Symplectic spatial integration schemes for systems of balance equations

Authors

Ngoc Minh Trang Vu, Laurent Lefèvre, Rémy Nouailletas, Sylvain Brémond

Abstract

A method to generate geometric pseudo-spectral spatial discretization schemes for hyperbolic or parabolic partial differential equations is presented. It applies to the spatial discretization of systems of conservation laws with boundary energy flows and/or distributed source terms. The symplecticity of the proposed spatial discretization schemes is defined with respect to the natural power pairing (form) used to define the port-Hamiltonian formulation for the considered systems of balance equations. The method is applied to the resistive diffusion model, a parabolic equation describing the plasma dynamics in tokamaks. A symplectic Galerkin scheme with Bessel conjugated bases is derived from the usual Galerkin method, using the proposed method. Besides the spectral and energetic properties expected from the symplecticity of the method, it is shown that more accurate approximation of eigenfunctions and reduced numerical oscillations result from this choice of conjugated approximation bases. Finally, the obtained numerical results are validated against experimental data from the tokamak Tore Supra facility.

Keywords

Symplectic spatial integration; Pseudo-spectral methods; Balance equations; Port-Hamiltonian systems; Resistive diffusion equation

Citation

• Journal: Journal of Process Control
• Year: 2017
• Volume: 51
• Issue:
• Pages: 1–17
• Publisher: Elsevier BV
• DOI: 10.1016/j.jprocont.2016.12.005

BibTeX

@article{Vu_2017,
  title={{Symplectic spatial integration schemes for systems of balance equations}},
  volume={51},
  ISSN={0959-1524},
  DOI={10.1016/j.jprocont.2016.12.005},
  journal={Journal of Process Control},
  publisher={Elsevier BV},
  author={Vu, Ngoc Minh Trang and Lefèvre, Laurent and Nouailletas, Rémy and Brémond, Sylvain},
  year={2017},
  pages={1--17}
}

Download the bib file

References

• Argomedo, A strict control Lyapunov function for a diffusion equation with time-varying distributed coefficients. IEEE Trans. Autom. Control (2012)
• Artaud, The cronos suite of codes for integrated tokamak modelling. Nucl. Fusion (2010)
• Baaiu, Port-based modelling of mass transfer phenomena. Math. Comput. Modell. Dyn. Syst. (2009)
• Baaiu, A., Couenne, F., Lefevre, L., Le Gorrec, Y. & Tayakout, M. Structure-preserving infinite dimensional model reduction: Application to adsorption processes. Journal of Process Control vol. 19 394–404 (2009) – 10.1016/j.jprocont.2008.07.002
• Blum, (1989)
• Bossavit, (1998)
• Courant, T. J. Dirac manifolds. Transactions of the American Mathematical Society vol. 319 631–661 (1990) – 10.1090/s0002-9947-1990-0998124-1
• Dalsmo, M. & van der Schaft, A. On Representations and Integrability of Mathematical Structures in Energy-Conserving Physical Systems. SIAM Journal on Control and Optimization vol. 37 54–91 (1998) – 10.1137/s0363012996312039
• Finlayson, (1972)
• Fornberg, (1996)
• Franco, A. A., Schott, P., Jallut, C. & Maschke, B. A Multi‐Scale Dynamic Mechanistic Model for the Transient Analysis of PEFCs. Fuel Cells vol. 7 99–117 (2007) – 10.1002/fuce.200500204
• Gaye, Sliding mode stabilization of the current profile in tokamak plasmas. (2011)
• 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
• Hairer, Geometric numerical integration: structure-preserving algorithms for ordinary differential equations. (2002)
• Hamroun, Port-based modelling for open channel irrigation systems. Trans. Fluid Mech. (2006)
• 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
• Macchelli, Modeling and control of complex physical systems – the port-Hamiltonian approach. (2009)
• Macchelli, A. & Melchiorri, C. Modeling and Control of the Timoshenko Beam. The Distributed Port Hamiltonian Approach. SIAM Journal on Control and Optimization vol. 43 743–767 (2004) – 10.1137/s0363012903429530
• Maschke, B. & van der Schaft, A. 4 Compositional Modelling of Distributed-Parameter Systems. Lecture Notes in Control and Information Sciences 115–154 (2005) doi:10.1007/11334774_4 – 10.1007/11334774_4
• Maschke, Canonical interdomain coupling in distributed parameter systems: an extension of the symplectic gyrator. (2001)
• Moreau, Ph. et al. Plasma Control in Tore Supra. Fusion Science and Technology vol. 56 1284–1299 (2009) – 10.13182/fst09-a9178
• Moulla, Pseudo-spectral methods for the spatial symplectic reduction of open systems of conservation laws. J. Comput. Phys. (2012)
• Nishida, Multiscale distributed port-Hamiltonian representation of ionic polymer-metal composite (ipmc). (2008)
• Olver, P. J. Symmetry Groups of Differential Equations. Graduate Texts in Mathematics 75–182 (1993) doi:10.1007/978-1-4612-4350-2_2 – 10.1007/978-1-4612-4350-2_2
• Ralston, (2001)
• Reich, S. Multi-Symplectic Runge–Kutta Collocation Methods for Hamiltonian Wave Equations. Journal of Computational Physics vol. 157 473–499 (2000) – 10.1006/jcph.1999.6372
• Sauter, Neoclassical conductivity and bootstrap current formulas for general axisymmetric equilibria and arbitrary collisionality regime. Phys. Plasma (1999)
• van der Schaft, The Hamiltonian formulation of energy conserving physical systems with external ports. Arch. Elektron. Übertragungstechnik (1995)
• 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
• Vu, The port-Hamiltonian approach for the modelling,. (2014)
• Trang, Structure preserving reduction for thermo-magneto plasma control model. (2014)
• Wesson, (2004)
• Witrant, E. et al. A control-oriented model of the current profile in tokamak plasma. Plasma Physics and Controlled Fusion vol. 49 1075–1105 (2007) – 10.1088/0741-3335/49/7/009