Metriplectic Euler-Poincaré equations: smooth and discrete dynamics

Anthony Bloch; Marta Farré Puiggalí; David Martín de Diego; Anthony Bloch; Marta Farré Puiggalí; David Martín de Diego

doi:10.3934/cam.2024040

Communications in Analysis and Mechanics

2024, Volume 16, Issue 4: 910-927. doi: 10.3934/cam.2024040

Previous Article Next Article

Research article Special Issues

Metriplectic Euler-Poincaré equations: smooth and discrete dynamics

1.
Department of Mathematics, University of Michigan 530 Church Street, Ann Arbor, MI, USA
2.
Instituto de Ciencias Matemáticas, ICMAT (CSIC-UAM-UC3M-UCM) Madrid, Spain

Received: 10 January 2024 Revised: 03 November 2024 Accepted: 05 December 2024 Published: 30 December 2024
70G45, 37J37

In this paper we will introduce a discrete version of systems obtained by modifications of the Euler-Poincaré equations when we add a special type of dissipative force, so that the equations of motion can be described using the metriplectic formalism. The metriplectic representation of the dynamics allows us to describe the conservation of energy, as well as to guarantee entropy production. For deriving the discrete equations we use discrete gradients to numerically simulate the evolution of the continuous metriplectic equations preserving their main properties: preservation of energy and correct entropy production rate.
- metriplectic system,
- Poisson manifold,
- discrete gradient,
- Euler-Poincaréequations
Citation: Anthony Bloch, Marta Farré Puiggalí, David Martín de Diego. Metriplectic Euler-Poincaré equations: smooth and discrete dynamics[J]. Communications in Analysis and Mechanics, 2024, 16(4): 910-927. doi: 10.3934/cam.2024040

Related Papers:

Abstract

In this paper we will introduce a discrete version of systems obtained by modifications of the Euler-Poincaré equations when we add a special type of dissipative force, so that the equations of motion can be described using the metriplectic formalism. The metriplectic representation of the dynamics allows us to describe the conservation of energy, as well as to guarantee entropy production. For deriving the discrete equations we use discrete gradients to numerically simulate the evolution of the continuous metriplectic equations preserving their main properties: preservation of energy and correct entropy production rate.

References

[1]	P. J. Morrison, A paradigm for joined Hamiltonian and dissipative systems, Phys. D, 18 (1986), 410–419. https://doi.org/10.1016/0167-2789(86)90209-5 doi: 10.1016/0167-2789(86)90209-5
[2]	A. M. Bloch, P. J. Morrison, T. S. Ratiu, Gradient flows in the normal and Kähler metrics and triple bracket generated metriplectic systems, in Recent trends in dynamical systems, Springer Basel, (2013), 371–415. https://doi.org/10.1007/978-3-0348-0451-6_15
[3]	A. M. Bloch, P. J. Morrison, Algebraic structure of the plasma quasilinear equations, Phys. Lett. A, 88 (1982), 405–406. https://doi.org/10.1016/0375-9601(82)90664-8 doi: 10.1016/0375-9601(82)90664-8
[4]	M. Grmela, H. C.Öttinge, Dynamics and thermodynamics of complex fluids. Ⅰ. Development of a general formalism, Phys. Rev. E, 56 (1987), 6620–6632. https://doi.org/10.1103/PhysRevE.56.6620 doi: 10.1103/PhysRevE.56.6620
[5]	P. J. Morrison, M. H. Updike, Inclusive curvaturelike framework for describing dissipation: metriplectic 4-bracket dynamics, Phys. Rev. E, 109 (2024), 045202. https://doi.org/10.1103/physreve.109.045202 doi: 10.1103/physreve.109.045202
[6]	A. Bloch, P. S. Krishnaprasad, J. E. Marsden, T. S. Ratiu, The Euler-Poincaré equations and double bracket dissipation, Commun. Math. Phys., 175 (1996), 1–42. https://doi.org/10.1007/BF02101622 doi: 10.1007/BF02101622
[7]	O. Esen, G. Özcan, S. Sütlü, On extensions, Lie-Poisson systems, and dissipation, J. Lie Theory, 32 (2022), 327–382.
[8]	R. Abraham, J. E. Marsden, Foundations of Mechanics, AMS Chelsea Publishing, 1978.
[9]	P. Libermann, C. Marle, Symplectic geometry and analytical mechanics, volume 35 of Mathematics and its Applications. Springer Dordrecht, 1987. https://doi.org/10.1007/978-94-009-3807-6
[10]	I. Romero, Algorithms for coupled problems that preserve symmetries and the laws of thermodynamics Part Ⅰ: Monolithic integrators and their application to finite strain thermoelasticity, Comput. Methods Appl. Mech. Engrg., 199 (2010), 1841–1858. https://doi.org/10.1016/j.cma.2010.02.014 doi: 10.1016/j.cma.2010.02.014
[11]	D. D. Holm, J. E. Marsden, T. S. Ratiu, The Euler–Poincaré equations and semidirect products with applications to continuum theories, Adv Math, 137 (1998), 1–81. https://doi.org/10.1006/aima.1998.1721 doi: 10.1006/aima.1998.1721
[12]	D. D. Holm, T. Schmah, C. Stoica, Geometric mechanics and symmetry: from finite to infinite dimensions, Oxford University Press, 2009.
[13]	M. Materassi, P. J. Morrison, Metriplectic torque for rotation control of a rigid body, Cybernetics and Physics, 7 (2018), 78–86. https://doi.org/10.35470/2226-4116-2018-7-2-78-86 doi: 10.35470/2226-4116-2018-7-2-78-86
[14]	H. C. Öttinger, GENERIC Integrators: Structure Preserving Time Integration for Thermodynamic Systems, J Non-Equil Thermody, 43 (2018), 89–100. https://doi.org/10.1515/jnet-2017-0034 doi: 10.1515/jnet-2017-0034
[15]	X. Shang, H. C. Öttinger, Structure-preserving integrators for dissipative systems based on reversible-irreversible splitting, Proc. A., 476 (2020), 20190446. https://doi.org/10.1098/rspa.2019.0446 doi: 10.1098/rspa.2019.0446
[16]	A. Harten, P. D. Lax, B. Leer, On upstream differencing and Godunov-type schemes for hyperbolic conservation laws, SIAM Rev., 25 (1983), 35–61. https://doi.org/10.1137/1025002 doi: 10.1137/1025002
[17]	O. Gonzalez, Time integration and discrete Hamiltonian systems, J. Nonlinear Sci., 6 (1996), 449–467. http://dx.doi.org/10.1007/s003329900018 doi: 10.1007/s003329900018
[18]	T. Itoh, K. Abe, Hamiltonian-conserving discrete canonical equations based on variational difference quotients, J. Comput. Phys., 76 (1988), 85–102. http://dx.doi.org/10.1016/0021-9991(88)90132-5 doi: 10.1016/0021-9991(88)90132-5
[19]	E. Hairer, C. Lubich, G. Wanner, Geometric numerical integratio, volume 31 of Springer Series in Computational Mathematics. Springer, Heidelberg, 2010.
[20]	M. A. Austin, P. S. Krishnaprasad, L. Wang, Almost Poisson integration of rigid body systems, J. Comput. Phys., 107 (1993), 105–117. https://doi.org/10.1006/jcph.1993.1128 doi: 10.1006/jcph.1993.1128
[21]	P. A. Absil, R. Mahony, R. Sepulchre, Optimization algorithms on matrix manifolds, Princeton University Press, 2008. https://doi.org/10.1515/9781400830244
[22]	E. Celledoni, S. Eidnes, B. Owren, T. Ringholm, Energy-preserving methods on Riemannian manifolds, Math. Comp., 89 (2020), 699–716. https://doi.org/10.1090/mcom/3470 doi: 10.1090/mcom/3470

Reader Comments

Your name:*

Email:*
© 2024 the Author(s), licensee AIMS Press. This is an open access article distributed under the terms of the Creative Commons Attribution License (http://creativecommons.org/licenses/by/4.0)