Discrete gradient methods for port-Hamiltonian differential-algebraic equations

Authors

Philipp L. Kinon, Riccardo Morandin, Philipp Schulze

Abstract

Discrete gradient methods are a powerful tool for the time discretization of dynamical systems, since they are structure-preserving regardless of the form of the total energy. In this work, we discuss the application of discrete gradient methods to the system class of nonlinear port-Hamiltonian differential-algebraic equations - as they emerge from the port- and energy-based modeling of physical systems in various domains. We introduce a novel numerical scheme tailored for semi-explicit differential-algebraic equations and further address more general settings using the concepts of discrete gradient pairs and Dirac-dissipative structures. Additionally, the behavior under system transformations is investigated and we demonstrate that under suitable assumptions port-Hamiltonian differential-algebraic equations admit a representation which consists of a parametrized port-Hamiltonian semi-explicit system and an unstructured equation. Finally, we present the application to multibody system dynamics and discuss numerical results to demonstrate the capabilities of our approach.

Keywords

differential-algebraic equations, discrete gradients, port-hamiltonian systems, structure-preserving discretization, time integration methods

Citation

• Journal: Applied Numerical Mathematics
• Year: 2026
• Volume: 223
• Issue:
• Pages: 45–75
• Publisher: Elsevier BV
• DOI: 10.1016/j.apnum.2025.12.006

BibTeX

@article{Kinon_2026,
  title={{Discrete gradient methods for port-Hamiltonian differential-algebraic equations}},
  volume={223},
  ISSN={0168-9274},
  DOI={10.1016/j.apnum.2025.12.006},
  journal={Applied Numerical Mathematics},
  publisher={Elsevier BV},
  author={Kinon, Philipp L. and Morandin, Riccardo and Schulze, Philipp},
  year={2026},
  pages={45--75}
}

Download the bib file

References

• Duindam, (2009)
• van der Schaft A, Jeltsema D (2014) Port-Hamiltonian Systems Theory: An Introductory Overview. Foundations and Trends® in Systems and Control 1(2–3):173–378. https://doi.org/10.1561/260000000 – 10.1561/2600000002
• Beattie C, Mehrmann V, Xu H, Zwart H (2018) Linear port-Hamiltonian descriptor systems. Math Control Signals Syst 30(4). https://doi.org/10.1007/s00498-018-0223- – 10.1007/s00498-018-0223-3
• Mehrmann, Structure-preserving discretization for port-Hamiltonian descriptor systems. (2019)
• van der Schaft, Port-Hamiltonian differential-algebraic systems. (2013)
• van der Schaft A, Maschke B (2020) Dirac and Lagrange Algebraic Constraints in Nonlinear Port-Hamiltonian Systems. Vietnam J Math 48(4):929–939. https://doi.org/10.1007/s10013-020-00419- – 10.1007/s10013-020-00419-x
• van der Schaft A, Maschke B (2018) Generalized port-Hamiltonian DAE systems. Systems & Control Letters 121:31–37. https://doi.org/10.1016/j.sysconle.2018.09.00 – 10.1016/j.sysconle.2018.09.008
• Hairer, (2006)
• Hirsch, (1974)
• Arnold, (1989)
• Gonzalez O (1999) Mechanical systems subject to holonomic constraints: Differential–algebraic formulations and conservative integration. Physica D: Nonlinear Phenomena 132(1–2):165–174. https://doi.org/10.1016/s0167-2789(99)00054- – 10.1016/s0167-2789(99)00054-8
• Gonzalez O (2000) Exact energy and momentum conserving algorithms for general models in nonlinear elasticity. Computer Methods in Applied Mechanics and Engineering 190(13–14):1763–1783. https://doi.org/10.1016/s0045-7825(00)00189- – 10.1016/s0045-7825(00)00189-4
• Simo JC, Tarnow N (1992) The discrete energy-momentum method. Conserving algorithms for nonlinear elastodynamics. Z angew Math Phys 43(5):757–792. https://doi.org/10.1007/bf0091340 – 10.1007/bf00913408
• Structure-Preserving Integrators in Nonlinear Structural Dynamics and Flexible Multibody Dynamics. (2016)
• Kinon PL, Betsch P, Schneider S (2023) Structure-preserving integrators based on a new variational principle for constrained mechanical systems. Nonlinear Dyn 111(15):14231–14261. https://doi.org/10.1007/s11071-023-08522- – 10.1007/s11071-023-08522-7
• Marsden JE, West M (2001) Discrete mechanics and variational integrators. Acta Numerica 10:357–514. https://doi.org/10.1017/s096249290100006 – 10.1017/s096249290100006x
• Leyendecker S, Marsden JE, Ortiz M (2008) Variational integrators for constrained dynamical systems. Z Angew Math Mech 88(9):677–708. https://doi.org/10.1002/zamm.20070017 – 10.1002/zamm.200700173
• Leimkuhler BJ, Skeel RD (1994) Symplectic Numerical Integrators in Constrained Hamiltonian Systems. Journal of Computational Physics 112(1):117–125. https://doi.org/10.1006/jcph.1994.108 – 10.1006/jcph.1994.1085
• May M, Betsch P (2025) Galerkin-based time integration approaches to rigid body dynamics in terms of unit quaternion – 10.21203/rs.3.rs-6354821/v1
• Betsch P, Steinmann P (2000) Conservation properties of a time FE method. Part I: time-stepping schemes forN-body problems. Int J Numer Meth Engng 49(5):599–638. https://doi.org/10.1002/1097-0207(20001020)49:5<599::aid-nme960>3.0.co;2- – 10.1002/1097-0207(20001020)49:5<599::aid-nme960>3.0.co;2-9
• Betsch P, Steinmann P (2001) Conservation properties of a time FE method—part II: Time‐stepping schemes for non‐linear elastodynamics. Numerical Meth Engineering 50(8):1931–1955. https://doi.org/10.1002/nme.10 – 10.1002/nme.103
• Betsch P, Steinmann P (2002) Conservation properties of a time FE method—part III: Mechanical systems with holonomic constraints. Numerical Meth Engineering 53(10):2271–2304. https://doi.org/10.1002/nme.34 – 10.1002/nme.347
• Egger H, Habrich O, Shashkov V (2020) On the Energy Stable Approximation of Hamiltonian and Gradient Systems. Computational Methods in Applied Mathematics 21(2):335–349. https://doi.org/10.1515/cmam-2020-002 – 10.1515/cmam-2020-0025
• Jüngel A, Stefanelli U, Trussardi L (2019) Two Structure-Preserving Time Discretizations for Gradient Flows. Appl Math Optim 80(3):733–764. https://doi.org/10.1007/s00245-019-09605- – 10.1007/s00245-019-09605-x
• Kunkel P, Mehrmann V (2023) Discretization of inherent ODEs and the geometric integration of DAEs with symmetries. Bit Numer Math 63(2). https://doi.org/10.1007/s10543-023-00966- – 10.1007/s10543-023-00966-y
• Öttinger HC (2018) GENERIC Integrators: Structure Preserving Time Integration for Thermodynamic Systems. Journal of Non-Equilibrium Thermodynamics 43(2):89–100. https://doi.org/10.1515/jnet-2017-003 – 10.1515/jnet-2017-0034
• Kotyczka P, Lefèvre L (2019) Discrete-time port-Hamiltonian systems: A definition based on symplectic integration. Systems & Control Letters 133:104530. https://doi.org/10.1016/j.sysconle.2019.10453 – 10.1016/j.sysconle.2019.104530
• Giesselmann J, Karsai A, Tscherpel T (2025) Energy-consistent Petrov–Galerkin time discretization of port-Hamiltonian systems. The SMAI Journal of computational mathematics 11:335–367. https://doi.org/10.5802/smai-jcm.12 – 10.5802/smai-jcm.127
• Bartel A, Diab M, Frommer A, Günther M, Marheineke N (2025) Splitting techniques for DAEs with port-Hamiltonian applications. Applied Numerical Mathematics 214:28–53. https://doi.org/10.1016/j.apnum.2025.03.00 – 10.1016/j.apnum.2025.03.004
• Mönch M, Marheineke N (2025) Commutator-based operator splitting for linear port-Hamiltonian systems. Applied Numerical Mathematics 210:25–38. https://doi.org/10.1016/j.apnum.2024.12.00 – 10.1016/j.apnum.2024.12.007
• Bartel A, Schaller M (2025) Goal-oriented time adaptivity for port-Hamiltonian systems. Journal of Computational and Applied Mathematics 461:116450. https://doi.org/10.1016/j.cam.2024.11645 – 10.1016/j.cam.2024.116450
• Eidnes S (2022) Order theory for discrete gradient methods. Bit Numer Math 62(4):1207–1255. https://doi.org/10.1007/s10543-022-00909- – 10.1007/s10543-022-00909-z
• Sato S (2019) Linear gradient structures and discrete gradient methods for conservative/dissipative differential-algebraic equations. Bit Numer Math 59(4):1063–1091. https://doi.org/10.1007/s10543-019-00759- – 10.1007/s10543-019-00759-2
• Aoues S, Di Loreto M, Eberard D, Marquis-Favre W (2017) Hamiltonian systems discrete-time approximation: Losslessness, passivity and composability. Systems & Control Letters 110:9–14. https://doi.org/10.1016/j.sysconle.2017.10.00 – 10.1016/j.sysconle.2017.10.003
• Falaize A, Hélie T (2016) Passive Guaranteed Simulation of Analog Audio Circuits: A Port-Hamiltonian Approach. Applied Sciences 6(10):273. https://doi.org/10.3390/app610027 – 10.3390/app6100273
• Gören-Sümer L, Yalçιn Y (2008) Gradient Based Discrete-Time Modeling and Control of Hamiltonian Systems. IFAC Proceedings Volumes 41(2):212–217. https://doi.org/10.3182/20080706-5-kr-1001.0003 – 10.3182/20080706-5-kr-1001.00036
• Kinon, Discrete nonlinear elastodynamics in a port-Hamiltonian framework. PAMM (Proc. Appl. Math. Mech.) (2023)
• Moreschini, Discrete port-controlled Hamiltonian dynamics and average passivation. (2019)
• Rabier PJ, Rheinboldt WC (1994) On Impasse Points of Quasilinear Differential-Algebraic Equations. Journal of Mathematical Analysis and Applications 181(2):429–454. https://doi.org/10.1006/jmaa.1994.103 – 10.1006/jmaa.1994.1033
• Steinbrecher, (2006)
• Kunkel P, Mehrmann V (2006) Differential-Algebraic Equations. EMS Textbooks in Mathematic – 10.4171/017
• Mehrmann V, Unger B (2023) Control of port-Hamiltonian differential-algebraic systems and applications. Acta Numerica 32:395–515. https://doi.org/10.1017/s096249292200008 – 10.1017/s0962492922000083
• Morandin, (2023)
• Moreschini A, Monaco S, Normand-Cyrot D (2024) Dirac Structures for a Class of Port-Hamiltonian Systems in Discrete Time. IEEE Trans Automat Contr 69(3):1999–2006. https://doi.org/10.1109/tac.2023.331332 – 10.1109/tac.2023.3313327
• McLachlan RI, Quispel GRW, Robidoux N (1999) Geometric integration using discrete gradients. Philosophical Transactions of the Royal Society of London Series A: Mathematical, Physical and Engineering Sciences 357(1754):1021–1045. https://doi.org/10.1098/rsta.1999.036 – 10.1098/rsta.1999.0363
• Gonzalez O (1996) Time integration and discrete Hamiltonian systems. J Nonlinear Sci 6(5):449–467. https://doi.org/10.1007/bf0244016 – 10.1007/bf02440162
• Kundur, Power System Stability and Control. (1994)
• Fiaz S, Zonetti D, Ortega R, Scherpen JMA, van der Schaft AJ (2013) A port-Hamiltonian approach to power network modeling and analysis. European Journal of Control 19(6):477–485. https://doi.org/10.1016/j.ejcon.2013.09.00 – 10.1016/j.ejcon.2013.09.002
• Kinon, Port-Hamiltonian formulation and structure-preserving discretization of hyperelastic strings. (2023)
• Kinon PL, Thoma T, Betsch P, Kotyczka P (2024) Generalized Maxwell viscoelasticity for geometrically exact strings: Nonlinear port-Hamiltonian formulation and structure-preserving discretization. IFAC-PapersOnLine 58(6):101–106. https://doi.org/10.1016/j.ifacol.2024.08.26 – 10.1016/j.ifacol.2024.08.264
• Udwadia FE, Phohomsiri P (2006) Explicit equations of motion for constrained mechanical systems with singular mass matrices and applications to multi-body dynamics. Proc R Soc A 462(2071):2097–2117. https://doi.org/10.1098/rspa.2006.166 – 10.1098/rspa.2006.1662
• Kinon, Conserving integration of multibody systems with singular and non-constant mass matrix including quaternion-based rigid body dynamics. Multibody Sys. Dyn. (2024)
• Holm, (2009)
• Gonzalez O, Simo JC (1996) On the stability of symplectic and energy-momentum algorithms for non-linear Hamiltonian systems with symmetry. Computer Methods in Applied Mechanics and Engineering 134(3–4):197–222. https://doi.org/10.1016/0045-7825(96)01009- – 10.1016/0045-7825(96)01009-2
• Jonasson, (2007)
• Goursat, (1959)