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Asymptotic Methods for Kinetic and Hyperbolic Evolution Equations on Networks

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Trails in Kinetic Theory

Part of the book series: SEMA SIMAI Springer Series ((SEMA SIMAI,volume 25))

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Abstract

We consider kinetic and associated macroscopic equations on networks. A general approach to derive coupling conditions for the macroscopic equations from coupling conditions of the underlying kinetic problem is presented using an asymptotic analysis near the nodes of the network. This analysis leads to the consideration of a fixpoint problem involving the coupled solutions of kinetic half-space problems. The procedure is explained for two simplified situations. The linear case is discussed for a linear kinetic BGK-type model leading in the macroscopic limit to a linear hyperbolic problem. The nonlinear situation is investigated for a kinetic relaxation model and an associated macroscopic scalar nonlinear hyperbolic conservation law on a network. Numerical comparisons between the solutions of the macroscopic equation with different coupling conditions and the kinetic solution are presented for the case of tripod and more complicated networks.

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Authors and Affiliations

  1. Department of Mathematics, TU Kaiserslautern, Kaiserslautern, Germany

    Raul Borsche & Axel Klar

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  1. Raul Borsche
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  2. Axel Klar
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Correspondence to Axel Klar .

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Editors and Affiliations

  1. Department of Computer Science, University of Verona, Verona, Italy

    Giacomo Albi

  2. Faculty of Mathematics, University of Vienna, Vienna, Austria

    Sara Merino-Aceituno

  3. Department of Information Engineering Computer Science and Mathematics, University of L’Aquila, L’Aquila, Italy

    Alessia Nota

  4. Department of Mathematics, University of Pavia, Pavia, Italy

    Mattia Zanella

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Borsche, R., Klar, A. (2021). Asymptotic Methods for Kinetic and Hyperbolic Evolution Equations on Networks. In: Albi, G., Merino-Aceituno, S., Nota, A., Zanella, M. (eds) Trails in Kinetic Theory. SEMA SIMAI Springer Series, vol 25. Springer, Cham. https://doi.org/10.1007/978-3-030-67104-4_2

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