Coherence of coupling Riemann solvers for gas flows through flux-maximizing valves. (English) Zbl 1437.35479
Summary: In this paper we propose a model based on the strictly hyperbolic system of isothermal Euler equations for the gas flow in a straight pipe with a valve. We are then faced with an initial value problem with coupling conditions at the valve position. The valves under consideration are requested to maximize the flux; moreover, the flow is imposed to occur within prescribed bounds of pressure and flow. The issue is the mathematical characterization of the coherence of the corresponding coupling Riemann solvers; this property is related to the phenomenon of chattering, the rapid switching on and off of the valve. Within this framework we describe three kinds of valves, which differ for their action; two of them lead to a coherent solver, the third one does not. Proofs involve geometric and analytic properties of the Lax curves.
MSC:
| 35L65 | Hyperbolic conservation laws |
| 35L67 | Shocks and singularities for hyperbolic equations |
| 76B75 | Flow control and optimization for incompressible inviscid fluids |
References:
| [1] | D. Amadori, Initial-boundary value problems for nonlinear systems of conservation laws, NoDEA Nonlinear Differential Equations Appl., 4 (1997), pp. 1-42, https://doi.org/10.1007/PL00001406. · Zbl 0868.35069 |
| [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), pp. 295-314, https://doi.org/10.3934/nhm.2006.1.295. · Zbl 1109.76052 |
| [3] | M. K. Banda, M. Herty, and A. Klar, Gas flow in pipeline networks, Netw. Heterog. Media, 1 (2006), pp. 41-56, https://doi.org/10.3934/nhm.2006.1.41. · Zbl 1108.76063 |
| [4] | C. Bardos, A. Y. le Roux, and J.-C. Nédélec, First order quasilinear equations with boundary conditions, Comm. Partial Differential Equations, 4 (1979), pp. 1017-1034, https://doi.org/10.1080/03605307908820117. · Zbl 0418.35024 |
| [5] | A. Bressan, Hyperbolic Systems of Conservation Laws, vol. 20, Oxford University Press, Oxford, 2000. · Zbl 0987.35105 |
| [6] | R. M. Colombo, Hyperbolic phase transitions in traffic flow, SIAM J. Appl. Math., 63 (2002), pp. 708-721, https://doi.org/10.1137/S0036139901393184. · Zbl 1037.35043 |
| [7] | R. M. Colombo and M. Garavello, Comparison among different notions of solution for the \(p\)-system at a junction, Discrete Contin. Dyn. Syst., Supplement (2009), pp. 181-190. · Zbl 1188.35114 |
| [8] | A. Corli, M. Figiel, A. Futa, and M. D. Rosini, Coupling conditions for isothermal gas flow and applications to valves, Nonlinear Anal. Real World Appl., 40 (2018), pp. 403-427, https://doi.org/10.1016/j.nonrwa.2017.09.005. · Zbl 1382.35214 |
| [9] | A. Corli and M. D. Rosini, Coherence and chattering of a one-way valve, ZAMM Z. Angew. Math. Mech., 99 (2019), e201800250, https://doi.org/10.1002/zamm.201800250. · Zbl 1498.35357 |
| [10] | C. M. Dafermos, Hyperbolic Conservation Laws in Continuum Physics, 4th ed., Springer-Verlag, Berlin, 2016, https://doi.org/10.1007/978-3-662-49451-6. · Zbl 1364.35003 |
| [11] | E. Dal Santo, C. Donadello, S. F. Pellegrino, and M. D. Rosini, Representation of capacity drop at a road merge via point constraints in a first order traffic model, ESAIM Math. Model. Numer. Anal., 53 (2019), pp. 1-34, https://doi.org/10.1051/m2an/2019002. · Zbl 1420.35156 |
| [12] | E. Dal Santo, M. D. Rosini, N. Dymski, and M. Benyahia, General phase transition models for vehicular traffic with point constraints on the flow, Math. Methods Appl. Sci., 40 (2017), pp. 6623-6641, https://doi.org/10.1002/mma.4478. · Zbl 1403.35168 |
| [13] | N. S. Dymski, P. Goatin, and M. D. Rosini, Existence of \(\bold{BV}\) solutions for a non-conservative constrained Aw-Rascle-Zhang model for vehicular traffic, J. Math. Anal. Appl., 467 (2018), pp. 45-66, https://doi.org/10.1016/j.jmaa.2018.07.025. · Zbl 1403.90224 |
| [14] | M. Garavello and B. Piccoli, Traffic flow on a road network using the Aw-Rascle model, Comm. Partial Differential Equations, 31 (2006), pp. 243-275, https://doi.org/10.1080/03605300500358053,. · Zbl 1090.90032 |
| [15] | M. Garavello and B. Piccoli, Traffic Flow on Networks, American Institute of Mathematical Sciences (AIMS), Springfield, MO, 2006. · Zbl 1136.90012 |
| [16] | M. Gugat, G. Leugering, A. Martin, M. Schmidt, M. Sirvent, and D. Wintergerst, MIP-based instantaneous control of mixed-integer PDE-constrained gas transport problems, Comput. Optim. Appl., 70 (2018), pp. 267-294, https://doi.org/10.1007/s10589-017-9970-1. · Zbl 1391.49040 |
| [17] | M. Herty, Modeling, simulation and optimization of gas networks with compressors, Netw. Heterog. Media, 2 (2007), pp. 81-97, https://doi.org/10.3934/nhm.2007.2.81. · Zbl 1132.76045 |
| [18] | C. Hös and A. R. Champneys, Grazing bifurcations and chatter in a pressure relief valve model, Phys. D, 241 (2012), pp. 2068-2076, https://doi.org/10.1016/j.physd.2011.05.013. |
| [19] | D. Hoff, Invariant regions for systems of conservation laws, Trans. Amer. Math. Soc., 289 (1985), pp. 591-610, https://doi.org/10.2307/2000254. · Zbl 0535.35056 |
| [20] | T. Koch, B. Hiller, M. E. Pfetsch, and L. Schewe, eds., Evaluating Gas Network Capacities, MOS-SIAM Series on Optimization 21, SIAM, Philadelphia, 2015. · Zbl 1322.90007 |
| [21] | P. G. LeFloch, Hyperbolic Systems of Conservation Laws, Lectures in Mathematics ETH Zürich, Birkhäuser Verlag, Basel, 2002, https://doi.org/10.1007/978-3-0348-8150-0. · Zbl 1019.35001 |
| [22] | R. J. LeVeque, Numerical Methods for Conservation Laws, Birkhäuser Verlag, Basel, 1990. · Zbl 0723.65067 |
| [23] | G. Licskó, A. Champneys, and C. Hös, Nonlinear analysis of a single stage pressure relief valve, IAENG Int. J. Appl. Math., 39 (2009), pp. 286-299. |
| [24] | A. Martin, M. Möller, and S. Moritz, Mixed integer models for the stationary case of gas network optimization, Math. Program., 105 (2006), pp. 563-582, https://doi.org/10.1007/s10107-005-0665-5. · Zbl 1085.90035 |
| [25] | M. Möller, Mixed Integer Models for the Optimisation of Gas Networks in the Stationary Case, Ph.D. thesis, Fach. Mathematik T.U. Darmstadt, 2004. · Zbl 1069.90120 |
This reference list is based on information provided by the publisher or from digital mathematics libraries. Its items are heuristically matched to zbMATH identifiers and may contain data conversion errors. In some cases that data have been complemented/enhanced by data from zbMATH Open. This attempts to reflect the references listed in the original paper as accurately as possible without claiming completeness or a perfect matching.
