Geometric integrators for adiabatically closed simple thermodynamic systems

TL;DR

This paper develops a discrete variational framework for adiabatically closed simple thermodynamic systems using partially cosymplectic geometry, enabling structure-preserving numerical integrators. By introducing a discrete Lagrangian $L_d$ and a discrete friction form, it derives discrete thermodynamic Euler–Lagrange equations and a corresponding discrete flow via Legendre transforms, and proves a discrete Noether theorem for conserved quantities under symmetries. The approach is validated through three piston-cylinder examples (damped harmonic oscillator, ideal gas, Van der Waals gas), where the discrete integrators preserve key invariants and closely approximate the continuous dynamics with favorable stability properties compared to standard RK methods. The work thus provides a geometry-based method for robust simulations of thermo-mechanical systems and lays the groundwork for extensions to more complex thermodynamic settings and reductions.

Abstract

A variational formulation for non-equilibrium thermodynamics was developed by Gay-Balmaz and Yoshimura. In a recent article, the first two authors of the present paper introduced partially cosymplectic structures as a geometric framework for thermodynamic systems, recovering the evolution equations obtained variationally. In this paper, we develop a discrete variational principle for adiabatically closed simple thermodynamic systems, which can be utilised to construct numerical integrators for the dynamics of such systems. The effectiveness of our method is illustrated with several examples.

Figures (7)

• Figure 1: Integration results for the damped harmonic oscillator with a time step $h=0.01$ using the Discrete Thermodynamic Euler--Lagrange equations solution (in green and dashed) and the method of the midpoint rule (in blue and dotted), compared with the exact continuous solution (in black and solid). A detailed section is shown to allow the distinction of the curves, and to show the numeric errors made in the calculation of the continuous solution for $S$.
• Figure 2: Estimation of the value of the Hamiltonian for the damped harmonic oscillator with a time step $h=0.01$ using the Discrete Thermodynamic Euler--Lagrange equations solution and the Legendre transforms $\mathbb{F}^{f+}$ (in green and dashed), $\mathbb{F}^{f-}$ (in red and dashed), as well as the estimation for the velocity (in magenta and dashed) and the method of the midpoint rule (in blue and dotted), compared with the exact continuous solution (in black and solid). The green and red curves are overlapping so they cannot be distinguished.
• Figure 3: Diagram of the cylinder containing the gas (dashed region) and the piston that closes it.
• Figure 4: Integration of $x$ for a perfect monoatomic gas contained in a cylinder with a time step $h=0.01$ using the Discrete thermodynamic Euler--Lagrange equations solution (in green and dashed) and the method of the midpoint rule (in blue and dotted), compared with the exact continuous solution (in black and solid).
• Figure 5: Estimation of the value of the Hamiltonian for a perfect monoatomic gas contained in a cylinder with a time step $h=0.01$ using the Discrete thermodynamic Euler--Lagrange equations solution and the Legendre transforms $\mathbb{F}^{f+}$ (in green and dashed), $\mathbb{F}^{f-}$ (in red and dashed), as well as the estimation for the velocity (in magenta and dashed) and the method of the midpoint rule (in blue and dotted), compared with the exact continuous solution (in black and solid). The green and red curves are overlapping so they cannot be distinguished.

Theorems & Definitions (28)

• Proposition 1
• proof
• Definition 1
• Definition 2
• Proposition 2
• Theorem 3: Noether' s theorem
• proof
• Proposition 4
• proof
• Proposition 5