Efficient numerical methods for gas network modeling and simulation

Yue Qiu; Sara Grundel; Martin Stoll; Peter Benner

doi:10.3934/nhm.2020018

Previous Article
The selection problem for some first-order stationary Mean-field games
NHM Home
This Issue
Next Article
Simultaneous observability of infinitely many strings and beams

December 2020, 15(4): 653-679. doi: 10.3934/nhm.2020018

Efficient numerical methods for gas network modeling and simulation

Yue Qiu ^1,2,, , Sara Grundel ^2, , Martin Stoll ^3, and Peter Benner ^2,

1.	School of Information Science and Technology, ShanghaiTech University, 393 Middle Huaxia Road, 201210, Shanghai, China
2.	Computational Methods for Systems and Control Theory, Max Planck Institute for Dynamics of Complex Technical Systems, Sandtorstraße 1, 39106, Magdeburg, Germany
3.	Faculty of Mathematics, Technische Universität Chemnitz, Reichenhainer Straße 41, 09126, Chemnitz, Germany

^* Corresponding author: Yue Qiu

Received August 2019 Revised August 2020 Published December 2020 Early access August 2020

Fund Project: This work is partially funded by the European Regional Development Fund (ERDF/EFRE: ZS/2016/04/78156) within the Center Dynamic Systems (CDS)

We study the modeling and simulation of gas pipeline networks, with a focus on fast numerical methods for the simulation of transient dynamics. The obtained mathematical model of the underlying network is represented by a system of nonlinear differential algebraic equations (DAEs). With our modeling approach, we reduce the number of algebraic constraints, which correspond to the $ (2,2) $ block in our semi-explicit DAE model, to the order of junction nodes in the network, where a junction node couples at least three pipelines. We can furthermore ensure that the $ (1, 1) $ block of all system matrices including the Jacobian is block lower triangular by using a specific ordering of the pipes of the network. We then exploit this structure to propose an efficient preconditioner for the fast simulation of the network. We test our numerical methods on benchmark problems of (well-)known gas networks and the numerical results show the efficiency of our methods.

Keywords: Gas networks modeling, isothermal Euler equation, directed acyclic graph (DAG), differential algebraic equation (DAE), preconditioning.

Mathematics Subject Classification: Primary: 65F08, 37M05, 37N30, 94C15.

Citation: Yue Qiu, Sara Grundel, Martin Stoll, Peter Benner. Efficient numerical methods for gas network modeling and simulation. Networks and Heterogeneous Media, 2020, 15 (4) : 653-679. doi: 10.3934/nhm.2020018

References:

[1]	N. Banagaaya, S. Grundel and P. Benner, Index-aware MOR for gas transport networks with many supply inputs, in IUTAM Symposium on Model Order Reduction of Coupled Systems (eds. J. Fehr and B. Haasdonk), Springer International Publishing, Cham, 2020,191–207.
[2]	J. Bang-Jensen and G. Z. Gutin, Digraphs: Theory, Algorithms and Applications, Springer-Verlag, London, 2008. doi: 10.1007/978-1-84800-998-1.
[3]	P. Benner, S. Grundel, C. Himpe, C. Huck, T. Streubel and C. Tischendorf, Gas network benchmark models, in Differential-Algebraic Equations Forum, Springer, Berlin, Heidelberg, 2018.
[4]	M. Benzi, G. H. Golub and J. Liesen, Numerical solution of saddle point problems, Acta Numer., 14 (2005), 1-137. doi: 10.1017/S0962492904000212.
[5]	A. Bermúdez, X. López and M. E. Vázquez-Cendón, Finite volume methods for multi-component Euler equations with source terms, Comput. Fluids, 156 (2017), 113-134. doi: 10.1016/j.compfluid.2017.07.004.
[6]	M. Chaczykowski, Sensitivity of pipeline gas flow model to the selection of the equation of state, Chem. Eng. Res. Des., 87 (2009), 1596-1603. doi: 10.1016/j.cherd.2009.06.008.
[7]	R. Dembo, S. Eisenstat and T. Steihaug, Inexact Newton methods, SIAM J. Numer. Anal., 19 (1982), 400-408. doi: 10.1137/0719025.
[8]	H. Egger, A robust conservative mixed finite element method for isentropic compressible flow on pipe networks, SIAM J. Sci. Comput., 40 (2018), A108–A129. doi: 10.1137/16M1094373.
[9]	H. Egger, T. Kugler and N. Strogies, Parameter identification in a semilinear hyperbolic system, Inverse Probl., 33 (2017), 055022. doi: 10.1088/1361-6420/aa648c.
[10]	H. Elman, D. Silvester and A. Wathen, Finite Elements and Fast Iterative Solvers, Oxford University Press, New York, 2014. doi: 10.1093/acprof:oso/9780199678792.001.0001.
[11]	A. Fügenschuh, et al., Physical and technical fundamentals of gas networks, in Evaluating Gas Network Capacities
[12]	T. G. Grandón, H. Heitsch and R. Henrion, A joint model of probabilistic/robust constraints for gas transport management in stationary networks, Comput. Manag. Sci., 14 (2017), 443-460. doi: 10.1007/s10287-017-0284-7.
[13]	S. Grundel, N. Hornung, B. Klaassen, P. Benner and T. Clees, Computing surrogates for gas network simulation using model order reduction, in Surrogate-Based Modeling and Optimization doi: 10.1007/978-1-4614-7551-4_9.
[14]	S. Grundel, N. Hornung and S. Roggendorf, Numerical aspects of model order reduction for gas transportation networks, in Simulation-Driven Modeling and Optimization, Springer Proceedings in Mathematics & Statistics, 153, 2016, 1–28.
[15]	S. Grundel and L. Jansen, Efficient simulation of transient gas networks using IMEX integration schemes and MOR methods, in 2015 54th IEEE Conference on Decision and Control (CDC), 2015, 4579–4584. doi: 10.1109/CDC.2015.7402934.
[16]	S. Grundel, L. Jansen, N. Hornung, T. Clees, C. Tischendorf and P. Benner, Model order reduction of differential algebraic equations arising from the simulation of gas transport networks, in Progress in Differential-Algebraic Equations, Differential-Algebraic Equations Forum, Springer Berlin Heidelberg, 2014,183–205. doi: 10.1007/978-3-642-34928-7_2.
[17]	M. Gugat, F. M. Hante, M. Hirsch-Dick and G. Leugering, Stationary states in gas networks, Netw. Heterog. Media, 10 (2015), 295-320. doi: 10.3934/nhm.2015.10.295.
[18]	F. M. Hante, G. Leugering, A. Martin, L. Schewe and M. Schmidt, Challenges in Optimal Control Problems for Gas and Fluid Flow in Networks of Pipes and Canals: From Modeling to Industrial Applications, Springer Verlag, Singapore, 2017, 77–122.
[19]	A. Herrán-González, J. M. D. L. Cruz, B. D. Andrés-Toro and J. L. Risco-Martín, Modeling and simulation of a gas distribution pipeline network, Appl. Math. Model., 33 (2009), 1584-1600.
[20]	M. Herty, Modeling, simulation and optimization of gas networks with compressors, Netw. Heterog. Media, 2 (2007), 81-97. doi: 10.3934/nhm.2007.2.81.
[21]	M. Herty, Coupling conditions for networked systems of Euler equations, SIAM J. Sci. Comput., 30 (2008), 1596-1612. doi: 10.1137/070688535.
[22]	M. Herty, J. Mohring and V. Sachers, A new model for gas flow in pipe networks, Math. Methods Appl. Sci., 33 (2010), 845-855. doi: 10.1002/mma.1197.
[23]	C. Huck, L. Jansen and C. Tischendorf, A topology based discretization of PDAEs describing water transportation networks, Proc. Appl. Math. Mech., 14 (2014), 923-924. doi: 10.1002/pamm.201410442.
[24]	C. Johnson and J. Pitkäranta, An analysis of the discontinuous Galerkin method for a scalar hyperbolic equation, Math. Comp., 46 (1986), 1-26. doi: 10.1090/S0025-5718-1986-0815828-4.
[25]	C. Kelley, Solving Nonlinear Equations with Newton's Method, Society for Industrial and Applied Mathematics, Philadelphia, 2003. doi: 10.1137/1.9780898718898.
[26]	A. Osiadacz, Simulation of transient gas flows in networks, Internat. J. Numer. Methods Fluids, 4 (1984), 13-24. doi: 10.1002/fld.1650040103.
[27]	A. Osiadacz, Simulation and Analysis of Gas Networks, Gulf Publishing, Houston, TX, 1987.
[28]	A. J. Osiadacz and M. Yedroudj, A comparison of a finite element method and a finite difference method for transient simulation of a gas pipeline, Appl. Math. Model., 13 (1989), 79-85. doi: 10.1016/0307-904X(89)90018-8.
[29]	J. W. Pearson, On the development of parameter-robust preconditioners and commutator arguments for solving Stokes control problems, Electron. Trans. Numer. Anal., 44 (2015), 53-72.
[30]	J. Pestana and A. J. Wathen, Natural preconditioning and iterative methods for saddle point systems, SIAM Rev., 57 (2015), 71-91. doi: 10.1137/130934921.
[31]	M. Porcelli, V. Simoncini and M. Tani, Preconditioning of active-set Newton methods for PDE-constrained optimal control problems, SIAM J. Sci. Comput., 37 (2015), S472–S502. doi: 10.1137/140975711.
[32]	Y. Qiu, Preconditioning Optimal Flow Control Problems Using Multilevel Sequentially Semiseparable Matrix Computations, Ph.D thesis, Delft Institute of Applied Mathematics, Delft University of Technology, 2015.
[33]	T. Rees, Preconditioning Iterative Methods for PDE-Constrained Optimization, Ph.D thesis, University of Oxford, 2010.
[34]	S. Roggendorf, Model Order Reduction for Linearized Systems Arising from the Simulation of Gas Transportation Networks, Master's thesis, Rheinischen Friedrich-Wilhelms-Universität Bonn, Germany, 2015.
[35]	Y. Saad, Iterative Methods for Sparse Linear Systems, Society for Industrial and Applied Mathematics, Philadelphia, 2003. doi: 10.1137/1.9780898718003.
[36]	P. Sonneveld and M. B. van Gijzen, IDR(s): A family of simple and fast algorithms for solving large nonsymmetric systems of linear equations, SIAM J. Sci. Comput., 31 (2008), 1035-1062. doi: 10.1137/070685804.
[37]	M. C. Steinbach, On PDE solution in transient optimization of gas networks, J. Comput. Appl. Math., 203 (2007), 345-361. doi: 10.1016/j.cam.2006.04.018.
[38]	M. Stoll and T. Breiten, A low-rank in time approach to PDE-constrained optimization, SIAM J. Sci. Comput., 37 (2015), B1–B29. doi: 10.1137/130926365.
[39]	W. Q. Tao and H. C. Ti, Transient analysis of gas pipeline network, Chem. Eng. J., 69 (1998), 47-52. doi: 10.1016/S1385-8947(97)00109-5.
[40]	E. F. Toro and S. J. Billett, Centred TVD schemes for hyperbolic conservation laws, IMA J. Numer. Anal., 20 (2000), 47-79. doi: 10.1093/imanum/20.1.47.
[41]	M. B. van Gijzen and P. Sonneveld, Algorithm 913: An elegant IDR(s) variant that efficiently exploits biorthogonality properties, ACM Trans. Math. Software, 38 (2011), 5: 1–5: 19. doi: 10.1145/2049662.2049667.
[42]	A. J. Wathen, Preconditioning, Acta Numer., 24 (2015), 329-376. doi: 10.1017/S0962492915000021.
[43]	M. Wathen, C. Greif and D. Schötzau, Preconditioners for mixed finite element discretizations of incompressible MHD equations, SIAM J. Sci. Comput., 39 (2017), A2993–A3013. doi: 10.1137/16M1098991.
[44]	J. Zhou and M. A. Adewumi, Simulation of transients in natural gas pipelines using hybrid TVD schemes, Internat. J. Numer. Methods Fluids, 32 (2000), 407-437. doi: 10.1002/(SICI)1097-0363(20000229)32:4<407::AID-FLD945>3.0.CO;2-9.
[45]	A. Zlotnik, M. Chertkov and S. Backhaus, Optimal control of transient flow in natural gas networks, in 2015 54th IEEE Conference on Decision and Control (CDC), 2015, 4563–4570. doi: 10.1109/TCNS.2014.2367360.

show all references

References:

[1]	N. Banagaaya, S. Grundel and P. Benner, Index-aware MOR for gas transport networks with many supply inputs, in IUTAM Symposium on Model Order Reduction of Coupled Systems (eds. J. Fehr and B. Haasdonk), Springer International Publishing, Cham, 2020,191–207.
[2]	J. Bang-Jensen and G. Z. Gutin, Digraphs: Theory, Algorithms and Applications, Springer-Verlag, London, 2008. doi: 10.1007/978-1-84800-998-1.
[3]	P. Benner, S. Grundel, C. Himpe, C. Huck, T. Streubel and C. Tischendorf, Gas network benchmark models, in Differential-Algebraic Equations Forum, Springer, Berlin, Heidelberg, 2018.
[4]	M. Benzi, G. H. Golub and J. Liesen, Numerical solution of saddle point problems, Acta Numer., 14 (2005), 1-137. doi: 10.1017/S0962492904000212.
[5]	A. Bermúdez, X. López and M. E. Vázquez-Cendón, Finite volume methods for multi-component Euler equations with source terms, Comput. Fluids, 156 (2017), 113-134. doi: 10.1016/j.compfluid.2017.07.004.
[6]	M. Chaczykowski, Sensitivity of pipeline gas flow model to the selection of the equation of state, Chem. Eng. Res. Des., 87 (2009), 1596-1603. doi: 10.1016/j.cherd.2009.06.008.
[7]	R. Dembo, S. Eisenstat and T. Steihaug, Inexact Newton methods, SIAM J. Numer. Anal., 19 (1982), 400-408. doi: 10.1137/0719025.
[8]	H. Egger, A robust conservative mixed finite element method for isentropic compressible flow on pipe networks, SIAM J. Sci. Comput., 40 (2018), A108–A129. doi: 10.1137/16M1094373.
[9]	H. Egger, T. Kugler and N. Strogies, Parameter identification in a semilinear hyperbolic system, Inverse Probl., 33 (2017), 055022. doi: 10.1088/1361-6420/aa648c.
[10]	H. Elman, D. Silvester and A. Wathen, Finite Elements and Fast Iterative Solvers, Oxford University Press, New York, 2014. doi: 10.1093/acprof:oso/9780199678792.001.0001.
[11]	A. Fügenschuh, et al., Physical and technical fundamentals of gas networks, in Evaluating Gas Network Capacities
[12]	T. G. Grandón, H. Heitsch and R. Henrion, A joint model of probabilistic/robust constraints for gas transport management in stationary networks, Comput. Manag. Sci., 14 (2017), 443-460. doi: 10.1007/s10287-017-0284-7.
[13]	S. Grundel, N. Hornung, B. Klaassen, P. Benner and T. Clees, Computing surrogates for gas network simulation using model order reduction, in Surrogate-Based Modeling and Optimization doi: 10.1007/978-1-4614-7551-4_9.
[14]	S. Grundel, N. Hornung and S. Roggendorf, Numerical aspects of model order reduction for gas transportation networks, in Simulation-Driven Modeling and Optimization, Springer Proceedings in Mathematics & Statistics, 153, 2016, 1–28.
[15]	S. Grundel and L. Jansen, Efficient simulation of transient gas networks using IMEX integration schemes and MOR methods, in 2015 54th IEEE Conference on Decision and Control (CDC), 2015, 4579–4584. doi: 10.1109/CDC.2015.7402934.
[16]	S. Grundel, L. Jansen, N. Hornung, T. Clees, C. Tischendorf and P. Benner, Model order reduction of differential algebraic equations arising from the simulation of gas transport networks, in Progress in Differential-Algebraic Equations, Differential-Algebraic Equations Forum, Springer Berlin Heidelberg, 2014,183–205. doi: 10.1007/978-3-642-34928-7_2.
[17]	M. Gugat, F. M. Hante, M. Hirsch-Dick and G. Leugering, Stationary states in gas networks, Netw. Heterog. Media, 10 (2015), 295-320. doi: 10.3934/nhm.2015.10.295.
[18]	F. M. Hante, G. Leugering, A. Martin, L. Schewe and M. Schmidt, Challenges in Optimal Control Problems for Gas and Fluid Flow in Networks of Pipes and Canals: From Modeling to Industrial Applications, Springer Verlag, Singapore, 2017, 77–122.
[19]	A. Herrán-González, J. M. D. L. Cruz, B. D. Andrés-Toro and J. L. Risco-Martín, Modeling and simulation of a gas distribution pipeline network, Appl. Math. Model., 33 (2009), 1584-1600.
[20]	M. Herty, Modeling, simulation and optimization of gas networks with compressors, Netw. Heterog. Media, 2 (2007), 81-97. doi: 10.3934/nhm.2007.2.81.
[21]	M. Herty, Coupling conditions for networked systems of Euler equations, SIAM J. Sci. Comput., 30 (2008), 1596-1612. doi: 10.1137/070688535.
[22]	M. Herty, J. Mohring and V. Sachers, A new model for gas flow in pipe networks, Math. Methods Appl. Sci., 33 (2010), 845-855. doi: 10.1002/mma.1197.
[23]	C. Huck, L. Jansen and C. Tischendorf, A topology based discretization of PDAEs describing water transportation networks, Proc. Appl. Math. Mech., 14 (2014), 923-924. doi: 10.1002/pamm.201410442.
[24]	C. Johnson and J. Pitkäranta, An analysis of the discontinuous Galerkin method for a scalar hyperbolic equation, Math. Comp., 46 (1986), 1-26. doi: 10.1090/S0025-5718-1986-0815828-4.
[25]	C. Kelley, Solving Nonlinear Equations with Newton's Method, Society for Industrial and Applied Mathematics, Philadelphia, 2003. doi: 10.1137/1.9780898718898.
[26]	A. Osiadacz, Simulation of transient gas flows in networks, Internat. J. Numer. Methods Fluids, 4 (1984), 13-24. doi: 10.1002/fld.1650040103.
[27]	A. Osiadacz, Simulation and Analysis of Gas Networks, Gulf Publishing, Houston, TX, 1987.
[28]	A. J. Osiadacz and M. Yedroudj, A comparison of a finite element method and a finite difference method for transient simulation of a gas pipeline, Appl. Math. Model., 13 (1989), 79-85. doi: 10.1016/0307-904X(89)90018-8.
[29]	J. W. Pearson, On the development of parameter-robust preconditioners and commutator arguments for solving Stokes control problems, Electron. Trans. Numer. Anal., 44 (2015), 53-72.
[30]	J. Pestana and A. J. Wathen, Natural preconditioning and iterative methods for saddle point systems, SIAM Rev., 57 (2015), 71-91. doi: 10.1137/130934921.
[31]	M. Porcelli, V. Simoncini and M. Tani, Preconditioning of active-set Newton methods for PDE-constrained optimal control problems, SIAM J. Sci. Comput., 37 (2015), S472–S502. doi: 10.1137/140975711.
[32]	Y. Qiu, Preconditioning Optimal Flow Control Problems Using Multilevel Sequentially Semiseparable Matrix Computations, Ph.D thesis, Delft Institute of Applied Mathematics, Delft University of Technology, 2015.
[33]	T. Rees, Preconditioning Iterative Methods for PDE-Constrained Optimization, Ph.D thesis, University of Oxford, 2010.
[34]	S. Roggendorf, Model Order Reduction for Linearized Systems Arising from the Simulation of Gas Transportation Networks, Master's thesis, Rheinischen Friedrich-Wilhelms-Universität Bonn, Germany, 2015.
[35]	Y. Saad, Iterative Methods for Sparse Linear Systems, Society for Industrial and Applied Mathematics, Philadelphia, 2003. doi: 10.1137/1.9780898718003.
[36]	P. Sonneveld and M. B. van Gijzen, IDR(s): A family of simple and fast algorithms for solving large nonsymmetric systems of linear equations, SIAM J. Sci. Comput., 31 (2008), 1035-1062. doi: 10.1137/070685804.
[37]	M. C. Steinbach, On PDE solution in transient optimization of gas networks, J. Comput. Appl. Math., 203 (2007), 345-361. doi: 10.1016/j.cam.2006.04.018.
[38]	M. Stoll and T. Breiten, A low-rank in time approach to PDE-constrained optimization, SIAM J. Sci. Comput., 37 (2015), B1–B29. doi: 10.1137/130926365.
[39]	W. Q. Tao and H. C. Ti, Transient analysis of gas pipeline network, Chem. Eng. J., 69 (1998), 47-52. doi: 10.1016/S1385-8947(97)00109-5.
[40]	E. F. Toro and S. J. Billett, Centred TVD schemes for hyperbolic conservation laws, IMA J. Numer. Anal., 20 (2000), 47-79. doi: 10.1093/imanum/20.1.47.
[41]	M. B. van Gijzen and P. Sonneveld, Algorithm 913: An elegant IDR(s) variant that efficiently exploits biorthogonality properties, ACM Trans. Math. Software, 38 (2011), 5: 1–5: 19. doi: 10.1145/2049662.2049667.
[42]	A. J. Wathen, Preconditioning, Acta Numer., 24 (2015), 329-376. doi: 10.1017/S0962492915000021.
[43]	M. Wathen, C. Greif and D. Schötzau, Preconditioners for mixed finite element discretizations of incompressible MHD equations, SIAM J. Sci. Comput., 39 (2017), A2993–A3013. doi: 10.1137/16M1098991.
[44]	J. Zhou and M. A. Adewumi, Simulation of transients in natural gas pipelines using hybrid TVD schemes, Internat. J. Numer. Methods Fluids, 32 (2000), 407-437. doi: 10.1002/(SICI)1097-0363(20000229)32:4<407::AID-FLD945>3.0.CO;2-9.
[45]	A. Zlotnik, M. Chertkov and S. Backhaus, Optimal control of transient flow in natural gas networks, in 2015 54th IEEE Conference on Decision and Control (CDC), 2015, 4563–4570. doi: 10.1109/TCNS.2014.2367360.

Figure 1. Separation of control volume $ C_i $

$ h $	$ \# D_F $	with DF	without DF
20	2.01e+05	$ 8.12 $	8.75
10	3.97e+05	$ 17.84 $	19.14
5	7.91e+05	$ 38.44 $	41.75
2.5	1.58e+06	$ 81.42 $	87.77

Newton iter.	1	2	3	4
FVM	1.56e+07	1.57e+07	1.57e+07	1.57e+07
FDM	1.24e+08	1.25e+08	1.25e+08	1.25e+08

$ h $	$ \# D_F $	$ t_{S^1} $	IDR($ 4 $)	backslash
40	1.03e+05	$ 3.85 $	0.25	0.13
20	2.01e+05	$ 8.12 $	0.52	0.36
10	3.97e+05	$ 17.84 $	1.06	1.18
5	7.91e+05	$ 38.44 $	2.13	1054.62
2.5	1.58e+06	$ 81.42 $	4.34	-

[1]	Mapundi K. Banda, Michael Herty, Axel Klar. Coupling conditions for gas networks governed by the isothermal Euler equations. Networks and Heterogeneous Media, 2006, 1 (2) : 295-314. doi: 10.3934/nhm.2006.1.295
[2]	M. D. König, Stefano Battiston, M. Napoletano, F. Schweitzer. On algebraic graph theory and the dynamics of innovation networks. Networks and Heterogeneous Media, 2008, 3 (2) : 201-219. doi: 10.3934/nhm.2008.3.201
[3]	Michael Herty. Modeling, simulation and optimization of gas networks with compressors. Networks and Heterogeneous Media, 2007, 2 (1) : 81-97. doi: 10.3934/nhm.2007.2.81
[4]	Mario Roy, Mariusz Urbański. Random graph directed Markov systems. Discrete and Continuous Dynamical Systems, 2011, 30 (1) : 261-298. doi: 10.3934/dcds.2011.30.261
[5]	Mario Roy, Mariusz Urbański. Multifractal analysis for conformal graph directed Markov systems. Discrete and Continuous Dynamical Systems, 2009, 25 (2) : 627-650. doi: 10.3934/dcds.2009.25.627
[6]	Loïs Boullu, Mostafa Adimy, Fabien Crauste, Laurent Pujo-Menjouet. Oscillations and asymptotic convergence for a delay differential equation modeling platelet production. Discrete and Continuous Dynamical Systems - B, 2019, 24 (6) : 2417-2442. doi: 10.3934/dcdsb.2018259
[7]	David Mumford, Peter W. Michor. On Euler's equation and 'EPDiff'. Journal of Geometric Mechanics, 2013, 5 (3) : 319-344. doi: 10.3934/jgm.2013.5.319
[8]	Mario Roy. A new variation of Bowen's formula for graph directed Markov systems. Discrete and Continuous Dynamical Systems, 2012, 32 (7) : 2533-2551. doi: 10.3934/dcds.2012.32.2533
[9]	Nataša Djurdjevac Conrad, Ralf Banisch, Christof Schütte. Modularity of directed networks: Cycle decomposition approach. Journal of Computational Dynamics, 2015, 2 (1) : 1-24. doi: 10.3934/jcd.2015.2.1
[10]	Ciprian G. Gal, Maurizio Grasselli. The non-isothermal Allen-Cahn equation with dynamic boundary conditions. Discrete and Continuous Dynamical Systems, 2008, 22 (4) : 1009-1040. doi: 10.3934/dcds.2008.22.1009
[11]	Monika Eisenmann, Etienne Emmrich, Volker Mehrmann. Convergence of the backward Euler scheme for the operator-valued Riccati differential equation with semi-definite data. Evolution Equations and Control Theory, 2019, 8 (2) : 315-342. doi: 10.3934/eect.2019017
[12]	Giovanni Bonfanti, Arrigo Cellina. The validity of the Euler-Lagrange equation. Discrete and Continuous Dynamical Systems, 2010, 28 (2) : 511-517. doi: 10.3934/dcds.2010.28.511
[13]	Mapundi K. Banda, Michael Herty, Axel Klar. Gas flow in pipeline networks. Networks and Heterogeneous Media, 2006, 1 (1) : 41-56. doi: 10.3934/nhm.2006.1.41
[14]	Martin Gugat, Falk M. Hante, Markus Hirsch-Dick, Günter Leugering. Stationary states in gas networks. Networks and Heterogeneous Media, 2015, 10 (2) : 295-320. doi: 10.3934/nhm.2015.10.295
[15]	Quentin Chauleur. The isothermal limit for the compressible Euler equations with damping. Discrete and Continuous Dynamical Systems - B, 2022 doi: 10.3934/dcdsb.2022059
[16]	Jaime Angulo Pava, Nataliia Goloshchapova. On the orbital instability of excited states for the NLS equation with the δ-interaction on a star graph. Discrete and Continuous Dynamical Systems, 2018, 38 (10) : 5039-5066. doi: 10.3934/dcds.2018221
[17]	El Miloud Zaoui, Marc Laforest. Stability and modeling error for the Boltzmann equation. Kinetic and Related Models, 2014, 7 (2) : 401-414. doi: 10.3934/krm.2014.7.401
[18]	Michael Herty, Veronika Sachers. Adjoint calculus for optimization of gas networks. Networks and Heterogeneous Media, 2007, 2 (4) : 733-750. doi: 10.3934/nhm.2007.2.733
[19]	Stéphane Brull, Pierre Charrier, Luc Mieussens. Gas-surface interaction and boundary conditions for the Boltzmann equation. Kinetic and Related Models, 2014, 7 (2) : 219-251. doi: 10.3934/krm.2014.7.219
[20]	Raffaele Esposito, Mario Pulvirenti. Rigorous validity of the Boltzmann equation for a thin layer of a rarefied gas. Kinetic and Related Models, 2010, 3 (2) : 281-297. doi: 10.3934/krm.2010.3.281

American Institute of Mathematical Sciences

Efficient numerical methods for gas network modeling and simulation

References:

References:

Tools

Metrics

Other articlesby authors

Tools

Tools

Tools

Metrics

Other articlesby authors

Other articles
by authors

Other articles
by authors