Structure preserving spatial discretization of 2D hyperbolic systems using staggered grids finite difference

Authors

Vincent Trenchant, Hector Ramirez, Yann Le Gorrec, Paul Kotyczka

Abstract

This paper proposes a finite difference spatial discretization scheme that preserve the port-Hamiltonian structure of 1D and 2D infinite dimensional hyperbolic systems. This scheme is based on the use of staggered grids for the discretization of the state and co state variables of the system. It is shown that, by an appropriate choice of the boundary port variables, the underlying geometric structure of the infinite-dimensional system, i.e. its Dirac structure, is preserved during the discretization step. The consistency of the spatial discretization scheme is evaluated and its accuracy is validated with numerical results.

Citation

• Journal: 2017 American Control Conference (ACC)
• Year: 2017
• Volume:
• Issue:
• Pages: 2491–2496
• Publisher: IEEE
• DOI: 10.23919/acc.2017.7963327

BibTeX

@inproceedings{Trenchant_2017,
  title={{Structure preserving spatial discretization of 2D hyperbolic systems using staggered grids finite difference}},
  DOI={10.23919/acc.2017.7963327},
  booktitle={{2017 American Control Conference (ACC)}},
  publisher={IEEE},
  author={Trenchant, Vincent and Ramirez, Hector and Le Gorrec, Yann and Kotyczka, Paul},
  year={2017},
  pages={2491--2496}
}

Download the bib file

References

• Macchelli, A. Boundary Energy Shaping of Linear Distributed Port-Hamiltonian Systems. IFAC Proceedings Volumes vol. 45 120–125 (2012) – 10.3182/20120829-3-it-4022.00041
• Trenchant, V., Fares, Y., Ramirez, H. & Le Gorrec, Y. A port-Hamiltonian formulation of a 2D boundary controlled acoustic system. IFAC-PapersOnLine vol. 48 235–240 (2015) – 10.1016/j.ifacol.2015.10.245
• 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 vol. 62 1509–1531 (2012) – 10.1016/j.geomphys.2012.02.006
• 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
• Hamroun, B., Lefevre, L. & Mendes, E. Port-based modelling and geometric reduction for open channel irrigation systems. 2007 46th IEEE Conference on Decision and Control 1578–1583 (2007) doi:10.1109/cdc.2007.4434237 – 10.1109/cdc.2007.4434237
• Moulla, R., Lefévre, L. & Maschke, B. Pseudo-spectral methods for the spatial symplectic reduction of open systems of conservation laws. Journal of Computational Physics vol. 231 1272–1292 (2012) – 10.1016/j.jcp.2011.10.008
• Macchelli, A. Energy shaping of distributed parameter port-Hamiltonian systems based on finite element approximation. Systems & Control Letters vol. 60 579–589 (2011) – 10.1016/j.sysconle.2011.04.016
• Wu, Y., Hamroun, B., Gorrec, Y. L. & Maschke, B. Power preserving model reduction of 2D vibro-acoustic system: A port Hamiltonian approach. IFAC-PapersOnLine vol. 48 206–211 (2015) – 10.1016/j.ifacol.2015.10.240
• Le Gorrec, Y., Peng, H., Lefèvre, L., Hamroun, B. & Couenne, F. Systèmes hamiltoniens à ports de dimension infinie. Réduction et propriétés spectrales. Journal Européen des Systèmes Automatisés vol. 45 645–664 (2011) – 10.3166/jesa.45.645-664
• mazumder, Numerical methods for partial differential equations Finite difference and finite volume methods (2015)
• van der Schaft, A. J. & Maschke, B. M. Hamiltonian formulation of distributed-parameter systems with boundary energy flow. Journal of Geometry and Physics vol. 42 166–194 (2002) – 10.1016/s0393-0440(01)00083-3
• 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
• Jacob, B. & Zwart, H. J. Linear Port-Hamiltonian Systems on Infinite-Dimensional Spaces. (Springer Basel, 2012). doi:10.1007/978-3-0348-0399-1 – 10.1007/978-3-0348-0399-1
• Le Gorrec, Y., Zwart, H. & Maschke, B. Dirac structures and Boundary Control Systems associated with Skew-Symmetric Differential Operators. SIAM Journal on Control and Optimization vol. 44 1864–1892 (2005) – 10.1137/040611677
• Ramirez, H., Le Gorrec, Y., Macchelli, A. & Zwart, H. Exponential Stabilization of Boundary Controlled Port-Hamiltonian Systems With Dynamic Feedback. IEEE Transactions on Automatic Control vol. 59 2849–2855 (2014) – 10.1109/tac.2014.2315754
• Villegas, J. A., Zwart, H., Le Gorrec, Y. & Maschke, B. Exponential Stability of a Class of Boundary Control Systems. IEEE Transactions on Automatic Control vol. 54 142–147 (2009) – 10.1109/tac.2008.2007176
• van der schaft, L2-Gain and Passivity Techniques in Nonlinear Control (2012)
• maschke, Port controlled Hamiltonian systems: modeling origins and system theoretic properties. Proceedings of the 3rd IFAC Symposium on Nonlinear Control Systems NOLCOS’92 (1992)
• macchelli, On the synthesis of boundary control laws for distributed port Hamiltonian systems (2016)
• ISERLES, A. Generalized Leapfrog Methods. IMA Journal of Numerical Analysis vol. 6 381–392 (1986) – 10.1093/imanum/6.4.381
• Zeng, Y. Q. & Liu, Q. H. A staggered-grid finite-difference method with perfectly matched layers for poroelastic wave equations. The Journal of the Acoustical Society of America vol. 109 2571–2580 (2001) – 10.1121/1.1369783
• Fornberg, B. High-Order Finite Differences and the Pseudospectral Method on Staggered Grids. SIAM Journal on Numerical Analysis vol. 27 904–918 (1990) – 10.1137/0727052
• munjal, Acoustics of ducts and muffiers with application to exhaust and ventilation system design (1987)
• kotyczka, Finite volume structure-preserving discretization of ld distributed-parameter port-hamiltonian systems. Proc of the 1st IFAC Workshop on Control of Systems Governed by Partial Differential Equations (2016)