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December  2020, 15(4): 605-631. doi: 10.3934/nhm.2020016

New coupling conditions for isentropic flow on networks

Yannick Holle 1,, , Michael Herty 2, and  Michael Westdickenberg 1,

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

Received  April 2020 Published  December 2020 Early access  July 2020

Fund Project: 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.

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: Hyperbolic conservation laws, networks, coupling condition, isentropic gas dynamics, kinetic model, maximum entropy dissipation.
Mathematics Subject Classification: 35L65, 76N15, 82C40.
Citation: Yannick Holle, Michael Herty, Michael Westdickenberg. New coupling conditions for isentropic flow on networks. Networks & Heterogeneous Media, 2020, 15 (4) : 605-631. doi: 10.3934/nhm.2020016
References:
[1]

M. K. Banda, M. Herty and A. Klar, Coupling conditions for gas networks governed by the isothermal Euler equations, Netw. Heterog. Media, 1 (2006), 295-314.  doi: 10.3934/nhm.2006.1.295.  Google Scholar

[2]

M. K. Banda, M. Herty and A. Klar, Gas flow in pipeline networks, Netw. Heterog. Media, 1 (2006), 41-46.  doi: 10.3934/nhm.2006.1.41.  Google Scholar

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F. Berthelin and F. Bouchut, Solution with finite energy to a BGK system relaxing to isentropic gas dynamics, Ann. Fac. Sci. Toulouse Math., 9 (2000), 605-630.  doi: 10.5802/afst.974.  Google Scholar

[4]

F. Berthelin and F. Bouchut, Kinetic invariant domains and relaxation limit from a BGK model to isentropic gas dynamics, Asymtotic Anal., 31 (2002), 153-176.   Google Scholar

[5]

F. Berthelin and F. Bouchut, Weak entropy boundary conditions for isentropic gas dynamics via kinetic relaxation, J. Differential Equations, 185 (2002), 251-270.  doi: 10.1006/jdeq.2001.4161.  Google Scholar

[6]

F. Bouchut, Construction of BGK models with a family of kinetic entropies for a given system of conservation laws, J. Statist. Phys., 95 (1999), 113-170.  doi: 10.1023/A:1004525427365.  Google Scholar

[7]

A. Bressan, The One-Dimensional Cauchy Problem, in Hyperbolic Systems of Conservation Laws, vol. 20 of Oxford Lecture Series in Mathematics and its Application, Oxford University Press, Oxford, 2000.  Google Scholar

[8]

A. Bressan, S. Čanić, M. Garavello, M. Herty and B. Piccoli, Flows on networks: recent results and perspectives, EMS Surv. Math. Sci., 1 (2014), 47-111.  doi: 10.4171/EMSS/2.  Google Scholar

[9]

G. M. Cocolite and M. Garavello, Vanishing viscosity for traffic on networks, SIAM J. Math. Anal., 42 (2010), 1761-1783.  doi: 10.1137/090771417.  Google Scholar

[10]

R. M. Colombo and M. Garavello, A well posed Riemann problem for the $p$-system at a junction, Netw. Heterog. Media, 1 (2006), 495-511.  doi: 10.3934/nhm.2006.1.495.  Google Scholar

[11]

R. M. Colombo and M. Garavello, On the Cauchy problem for the $p$-system at a junction, SIAM J. Math. Anal., 39 (2008), 1456-1471.  doi: 10.1137/060665841.  Google Scholar

[12]

R. M. Colombo, M. Herty and V. Sachers, On $2\times2$ conservation laws at a junction, SIAM J. Math. Anal., 40 (2008), 605-622.  doi: 10.1137/070690298.  Google Scholar

[13]

R. M. Colombo and H. Holden, Isentropic fluid dynamics in a curved pipe, Zeitschrift für angewandte Mathematik und Physik, 67 (2016), 1-10.  doi: 10.1007/s00033-016-0725-0.  Google Scholar

[14]

R. M. Colombo and F. Marcellini, Coupling conditions for the 3x3 Euler system, Netw. Heterog. Media, 5 (2010), 675-690.  doi: 10.3934/nhm.2010.5.675.  Google Scholar

[15]

R. M. Colombo and C. Mauri, Euler system for compressible fluids at a junction, J. Hyperbolic Differ. Equ., 5 (2008), 547-568.  doi: 10.1142/S0219891608001593.  Google Scholar

[16]

C. M. Dafermos, The entropy rate admissibility criterion for solutions of hyperbolic conservation laws, J. Differential Equations, 14 (1973), 202-212.  doi: 10.1016/0022-0396(73)90043-0.  Google Scholar

[17]

C. M. Dafermos, Hyperbolic Conservation Laws in Continuum Physics, vol. 325 of Grundlehren der mathematischen Wissenschaften, 3rd edition, Springer-Verlag Berlin Heidelberg, 2010. doi: 10.1007/978-3-642-04048-1.  Google Scholar

[18]

F. Dubois and P. LeFloch, Boundary conditions for nonlinear hyperbolic systems of conservation laws, J. Differential Equations, 71 (1988), 93-122.  doi: 10.1016/0022-0396(88)90040-X.  Google Scholar

[19]

M. Herty, Coupling conditions for networked systems of Euler equations, SIAM J. Sci. Comput., 30 (2008), 1596-1612.  doi: 10.1137/070688535.  Google Scholar

[20]

M. Herty and M. Rascle, Coupling conditions for a class of second-order models for traffic flow, SIAM J. Math. Anal., 38 (2006), 595-616.  doi: 10.1137/05062617X.  Google Scholar

[21]

H. Holden and N. H. Risebro, Riemann problems with a kink, SIAM J. Math. Anal., 30 (1999), 497-515.  doi: 10.1137/S0036141097327033.  Google Scholar

[22]

Y. Holle, Kinetic relaxation to entropy based coupling conditions for isentropic flow on networks, J. Differential Equations, 269 (2020), 1192-1225.  doi: 10.1016/j.jde.2020.01.005.  Google Scholar

[23]

P. T. Kan, M. M. Santos and Z. Xin, Initial-boundary value problem for conservation laws, Comm. Math. Phys., 186 (1997), 701-730.  doi: 10.1007/s002200050125.  Google Scholar

[24]

J. Lang and P. Mindt, Entropy-preserving coupling conditions for one-dimensional Euler systems at junctions, Netw. Heterog. Media, 13 (2018), 177-190.  doi: 10.3934/nhm.2018008.  Google Scholar

[25]

P.-L. Lions, B. Pertheme and E. Tadmor, Kinetic formulation of the isentropic gas dynamics and $p$-systems, Commun. Math. Phys., 163 (1994), 415-431.  doi: 10.1007/BF02102014.  Google Scholar

[26]

T. P. Liu and J. A. Smoller, On the vacuum state for the isentropic gas dynamics equations, Adv. in Appl. Math., 1 (1980), 345-359.  doi: 10.1016/0196-8858(80)90016-0.  Google Scholar

[27]

G. A. Reigstad, Existence and uniqueness of solutions to the generalized Riemann problem for isentropic flow, SIAM J. Appl. Math., 75 (2015), 679-702.  doi: 10.1137/140962759.  Google Scholar

[28]

G. A. Reigstad, T. Flåtten, N. E. Haugen and T. Ytrehus, Coupling constants and the generalized Riemann problem for isothermal junction flow, J. Hyperbolic Differ. Equ., 12 (2015), 37-59.  doi: 10.1142/S0219891615500022.  Google Scholar

show all references

References:
[1]

M. K. Banda, M. Herty and A. Klar, Coupling conditions for gas networks governed by the isothermal Euler equations, Netw. Heterog. Media, 1 (2006), 295-314.  doi: 10.3934/nhm.2006.1.295.  Google Scholar

[2]

M. K. Banda, M. Herty and A. Klar, Gas flow in pipeline networks, Netw. Heterog. Media, 1 (2006), 41-46.  doi: 10.3934/nhm.2006.1.41.  Google Scholar

[3]

F. Berthelin and F. Bouchut, Solution with finite energy to a BGK system relaxing to isentropic gas dynamics, Ann. Fac. Sci. Toulouse Math., 9 (2000), 605-630.  doi: 10.5802/afst.974.  Google Scholar

[4]

F. Berthelin and F. Bouchut, Kinetic invariant domains and relaxation limit from a BGK model to isentropic gas dynamics, Asymtotic Anal., 31 (2002), 153-176.   Google Scholar

[5]

F. Berthelin and F. Bouchut, Weak entropy boundary conditions for isentropic gas dynamics via kinetic relaxation, J. Differential Equations, 185 (2002), 251-270.  doi: 10.1006/jdeq.2001.4161.  Google Scholar

[6]

F. Bouchut, Construction of BGK models with a family of kinetic entropies for a given system of conservation laws, J. Statist. Phys., 95 (1999), 113-170.  doi: 10.1023/A:1004525427365.  Google Scholar

[7]

A. Bressan, The One-Dimensional Cauchy Problem, in Hyperbolic Systems of Conservation Laws, vol. 20 of Oxford Lecture Series in Mathematics and its Application, Oxford University Press, Oxford, 2000.  Google Scholar

[8]

A. Bressan, S. Čanić, M. Garavello, M. Herty and B. Piccoli, Flows on networks: recent results and perspectives, EMS Surv. Math. Sci., 1 (2014), 47-111.  doi: 10.4171/EMSS/2.  Google Scholar

[9]

G. M. Cocolite and M. Garavello, Vanishing viscosity for traffic on networks, SIAM J. Math. Anal., 42 (2010), 1761-1783.  doi: 10.1137/090771417.  Google Scholar

[10]

R. M. Colombo and M. Garavello, A well posed Riemann problem for the $p$-system at a junction, Netw. Heterog. Media, 1 (2006), 495-511.  doi: 10.3934/nhm.2006.1.495.  Google Scholar

[11]

R. M. Colombo and M. Garavello, On the Cauchy problem for the $p$-system at a junction, SIAM J. Math. Anal., 39 (2008), 1456-1471.  doi: 10.1137/060665841.  Google Scholar

[12]

R. M. Colombo, M. Herty and V. Sachers, On $2\times2$ conservation laws at a junction, SIAM J. Math. Anal., 40 (2008), 605-622.  doi: 10.1137/070690298.  Google Scholar

[13]

R. M. Colombo and H. Holden, Isentropic fluid dynamics in a curved pipe, Zeitschrift für angewandte Mathematik und Physik, 67 (2016), 1-10.  doi: 10.1007/s00033-016-0725-0.  Google Scholar

[14]

R. M. Colombo and F. Marcellini, Coupling conditions for the 3x3 Euler system, Netw. Heterog. Media, 5 (2010), 675-690.  doi: 10.3934/nhm.2010.5.675.  Google Scholar

[15]

R. M. Colombo and C. Mauri, Euler system for compressible fluids at a junction, J. Hyperbolic Differ. Equ., 5 (2008), 547-568.  doi: 10.1142/S0219891608001593.  Google Scholar

[16]

C. M. Dafermos, The entropy rate admissibility criterion for solutions of hyperbolic conservation laws, J. Differential Equations, 14 (1973), 202-212.  doi: 10.1016/0022-0396(73)90043-0.  Google Scholar

[17]

C. M. Dafermos, Hyperbolic Conservation Laws in Continuum Physics, vol. 325 of Grundlehren der mathematischen Wissenschaften, 3rd edition, Springer-Verlag Berlin Heidelberg, 2010. doi: 10.1007/978-3-642-04048-1.  Google Scholar

[18]

F. Dubois and P. LeFloch, Boundary conditions for nonlinear hyperbolic systems of conservation laws, J. Differential Equations, 71 (1988), 93-122.  doi: 10.1016/0022-0396(88)90040-X.  Google Scholar

[19]

M. Herty, Coupling conditions for networked systems of Euler equations, SIAM J. Sci. Comput., 30 (2008), 1596-1612.  doi: 10.1137/070688535.  Google Scholar

[20]

M. Herty and M. Rascle, Coupling conditions for a class of second-order models for traffic flow, SIAM J. Math. Anal., 38 (2006), 595-616.  doi: 10.1137/05062617X.  Google Scholar

[21]

H. Holden and N. H. Risebro, Riemann problems with a kink, SIAM J. Math. Anal., 30 (1999), 497-515.  doi: 10.1137/S0036141097327033.  Google Scholar

[22]

Y. Holle, Kinetic relaxation to entropy based coupling conditions for isentropic flow on networks, J. Differential Equations, 269 (2020), 1192-1225.  doi: 10.1016/j.jde.2020.01.005.  Google Scholar

[23]

P. T. Kan, M. M. Santos and Z. Xin, Initial-boundary value problem for conservation laws, Comm. Math. Phys., 186 (1997), 701-730.  doi: 10.1007/s002200050125.  Google Scholar

[24]

J. Lang and P. Mindt, Entropy-preserving coupling conditions for one-dimensional Euler systems at junctions, Netw. Heterog. Media, 13 (2018), 177-190.  doi: 10.3934/nhm.2018008.  Google Scholar

[25]

P.-L. Lions, B. Pertheme and E. Tadmor, Kinetic formulation of the isentropic gas dynamics and $p$-systems, Commun. Math. Phys., 163 (1994), 415-431.  doi: 10.1007/BF02102014.  Google Scholar

[26]

T. P. Liu and J. A. Smoller, On the vacuum state for the isentropic gas dynamics equations, Adv. in Appl. Math., 1 (1980), 345-359.  doi: 10.1016/0196-8858(80)90016-0.  Google Scholar

[27]

G. A. Reigstad, Existence and uniqueness of solutions to the generalized Riemann problem for isentropic flow, SIAM J. Appl. Math., 75 (2015), 679-702.  doi: 10.1137/140962759.  Google Scholar

[28]

G. A. Reigstad, T. Flåtten, N. E. Haugen and T. Ytrehus, Coupling constants and the generalized Riemann problem for isothermal junction flow, J. Hyperbolic Differ. Equ., 12 (2015), 37-59.  doi: 10.1142/S0219891615500022.  Google Scholar

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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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 $
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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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} $
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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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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