Variational discretization of thermodynamical simple systems on Lie groups

TL;DR

This paper develops a variational discretization framework for simple thermodynamical systems on Lie groups, deriving continuous and discrete evolution equations, and demonstrating the approach with a viscous fluid-immersed heavy top simulation.

Contribution

It introduces a novel variational discretization method for thermodynamical systems on Lie groups, including reduction and numerical implementation.

Findings

Derived discrete evolution equations on Lie groups.

Established energy balance and Kelvin-Noether theorem.

Validated framework with a viscous fluid and heavy top simulation.

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 \cite{GBYo2017a,GBYo2017b}, as well as its discrete counterpart whose foundations have been laid in \cite{GBYo2017c}. In a first part, starting from this variational formalism on the Lie group, we perform an Euler-Poincar\'e 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.