Variational discretization of thermodynamical simple systems on Lie groups

Benjamin Couéraud; François Gay-Balmaz

doi:10.3934/dcdss.2020064

Article Contents

2020, Volume 13, Issue 4: 1075-1102. Doi: 10.3934/dcdss.2020064

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Variational discretization of thermodynamical simple systems on Lie groups

Benjamin Couéraud^1, and
François Gay-Balmaz^2, ,

1.
LMD - IPSL, École Normale Supérieure de Paris - PSL, 24 rue Lhomond, 75005 Paris, France
2.
CNRS - LMD - IPSL, École Normale Supérieure de Paris - PSL, 24 rue Lhomond, 75005 Paris, France

^* Corresponding author
^* Corresponding author

Received: November 30, 2017

Revised: July 31, 2018

Published: April 2020

The authors are supported by the ANR project GEOMFLUID (ANR-14-CE23-0002)

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Abstract

This paper presents the continuous and discrete variational formulations of simple thermodynamical systems whose configuration space is a (finite dimensional) Lie group. We follow the variational approach to nonequilibrium thermodynamics developed in [12,13], as well as its discrete counterpart whose foundations have been laid in [14]. In a first part, starting from this variational formalism on the Lie group, we perform an Euler-Poincaré reduction in order to obtain the reduced evolution equations of the system on the Lie algebra of the configuration space. We obtain as corollaries the energy balance and a Kelvin-Noether theorem. In a second part, a compatible discretization is developed resulting in discrete evolution equations that take place on the Lie group. Then, these discrete equations are transported onto the Lie algebra of the configuration space with the help of a group difference map. Finally we illustrate our framework with a heavy top immersed in a viscous fluid modeled by a Stokes flow and proceed with a numerical simulation.

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Mathematics Subject Classification: Primary: 37D35, 37M15; Secondary: 49S05, 80M30.

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Figure 1. The trajectory of the center of mass of the heavy top

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Figure 2. The different energies of the system

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Figure 3. The relative total energy of the system. While the Runge-Kutta 2 method yields an increase in the total energy, our variational integrator displays the usual oscillatory behaviour until the system stops moving, even with a large time step

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