New coupling conditions for isentropic flow on networks

Yannick Holle; Michael Herty; Michael Westdickenberg

doi:10.3934/nhm.2020016

Article Contents

2020, Volume 15, Issue 4: 605-631. Doi: 10.3934/nhm.2020016

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New coupling conditions for isentropic flow on networks

1.
Institut für Mathematik, RWTH Aachen University, Templergraben 55, 52062 Aachen, Germany
2.
Institut für Geometrie und Praktische Mathematik, RWTH Aachen University, Templergraben 55, 52062 Aachen, Germany

^* Corresponding author: Yannick Holle
^* Corresponding author: Yannick Holle

Received: April 2020

Early access: July 2020

Published: December 2020

This work has been funded by the Deutsche Forschungsgemeinschaft (DFG, German Research Foundation) projects 320021702/GRK2326 Energy, Entropy, and Dissipative Dynamics (EDDy) and HE5386/18, 19.

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Abstract

We introduce new coupling conditions for isentropic flow on networks based on an artificial density at the junction. The new coupling conditions can be derived from a kinetic model by imposing a condition on energy dissipation. Existence and uniqueness of solutions to the generalized Riemann and Cauchy problem are proven. The result for the generalized Riemann problem is globally in state space. Furthermore, non-increasing energy at the junction and a maximum principle are proven. A numerical example is given in which the new conditions are the only known conditions leading to the physically correct wave types. The approach generalizes to full gas dynamics.

Keywords:

Mathematics Subject Classification: 35L65, 76N15, 82C40.

Citation:

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Figure 3. The boundary Riemann set $ \mathcal V(\rho_*, 0) $

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Figure 1. Construction of $ (\rho_\alpha, u_\alpha) $ and $ (\rho_\beta, u_\beta) $

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Figure 2. The sets $ \mathcal A_* $, $ \mathcal B_* $, $ \mathcal C_* $ and $ \mathcal J_* $

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Figure 4. Level sets for different coupling conditions

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Table 1. Initial data

pipeline	$ \hat\rho_{k} $	$ \hat\rho_{k} \hat u_{k} $
1	$ +1.0000 $	$ -1.0000 $
2	$ +1.0000 $	$ +0.5000 $
3	$ +1.0000 $	$ +0.5000 $

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Table 2. Numerical results

	Equal density		Equal momentum flux		Equal stagnation enthalpy		Equal artificial density
pipeline	$ \bar\rho_{k} $	$ \bar\rho_{k} \bar u_{k} $	$ \bar\rho_{k} $	$ \bar\rho_{k} \bar u_{k} $	$ \bar\rho_{k} $	$ \bar\rho_{k} \bar u_{k} $	$ \bar\rho_{k} $	$ \bar\rho_{k} \bar u_{k} $
1	$ +1.0000 $	$ -1.0000 $	$ +0.8964 $	$ -1.1981 $	$ +0.8518 $	$ -1.2670 $	$ +1.1776 $	$ -0.5417 $
2	$ +1.0000 $	$ +0.5000 $	$ +1.0266 $	$ +0.5991 $	$ +1.0356 $	$ +0.6335 $	$ +0.9346 $	$ +0.2708 $
3	$ +1.0000 $	$ +0.5000 $	$ +1.0266 $	$ +0.5991 $	$ +1.0356 $	$ +0.6335 $	$ +0.9346 $	$ +0.2708 $
Energy dissipation	$ -7.5000\times 10^{-2} $		$ -1.725\times 10^{-2} $		$ \approx 0 $		$ -1.3852\times 10^{-1} $

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Table 3. Wave types

pipeline	Equal density	Equal momentum flux	Equal stagnation enthalpy	Equal artificial density
1	no waves	rarefaction wave	rarefaction wave	shock
2	no waves	shock	shock	rarefaction wave
3	no waves	shock	shock	rarefaction wave

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