hyperbolic Scaling Limits of Linear Kinetic Models on Networks

Abstract

This paper studies linear discrete kinetic models on networks and their asymptotic behavior in the small Knudsen number limit. For coupling conditions at an n-edge junction under a symmetric formulation, we introduce a change of variables that reformulates the system into n independent initial-boundary value problems. The asymptotic expansions are then constructed and rigorously justified by deriving an error estimate based on the energy method.
Paper Structure (19 sections, 5 theorems, 160 equations, 1 figure, 1 table)

This paper contains 19 sections, 5 theorems, 160 equations, 1 figure, 1 table.

Table of Contents

  1. Introduction
  2. Kinetic model
  3. Moment systems
  4. Moment system with first type of collision term
  5. Moment system with second type of collision term
  6. Coupling conditions for moment equations
  7. Asymptotic expansion
  8. First-type collision term $Q_1$: IBVP I
  9. First-type collision term $Q_1$: IBVP II
  10. Second-type collision term $Q_2$: IBVP I
  11. Second-type collision term $Q_2$: IBVP II
  12. Error Estimate
  13. First-type collision term $Q_1$, IBVP I
  14. First-type collision term $Q_1$, IBVP II
  15. Conclusion for the first-type collision term $Q_1$
  16. ...and 4 more sections

Key Result

Lemma 1

\newlabelS0 Let $\{\phi_k(v)~|~k=1,2,...\}$ be the orthogonal Hermitian polynomials on $\mathbb{R}$ with the weight function $M(v)$. Let $-v_N,\cdots,-v_1,v_1,\cdots,v_N$ be roots of $\phi_{2N}(v)=0$. Then we have

Figures (1)

  • Figure 1: Node connecting $n$ edges.

Theorems & Definitions (15)

  • Lemma 1: BDKZ1
  • Lemma 1
  • Proof 1
  • Lemma 2
  • Proof 2
  • Remark 1
  • Remark 2
  • Proposition 1
  • Remark 3
  • Remark 4
  • ...and 5 more