Authors

H Egger, J Giesselmann, T Kunkel, N Philippi

Abstract

We consider the simulation of barotropic flow of gas in long pipes and pipe networks. Based on a Hamiltonian reformulation of the governing system, a fully discrete approximation scheme is proposed using mixed finite elements in space and an implicit Euler method in time. Assuming the existence of a smooth subsonic solution bounded away from vacuum, a full convergence analysis is presented based on relative energy estimates. Particular attention is paid to establishing error bounds that are uniform in the friction parameter. As a consequence, the method and results also cover the parabolic problem arising in the asymptotic large friction limit. The error estimates are derived in detail for a single pipe, but using appropriate coupling conditions and the particular structure of the problem and its discretization, the main results directly generalize to pipe networks. Numerical tests are presented for illustration.

Citation

  • Journal: IMA Journal of Numerical Analysis
  • Year: 2023
  • Volume: 43
  • Issue: 4
  • Pages: 2137–2168
  • Publisher: Oxford University Press (OUP)
  • DOI: 10.1093/imanum/drac032

BibTeX

@article{Egger_2022,
  title={{An asymptotic-preserving discretization scheme for gas transport in pipe networks}},
  volume={43},
  ISSN={1464-3642},
  DOI={10.1093/imanum/drac032},
  number={4},
  journal={IMA Journal of Numerical Analysis},
  publisher={Oxford University Press (OUP)},
  author={Egger, H and Giesselmann, J and Kunkel, T and Philippi, N},
  year={2022},
  pages={2137--2168}
}

Download the bib file

References

  • Bamberger, Analyse et contrôle d’un réseau de transport de gaz. Computing Methods in Applied Sciences and Engineering (Proc. Third Internat. Sympos., Versailles, 1977), II (1979)
  • Berthon, C., Bessemoulin-Chatard, M. & Mathis, H. Numerical convergence rate for a diffusive limit of hyperbolic systems: $p$-system with damping. The SMAI journal of computational mathematics vol. 2 99–119 (2016) – 10.5802/smai-jcm.10
  • Brenner, S. C. & Scott, L. R. The Mathematical Theory of Finite Element Methods. Texts in Applied Mathematics (Springer New York, 2008). doi:10.1007/978-0-387-75934-0 – 10.1007/978-0-387-75934-0
  • Bressan, A., Čanić, S., Garavello, M., Herty, M. & Piccoli, B. Flows on networks: recent results and perspectives. EMS Surveys in Mathematical Sciences vol. 1 47–111 (2014) – 10.4171/emss/2
  • Brouwer, J., Gasser, I. & Herty, M. Gas Pipeline Models Revisited: Model Hierarchies, Nonisothermal Models, and Simulations of Networks. Multiscale Modeling & Simulation vol. 9 601–623 (2011) – 10.1137/100813580
  • Burlacu, R. et al. Maximizing the storage capacity of gas networks: a global MINLP approach. Optimization and Engineering vol. 20 543–573 (2018) – 10.1007/s11081-018-9414-5
  • Cardoso-Ribeiro, A partitioned finite element method for power-preserving discretization of open systems of conservation laws. IMA J. Math. Control. Inf. (2019)
  • Dafermos, C. M. Hyberbolic Conservation Laws in Continuum Physics. Grundlehren der mathematischen Wissenschaften (Springer Berlin Heidelberg, 2005). doi:10.1007/3-540-29089-3 – 10.1007/3-540-29089-3
  • Egger, H. A Robust Conservative Mixed Finite Element Method for Isentropic Compressible Flow on Pipe Networks. SIAM Journal on Scientific Computing vol. 40 A108–A129 (2018) – 10.1137/16m1094373
  • Egger, H. Structure preserving approximation of dissipative evolution problems. Numerische Mathematik vol. 143 85–106 (2019) – 10.1007/s00211-019-01050-w
  • Egger, Stability and asymptotic analysis for instationary gas transport via relative energy estimates. (2020)
  • Feireisl, E., Lukáčová-Medviďová, M., Nečasová, Š., Novotný, A. & She, B. Asymptotic Preserving Error Estimates for Numerical Solutions of Compressible Navier–Stokes Equations in the Low Mach Number Regime. Multiscale Modeling & Simulation vol. 16 150–183 (2018) – 10.1137/16m1094233
  • Gallouët, T., Herbin, R., Maltese, D. & Novotny, A. Error estimates for a numerical approximation to the compressible barotropic Navier–Stokes equations. IMA Journal of Numerical Analysis vol. 36 543–592 (2015) – 10.1093/imanum/drv028
  • Geveci, T. On the application of mixed finite element methods to the wave equations. ESAIM: Mathematical Modelling and Numerical Analysis vol. 22 243–250 (1988) – 10.1051/m2an/1988220202431
  • Giesselmann, J., Lattanzio, C. & Tzavaras, A. E. Relative Energy for the Korteweg Theory and Related Hamiltonian Flows in Gas Dynamics. Archive for Rational Mechanics and Analysis vol. 223 1427–1484 (2016) – 10.1007/s00205-016-1063-2
  • Joly, P. Variational Methods for Time-Dependent Wave Propagation Problems. Lecture Notes in Computational Science and Engineering 201–264 (2003) doi:10.1007/978-3-642-55483-4_6 – 10.1007/978-3-642-55483-4_6
  • John, V. Finite Element Methods for Incompressible Flow Problems. Springer Series in Computational Mathematics (Springer International Publishing, 2016). doi:10.1007/978-3-319-45750-5 – 10.1007/978-3-319-45750-5
  • Jüngel, A. Entropy Methods for Diffusive Partial Differential Equations. SpringerBriefs in Mathematics (Springer International Publishing, 2016). doi:10.1007/978-3-319-34219-1 – 10.1007/978-3-319-34219-1
  • Kwon, Consistency, convergence and error estimates for a mixed finite element/finite volume scheme to compressible Navier–Stokes equations with general inflow/outflow boundary data. (2020)
  • Liljegren-Sailer, On port-Hamiltonian approximation of a nonlinear flow problem on networks. SIAM J. Sci. Comput. (2020)
  • Osiadacz, A. Simulation of transient gas flows in networks. International Journal for Numerical Methods in Fluids vol. 4 13–24 (1984) – 10.1002/fld.1650040103
  • Raviart, P. A. Sur la résolution de certaines equations paraboliques non linéaires. Journal of Functional Analysis vol. 5 299–328 (1970) – 10.1016/0022-1236(70)90031-5
  • Reigstad, G. A. Existence and Uniqueness of Solutions to the Generalized Riemann Problem for Isentropic Flow. SIAM Journal on Applied Mathematics vol. 75 679–702 (2015) – 10.1137/140962759
  • Schmidt, M. et al. GasLib—A Library of Gas Network Instances. Data vol. 2 40 (2017) – 10.3390/data2040040
  • Schöbel-Kröhn, Analysis and Numerical Approximation of Nonlinear Evolution Equations on Network Structures (2020)
  • Yee, Numerical approximation of boundary conditions with applications to inviscid equations of gas dynamics. Technical Report TM-18265 (1981)
  • Zeidler, E. Nonlinear Functional Analysis and Its Applications. (Springer New York, 1986). doi:10.1007/978-1-4612-4838-5 – 10.1007/978-1-4612-4838-5