# American Institute of Mathematical Sciences

March 2018, 13(1): 177-190. doi: 10.3934/nhm.2018008

## Entropy-preserving coupling conditions for one-dimensional Euler systems at junctions

Received  April 2017 Revised  September 2017 Published  March 2018

This paper is concerned with a set of novel coupling conditions for the 3× 3 one-dimensional Euler system with source terms at a junction of pipes with possibly different cross-sectional areas. Beside conservation of mass, we require the equality of the total enthalpy at the junction and that the specific entropy for pipes with outgoing flow equals the convex combination of all entropies that belong to pipes with incoming flow. Previously used coupling conditions include equality of pressure or dynamic pressure. They are restricted to the special case of a junction having only one pipe with outgoing flow direction. Recently, Reigstad [SIAM J. Appl. Math., 75:679-702,2015] showed that such pressure-based coupling conditions can produce non-physical solutions for isothermal flows through the production of mechanical energy. Our new coupling conditions ensure energy as well as entropy conservation and also apply to junctions connecting an arbitrary number of pipes with flexible flow directions. We prove the existence and uniqueness of solutions to the generalised Riemann problem at a junction in the neighbourhood of constant stationary states which belong to the subsonic region. This provides the basis for the well-posedness of the homogeneous and inhomogeneous Cauchy problems for initial data with sufficiently small total variation.

Citation: Jens Lang, Pascal Mindt. Entropy-preserving coupling conditions for one-dimensional Euler systems at junctions. Networks & Heterogeneous Media, 2018, 13 (1) : 177-190. doi: 10.3934/nhm.2018008
##### References:
 [1] M. K. Banda, M. Herty and A. Klar, Gas flow in pipeline networks, Netw. Heterog. Media, 1 (2006), 41-56. doi: 10.3934/nhm.2006.1.41. [2] 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. [3] A. Bressan, Hyberbolic Systems of Conservation Laws: The One-dimensional Cauchy Problem, Oxford Lecture Series in Mathematics and Its Application, Oxford University Press, 2000. [4] C. Chalons, P. A. Raviart and N. Seguin, The interface coupling of the gas dynamics equations, Quaterly Appl. Math., 66 (2008), 659-705. doi: 10.1090/S0033-569X-08-01087-X. [5] 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. [6] R. M. Colombo and G. Guerra, Hyperbolic balance laws with a non local source, Commun. Partial Differential Equations, 32 (2007), 1917-1939. doi: 10.1080/03605300701318849. [7] R. M. Colombo and G. Guerra, Hyperbolic balance laws with a dissipative non local source, Commun. Pure Appl. Anal., 7 (2008), 1077-1090. doi: 10.3934/cpaa.2008.7.1077. [8] R. M. Colombo, G. Guerra, M. Herty and V. Schleper, Optimal control in networks and pipes and canals, SIAM J. Control Optim., 48 (2009), 2032-2050. doi: 10.1137/080716372. [9] R. M. Colombo and F. Marcellini, Coupling conditions for the 3× 3 Euler system, Netw. Heterog. Media, 5 (2009), 675-690. doi: 10.3934/nhm.2010.5.675. [10] R. M. Colombo and C. Mauri, Euler systems for compressible fluids at a junction, J. Hyperbol. Differ. Eq., 5 (2008), 547-568. doi: 10.1142/S0219891608001593. [11] C. Donadello and A. Marson, Stability of front tracking solutions to the initial and boundary value problem for systems of conservation laws, Nonlinear Differ. Equ. Appl., 14 (2007), 569-592. doi: 10.1007/s00030-007-5010-7. [12] I. Gasser and M. Kraft, Modelling and simulation of fires in tunnel networks, Netw. Heterog. Media, 3 (2008), 691-707. doi: 10.3934/nhm.2008.3.691. [13] M. Herty, Coupling conditions for networked systems of Euler equations, SIAM J. Sci. Comput., 30 (2008), 1596-1612. doi: 10.1137/070688535. [14] R. J. LeVeque, Finite-Volume Methods for Hyperbolic Problems, Cambridge Texts in Applied Mathematics, Cambridge University Press, 2002. [15] A. Morin and G. A. Reigstad, Pipe networks: Coupling constants in a junction for the isentropic euler equations, Energy Procedia, 64 (2015), 140-149. doi: 10.1016/j.egypro.2015.01.017. [16] G. A. Reigstad, Numerical network models and entropy principles for isothermal junction flow, Netw. Heterog. Media, 9 (2014), 65-95. doi: 10.3934/nhm.2014.9.65. [17] 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. [18] G. A. Reigstad, T. Flåtten, N. E. Haugen and T. Ytrehus, Coupling constants and the generalized Riemann problem for isothermal junction flow, J. Hyperbol. Differ. Eq., 12 (2015), 37-59. doi: 10.1142/S0219891615500022. [19] M. Schmidt, M. C. Steinbach and B. M. Willert, High detail stationary optimization models for gas networks, Optim. Eng., 16 (2015), 131-164. doi: 10.1007/s11081-014-9246-x. [20] T. Toro, Riemann Solvers and Numerical Methods for Fluid Dynamics: A Practical Introduction, Springer, 2009.

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##### References:
 [1] M. K. Banda, M. Herty and A. Klar, Gas flow in pipeline networks, Netw. Heterog. Media, 1 (2006), 41-56. doi: 10.3934/nhm.2006.1.41. [2] 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. [3] A. Bressan, Hyberbolic Systems of Conservation Laws: The One-dimensional Cauchy Problem, Oxford Lecture Series in Mathematics and Its Application, Oxford University Press, 2000. [4] C. Chalons, P. A. Raviart and N. Seguin, The interface coupling of the gas dynamics equations, Quaterly Appl. Math., 66 (2008), 659-705. doi: 10.1090/S0033-569X-08-01087-X. [5] 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. [6] R. M. Colombo and G. Guerra, Hyperbolic balance laws with a non local source, Commun. Partial Differential Equations, 32 (2007), 1917-1939. doi: 10.1080/03605300701318849. [7] R. M. Colombo and G. Guerra, Hyperbolic balance laws with a dissipative non local source, Commun. Pure Appl. Anal., 7 (2008), 1077-1090. doi: 10.3934/cpaa.2008.7.1077. [8] R. M. Colombo, G. Guerra, M. Herty and V. Schleper, Optimal control in networks and pipes and canals, SIAM J. Control Optim., 48 (2009), 2032-2050. doi: 10.1137/080716372. [9] R. M. Colombo and F. Marcellini, Coupling conditions for the 3× 3 Euler system, Netw. Heterog. Media, 5 (2009), 675-690. doi: 10.3934/nhm.2010.5.675. [10] R. M. Colombo and C. Mauri, Euler systems for compressible fluids at a junction, J. Hyperbol. Differ. Eq., 5 (2008), 547-568. doi: 10.1142/S0219891608001593. [11] C. Donadello and A. Marson, Stability of front tracking solutions to the initial and boundary value problem for systems of conservation laws, Nonlinear Differ. Equ. Appl., 14 (2007), 569-592. doi: 10.1007/s00030-007-5010-7. [12] I. Gasser and M. Kraft, Modelling and simulation of fires in tunnel networks, Netw. Heterog. Media, 3 (2008), 691-707. doi: 10.3934/nhm.2008.3.691. [13] M. Herty, Coupling conditions for networked systems of Euler equations, SIAM J. Sci. Comput., 30 (2008), 1596-1612. doi: 10.1137/070688535. [14] R. J. LeVeque, Finite-Volume Methods for Hyperbolic Problems, Cambridge Texts in Applied Mathematics, Cambridge University Press, 2002. [15] A. Morin and G. A. Reigstad, Pipe networks: Coupling constants in a junction for the isentropic euler equations, Energy Procedia, 64 (2015), 140-149. doi: 10.1016/j.egypro.2015.01.017. [16] G. A. Reigstad, Numerical network models and entropy principles for isothermal junction flow, Netw. Heterog. Media, 9 (2014), 65-95. doi: 10.3934/nhm.2014.9.65. [17] 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. [18] G. A. Reigstad, T. Flåtten, N. E. Haugen and T. Ytrehus, Coupling constants and the generalized Riemann problem for isothermal junction flow, J. Hyperbol. Differ. Eq., 12 (2015), 37-59. doi: 10.1142/S0219891615500022. [19] M. Schmidt, M. C. Steinbach and B. M. Willert, High detail stationary optimization models for gas networks, Optim. Eng., 16 (2015), 131-164. doi: 10.1007/s11081-014-9246-x. [20] T. Toro, Riemann Solvers and Numerical Methods for Fluid Dynamics: A Practical Introduction, Springer, 2009.
Possible wave patterns in the solution of Riemann problems for the Euler equations: shock (S), contact (C) and rarefaction (R).
Connection of the regions $L$, $L\ast$, $R\ast$, and $R$ with the Lax curve $\mathcal{L}_3$ for incoming pipes (a) and the Lax curves $\mathcal{L}_2\!\circ\!\mathcal{L}_3$ for outgoing pipes (b).
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