On Port-Hamiltonian Approximation of a Nonlinear Flow Problem on Networks

Authors

Björn Liljegren-Sailer, Nicole Marheineke

Citation

• Journal: SIAM Journal on Scientific Computing
• Year: 2022
• Volume: 44
• Issue: 3
• Pages: B834–B859
• Publisher: Society for Industrial & Applied Mathematics (SIAM)
• DOI: 10.1137/21m1443480

BibTeX

@article{Liljegren_Sailer_2022,
  title={{On Port-Hamiltonian Approximation of a Nonlinear Flow Problem on Networks}},
  volume={44},
  ISSN={1095-7197},
  DOI={10.1137/21m1443480},
  number={3},
  journal={SIAM Journal on Scientific Computing},
  publisher={Society for Industrial & Applied Mathematics (SIAM)},
  author={Liljegren-Sailer, Björn and Marheineke, Nicole},
  year={2022},
  pages={B834--B859}
}

Download the bib file

References

• Absil, P.-A., Mahony, R. & Sepulchre, R. Optimization Algorithms on Matrix Manifolds. (2008) doi:10.1515/9781400830244 – 10.1515/9781400830244
• Afkham, B. M. & Hesthaven, J. S. Structure Preserving Model Reduction of Parametric Hamiltonian Systems. SIAM Journal on Scientific Computing vol. 39 A2616–A2644 (2017) – 10.1137/17m1111991
• Maboudi Afkham, B. & Hesthaven, J. S. Structure-Preserving Model-Reduction of Dissipative Hamiltonian Systems. Journal of Scientific Computing vol. 81 3–21 (2018) – 10.1007/s10915-018-0653-6
• An, S. S., Kim, T. & James, D. L. Optimizing cubature for efficient integration of subspace deformations. ACM Transactions on Graphics vol. 27 1–10 (2008) – 10.1145/1409060.1409118
• Antonelli, P. & Marcati, P. The Quantum Hydrodynamics System in Two Space Dimensions. Archive for Rational Mechanics and Analysis vol. 203 499–527 (2011) – 10.1007/s00205-011-0454-7
• Compatible Spatial Discretizations. The IMA Volumes in Mathematics and its Applications (Springer New York, 2006). doi:10.1007/0-387-38034-5 – 10.1007/0-387-38034-5
• Badlyan A. M., Open Physical Systems: From GENERIC to Port-Hamiltonian Systems, https://arxiv.org/abs/1804.04064 (2018)
• Baur, U., Benner, P. & Feng, L. Model Order Reduction for Linear and Nonlinear Systems: A System-Theoretic Perspective. Archives of Computational Methods in Engineering vol. 21 331–358 (2014) – 10.1007/s11831-014-9111-2
• Beattie C. A., Math. Control Signals Systems (2018)
• Dimension Reduction of Large-Scale Systems. Lecture Notes in Computational Science and Engineering (Springer Berlin Heidelberg, 2005). doi:10.1007/3-540-27909-1 – 10.1007/3-540-27909-1
• Braess, D. Finite Elements. (2007) doi:10.1017/cbo9780511618635 – 10.1017/cbo9780511618635
• Brezis, H. Functional Analysis, Sobolev Spaces and Partial Differential Equations. (Springer New York, 2011). doi:10.1007/978-0-387-70914-7 – 10.1007/978-0-387-70914-7
• Carlberg, K., Tuminaro, R. & Boggs, P. Preserving Lagrangian Structure in Nonlinear Model Reduction with Application to Structural Dynamics. SIAM Journal on Scientific Computing vol. 37 B153–B184 (2015) – 10.1137/140959602
• Celledoni, E. et al. Preserving energy resp. dissipation in numerical PDEs using the “Average Vector Field” method. Journal of Computational Physics vol. 231 6770–6789 (2012) – 10.1016/j.jcp.2012.06.022
• Chalot, F., Hughes, T. J. R. & Shakib, F. Symmetrization of conservation laws with entropy for high-temperature hypersonic computations. Computing Systems in Engineering vol. 1 495–521 (1990) – 10.1016/0956-0521(90)90032-g
• Chan, J. Entropy stable reduced order modeling of nonlinear conservation laws. Journal of Computational Physics vol. 423 109789 (2020) – 10.1016/j.jcp.2020.109789
• Chaturantabut, S., Beattie, C. & Gugercin, S. Structure-Preserving Model Reduction for Nonlinear Port-Hamiltonian Systems. SIAM Journal on Scientific Computing vol. 38 B837–B865 (2016) – 10.1137/15m1055085
• Christiansen, S. H., Munthe-Kaas, H. Z. & Owren, B. Topics in structure-preserving discretization. Acta Numerica vol. 20 1–119 (2011) – 10.1017/s096249291100002x
• Clees, T. et al. MathEnergy – Mathematical Key Technologies for Evolving Energy Grids. Mathematics in Industry 233–262 (2021) doi:10.1007/978-3-030-62732-4_11 – 10.1007/978-3-030-62732-4_11
• Domschke, P., Kolb, O. & Lang, J. Adjoint-based error control for the simulation and optimization of gas and water supply networks. Applied Mathematics and Computation vol. 259 1003–1018 (2015) – 10.1016/j.amc.2015.03.029
• 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 H., Stability and Asymptotic Analysis for Instationary Gas Transport via Relative Energy Estimates, https://arxiv.org/abs/2012.14135 (2020)
• Egger, H. & Kugler, T. Damped wave systems on networks: exponential stability and uniform approximations. Numerische Mathematik vol. 138 839–867 (2017) – 10.1007/s00211-017-0924-4
• Egger H., Hyperbolic Problems: Theory, Numerics, Applications, AIMS Ser. Appl. Math 10 (2020)
• Egger, H., Kugler, T., Liljegren-Sailer, B., Marheineke, N. & Mehrmann, V. On Structure-Preserving Model Reduction for Damped Wave Propagation in Transport Networks. SIAM Journal on Scientific Computing vol. 40 A331–A365 (2018) – 10.1137/17m1125303
• Farhat, C., Avery, P., Chapman, T. & Cortial, J. Dimensional reduction of nonlinear finite element dynamic models with finite rotations and energy‐based mesh sampling and weighting for computational efficiency. International Journal for Numerical Methods in Engineering vol. 98 625–662 (2014) – 10.1002/nme.4668
• Farle, O., Klis, D., Jochum, M., Floch, O. & Dyczij-Edlinger, R. A port-hamiltonian finite-element formulation for the maxwell equations. 2013 International Conference on Electromagnetics in Advanced Applications (ICEAA) 324–327 (2013) doi:10.1109/iceaa.2013.6632246 – 10.1109/iceaa.2013.6632246
• Fisher, T. C. & Carpenter, M. H. High-order entropy stable finite difference schemes for nonlinear conservation laws: Finite domains. Journal of Computational Physics vol. 252 518–557 (2013) – 10.1016/j.jcp.2013.06.014
• 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
• Golo, G., Talasila, V., van der Schaft, A. & Maschke, B. Hamiltonian discretization of boundary control systems. Automatica vol. 40 757–771 (2004) – 10.1016/j.automatica.2003.12.017
• Grundel S., Progress in Differential-Algebraic Equations: Deskriptor (2013)
• Gugat, M. & Ulbrich, S. The isothermal Euler equations for ideal gas with source term: Product solutions, flow reversal and no blow up. Journal of Mathematical Analysis and Applications vol. 454 439–452 (2017) – 10.1016/j.jmaa.2017.04.064
• Gugercin S., Proceedings of the 48th IEEE Conference on Decision and Control, and the 28th Chinese Control Conference (2009)
• Hairer E., Geometric Numerical Integration (2006)
• Harten, A. On the symmetric form of systems of conservation laws with entropy. Journal of Computational Physics vol. 49 151–164 (1983) – 10.1016/0021-9991(83)90118-3
• Hauschild, S.-A. et al. Port-Hamiltonian Modeling of District Heating Networks. Differential-Algebraic Equations Forum 333–355 (2020) doi:10.1007/978-3-030-53905-4_11 – 10.1007/978-3-030-53905-4_11
• Herrán-González, A., De La Cruz, J. M., De Andrés-Toro, B. & Risco-Martín, J. L. Modeling and simulation of a gas distribution pipeline network. Applied Mathematical Modelling vol. 33 1584–1600 (2009) – 10.1016/j.apm.2008.02.012
• Jameson, A. The Construction of Discretely Conservative Finite Volume Schemes that Also Globally Conserve Energy or Entropy. Journal of Scientific Computing vol. 34 152–187 (2007) – 10.1007/s10915-007-9171-7
• Kotyczka, P., Maschke, B. & Lefèvre, L. Weak form of Stokes–Dirac structures and geometric discretization of port-Hamiltonian systems. Journal of Computational Physics vol. 361 442–476 (2018) – 10.1016/j.jcp.2018.02.006
• Lee, D. & Palha, A. A mixed mimetic spectral element model of the rotating shallow water equations on the cubed sphere. Journal of Computational Physics vol. 375 240–262 (2018) – 10.1016/j.jcp.2018.08.042
• LeVeque R. J., Finite
• Liljegren-Sailer, B. & Marheineke, N. On Port-Hamiltonian Approximation of a Nonlinear Flow Problem on Networks. SIAM Journal on Scientific Computing vol. 44 B834–B859 (2022) – 10.1137/21m1443480
• Liljegren-Sailer B., Progress in Industrial Mathematics at ECMI (2016)
• Liljegren-Sailer B., On Snapshot-Based Model Reduction Under Compatibility Conditions for a Nonlinear Flow Problem on Networks, https://arxiv.org/abs/2110.04777 (2021)
• Maschke B. M., IFAC Proceedings Volumes (1992)
• Maschke B. M., Nonlinear Control in the Year (2000)
• McInerney A., Contact, Symplectic, Undergrad Texts Math. (2013)
• Mehrmann, V. & Morandin, R. Structure-preserving discretization for port-Hamiltonian descriptor systems. 2019 IEEE 58th Conference on Decision and Control (CDC) 6863–6868 (2019) doi:10.1109/cdc40024.2019.9030180 – 10.1109/cdc40024.2019.9030180
• Mei, M. Nonlinear diffusion waves for hyperbolic p-system with nonlinear damping. Journal of Differential Equations vol. 247 1275–1296 (2009) – 10.1016/j.jde.2009.04.004
• Mindt, P., Lang, J. & Domschke, P. Entropy-Preserving Coupling of Hierarchical Gas Models. SIAM Journal on Mathematical Analysis vol. 51 4754–4775 (2019) – 10.1137/19m1240034
• Mock, M. S. Systems of conservation laws of mixed type. Journal of Differential Equations vol. 37 70–88 (1980) – 10.1016/0022-0396(80)90089-3
• Badlyan A. Moses, Proceedings of the 23rd International Symposium on Mathematical Theory of Systems and Networks (2018)
• Novotny A., Oxford Lecture Ser. Math. Appl. 27 (2004)
• Pasumarthy, R., Ambati, V. R. & van der Schaft, A. J. Port-Hamiltonian discretization for open channel flows. Systems & Control Letters vol. 61 950–958 (2012) – 10.1016/j.sysconle.2012.05.003
• Peng, L. & Mohseni, K. Symplectic Model Reduction of Hamiltonian Systems. SIAM Journal on Scientific Computing vol. 38 A1–A27 (2016) – 10.1137/140978922
• 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
• Rockafellar, R. T. & Wets, R. J. B. Variational Analysis. Grundlehren der mathematischen Wissenschaften (Springer Berlin Heidelberg, 1998). doi:10.1007/978-3-642-02431-3 – 10.1007/978-3-642-02431-3
• Rockafellar, R. T. Convex Analysis. (1970) doi:10.1515/9781400873173 – 10.1515/9781400873173
• Ruggeri, T. Global existence of smooth solutions and stability of the constant state for dissipative hyperbolic systems with applications to extended thermodynamics. Trends and Applications of Mathematics to Mechanics 215–224 (2005) doi:10.1007/88-470-0354-7_17 – 10.1007/88-470-0354-7_17
• Schmidt, M. et al. GasLib—A Library of Gas Network Instances. Data vol. 2 40 (2017) – 10.3390/data2040040
• Svärd, M. & Nordström, J. Review of summation-by-parts schemes for initial–boundary-value problems. Journal of Computational Physics vol. 268 17–38 (2014) – 10.1016/j.jcp.2014.02.031
• van der Schaft, A. & Jeltsema, D. Port-Hamiltonian Systems Theory: An Introductory Overview. Foundations and Trends® in Systems and Control vol. 1 173–378 (2014) – 10.1561/2600000002
• van der Schaft, A. J. & Maschke, B. M. Port-Hamiltonian Systems on Graphs. SIAM Journal on Control and Optimization vol. 51 906–937 (2013) – 10.1137/110840091
• Winters, A. R., Czernik, C., Schily, M. B. & Gassner, G. J. Entropy stable numerical approximations for the isothermal and polytropic Euler equations. BIT Numerical Mathematics vol. 60 791–824 (2019) – 10.1007/s10543-019-00789-w
• Wolf, T., Lohmann, B., Eid, R. & Kotyczka, P. Passivity and Structure Preserving Order Reduction of Linear Port-Hamiltonian Systems Using Krylov Subspaces. European Journal of Control vol. 16 401–406 (2010) – 10.3166/ejc.16.401-406
• Zeidler E., Nonlinear Functional Analysis and Its Applications III: Variational Methods and Optimization (1985)