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In this paper, we develop variational integrators for the nonequilibriumthermodynamics of simple closed systems. These integrators are obtained by adiscretization of the Lagrangian variational formulation of nonequilibriumthermodynamics developed in \cite{GBYo2016a}, and thus extend the variationalintegrators of Lagrangian mechanics, to include irreversible processes. In thecontinuous setting, we derive the structure preserving property of the flow ofsuch systems. This property is an extension of the symplectic property of theflow of the Euler-Lagrange equations. In the discrete setting, we show that thediscrete flow solution of our numerical scheme verifies a discrete version ofthis property. We also present the regularity conditions which ensure theexistence of the discrete flow. We finally illustrate our discrete variationalschemes with the implementation of an example of a simple and closed system.

Variational discretization of the nonequilibrium thermodynamics of simple systems