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