Discontinuous Galerkin Finite Element Methods for Linear Port-Hamiltonian Dynamical Systems

Authors

Xiaoyu Cheng, J. J. W. van der Vegt, Yan Xu, H. J. Zwart

Abstract

In this paper, we present discontinuous Galerkin (DG) finite element discretizations for a class of linear hyperbolic port-Hamiltonian dynamical systems. The key point in constructing a port-Hamiltonian system is a Stokes-Dirac structure. Instead of following the traditional approach of defining the strong form of the Dirac structure, we define a Dirac structure in weak form, specifically in the input-state-output form. This is implemented within broken Sobolev spaces on a tessellation with polyhedral elements. After that, we state the weak port-Hamiltonian formulation and prove that it relates to a Poisson bracket. In our work, a crucial aspect of constructing the above-mentioned Dirac structure is that we provide a conservative relation between the boundary ports. Next, we state DG discretizations of the port-Hamiltonian system by using the weak form of the Dirac structure and broken polynomial spaces of differential forms, and we provide a priori error estimates for the structure-preserving port-Hamiltonian discontinuous Galerkin (PHDG) discretizations. The accuracy and capability of the methods developed in this paper are demonstrated by presenting several numerical experiments.

Keywords

Port-Hamiltonian systems; Dirac structure; Discontinuous Galerkin methods; Exterior calculus

Citation

• Journal: Journal of Scientific Computing
• Year: 2025
• Volume: 104
• Issue: 1
• Pages:
• Publisher: Springer Science and Business Media LLC
• DOI: 10.1007/s10915-025-02926-w

BibTeX

@article{Cheng_2025,
  title={{Discontinuous Galerkin Finite Element Methods for Linear Port-Hamiltonian Dynamical Systems}},
  volume={104},
  ISSN={1573-7691},
  DOI={10.1007/s10915-025-02926-w},
  number={1},
  journal={Journal of Scientific Computing},
  publisher={Springer Science and Business Media LLC},
  author={Cheng, Xiaoyu and van der Vegt, J. J. W. and Xu, Yan and Zwart, H. J.},
  year={2025}
}

Download the bib file

References

• Arnold, D., Falk, R. & Winther, R. Finite element exterior calculus: from Hodge theory to numerical stability. Bull. Amer. Math. Soc. 47, 281–354 (2010) – 10.1090/s0273-0979-10-01278-4
• Arnold, D. N. Spaces of Finite Element Differential Forms. Springer INdAM Series 117–140 (2013) doi:10.1007/978-88-470-2592-9_9 – 10.1007/978-88-470-2592-9_9
• Arnold, D. & Awanou, G. Finite element differential forms on cubical meshes. Math. Comp. 83, 1551–1570 (2013) – 10.1090/s0025-5718-2013-02783-4
• Arnold, D. N., Boffi, D. & Bonizzoni, F. Finite element differential forms on curvilinear cubic meshes and their approximation properties. Numer. Math. 129, 1–20 (2014) – 10.1007/s00211-014-0631-3
• Arnold, D. N., Falk, R. S. & Winther, R. Finite element exterior calculus, homological techniques, and applications. Acta Numerica 15, 1–155 (2006) – 10.1017/s0962492906210018
• Brugnoli, A., Rashad, R. & Stramigioli, S. Dual field structure-preserving discretization of port-Hamiltonian systems using finite element exterior calculus. Journal of Computational Physics 471, 111601 (2022) – 10.1016/j.jcp.2022.111601
• Cardoso-Ribeiro, F. L., Matignon, D. & Lefèvre, L. A partitioned finite element method for power-preserving discretization of open systems of conservation laws. IMA Journal of Mathematical Control and Information 38, 493–533 (2020) – 10.1093/imamci/dnaa038
• Cervera, J., van der Schaft, A. J. & Baños, A. Interconnection of port-Hamiltonian systems and composition of Dirac structures. Automatica 43, 212–225 (2007) – 10.1016/j.automatica.2006.08.014
• B Cockburn, II. General framework. Math. Comp. (1989)
• Cockburn, B. & Shu, C.-W. Journal of Scientific Computing 16, 173–261 (2001) – 10.1023/a:1012873910884
• Courant, T. J. Dirac manifolds. Trans. Amer. Math. Soc. 319, 631–661 (1990) – 10.1090/s0002-9947-1990-0998124-1
• Di Pietro, D. A. & Ern, A. Mathematical Aspects of Discontinuous Galerkin Methods. Mathématiques et Applications (Springer Berlin Heidelberg, 2012). doi:10.1007/978-3-642-22980-0 – 10.1007/978-3-642-22980-0
• Duindam, V., Macchelli, A., Stramigioli, S. & Bruyninckx, H. Modeling and Control of Complex Physical Systems. (Springer Berlin Heidelberg, 2009). doi:10.1007/978-3-642-03196-0 – 10.1007/978-3-642-03196-0
• T Frankel, The Geometry of Physics: An introduction (2012)
• Hsiao, G. C. & Wendland, W. L. Boundary Integral Equations. Applied Mathematical Sciences (Springer Berlin Heidelberg, 2008). doi:10.1007/978-3-540-68545-6 – 10.1007/978-3-540-68545-6
• Kalogiratou, Z., Monovasilis, Th. & Simos, T. E. Symplectic Partitioned Runge-Kutta Methods for the Numerical Integration of Periodic and Oscillatory Problems. Recent Advances in Computational and Applied Mathematics 169–208 (2011) doi:10.1007/978-90-481-9981-5_8 – 10.1007/978-90-481-9981-5_8
• P Kotyczka, Numerical methods for distributed parameter port-Hamiltonian Systems (2019)
• Kotyczka, P., Maschke, B. & Lefèvre, L. Weak form of Stokes–Dirac structures and geometric discretization of port-Hamiltonian systems. Journal of Computational Physics 361, 442–476 (2018) – 10.1016/j.jcp.2018.02.006
• Kumar, N., van der Vegt, J. J. W. & Zwart, H. J. Port-Hamiltonian discontinuous Galerkin finite element methods. IMA Journal of Numerical Analysis 45, 354–403 (2024) – 10.1093/imanum/drae008
• Maschke, B. M. J. & van der Schaft, A. J. Interconnected mechanical systems, part II: the dynamics of spatial mechanical networks. Modelling and Control of Mechanical Systems 17–30 (1997) doi:10.1142/9781848160873_0002 – 10.1142/9781848160873_0002
• Monk, P. Finite Element Methods for Maxwell’s Equations. (2003) doi:10.1093/acprof:oso/9780198508885.001.0001 – 10.1093/acprof:oso/9780198508885.001.0001
• BG Pachpatte, Inequalities for differential and integral equations (1998)
• Rashad, R., Califano, F., van der Schaft, A. J. & Stramigioli, S. Twenty years of distributed port-Hamiltonian systems: a literature review. IMA Journal of Mathematical Control and Information 37, 1400–1422 (2020) – 10.1093/imamci/dnaa018
• Schwarz, G. Hodge Decomposition—A Method for Solving Boundary Value Problems. Lecture Notes in Mathematics (Springer Berlin Heidelberg, 1995). doi:10.1007/bfb0095978 – 10.1007/bfb0095978
• Seslija, M., van der Schaft, A. & Scherpen, J. M. A. Discrete exterior geometry approach to structure-preserving discretization of distributed-parameter port-Hamiltonian systems. Journal of Geometry and Physics 62, 1509–1531 (2012) – 10.1016/j.geomphys.2012.02.006
• Shu, C.-W. Total-Variation-Diminishing Time Discretizations. SIAM J. Sci. and Stat. Comput. 9, 1073–1084 (1988) – 10.1137/0909073
• Struwe, M. Semi-linear wave equations. Bull. Amer. Math. Soc. 26, 53–85 (1992) – 10.1090/s0273-0979-1992-00225-2
• Sun, Z. & Xing, Y. Optimal error estimates of discontinuous Galerkin methods with generalized fluxes for wave equations on unstructured meshes. Math. Comp. 90, 1741–1772 (2021) – 10.1090/mcom/3605
• Thoma, T. & Kotyczka, P. Structure preserving discontinuous Galerkin approximation of one-dimensional port-Hamiltonian systems. IFAC-PapersOnLine 56, 6783–6788 (2023) – 10.1016/j.ifacol.2023.10.386
• Trenchant, V., Hu, W., Ramirez, H. & Gorrec, Y. L. Structure Preserving Finite Differences in Polar Coordinates for Heat and Wave Equations. IFAC-PapersOnLine 51, 571–576 (2018) – 10.1016/j.ifacol.2018.03.096
• Trenchant, V., Ramirez, H., Le Gorrec, Y. & Kotyczka, P. Finite differences on staggered grids preserving the port-Hamiltonian structure with application to an acoustic duct. Journal of Computational Physics 373, 673–697 (2018) – 10.1016/j.jcp.2018.06.051
• van der Schaft, A. & Jeltsema, D. Port-Hamiltonian Systems Theory: An Introductory Overview. FnT in Systems and Control 1, 173–378 (2014) – 10.1561/2600000002
• van der Schaft, A. & Maschke, B. Interconnected mechanical systems, part I: geometry of interconnection and implicit Hamiltonian systems. Modelling and Control of Mechanical Systems 1–15 (1997) doi:10.1142/9781848160873_0001 – 10.1142/9781848160873_0001
• van der Schaft, A. J. & Maschke, B. M. Hamiltonian formulation of distributed-parameter systems with boundary energy flow. Journal of Geometry and Physics 42, 166–194 (2002) – 10.1016/s0393-0440(01)00083-3
• Xu, Y., van der Vegt, J. J. W. & Bokhove, O. Discontinuous Hamiltonian Finite Element Method for Linear Hyperbolic Systems. J Sci Comput 35, 241–265 (2008) – 10.1007/s10915-008-9191-y