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Thermodynamically consistent modeling and simulation of two-fluid magnetohydrodynamic equations

Ting Xiao, Qiaolin He

TL;DR

The paper addresses the challenge of enforcing the first and second laws of thermodynamics in two-fluid MHD by formulating the model from a Helmholtz free energy functional. It derives thermodynamic quantities such as the chemical potential $\mu_\alpha$, entropy $s_\alpha$, and internal energy self-consistently from the free energy, with a representative density–temperature free energy $f_\alpha = k_B T_\alpha n \ln n - \frac{k_B}{r-1} T_\alpha^2 n$. A semi-implicit time-stepping scheme, built on the convex–concave structure of the free energy, preserves discrete energy conservation and entropy non-decrease, and a fully discrete mixed finite element method is developed for a 2D degenerate formulation with rigorous $L^2$ and $H^1$ error estimates. Numerical experiments demonstrate convergence and physically relevant phenomena such as magnetic reconnection, current-sheet dynamics, and entropy production, validating the framework and its numerical implementation. The results provide a principled pathway toward thermodynamically sound plasma simulations and motivate extensions to 2.5D and full 3D configurations across complex magnetic and thermal couplings.

Abstract

Based on a rigorous thermodynamic framework, this work develops a two-fluid magnetohydrodynamic model grounded in the Helmholtz free energy formalism. The model maintains full thermodynamic consistency by simultaneously satisfying energy conservation and entropy production laws in two-fluid systems. By analyzing the convex-concave structure of the Helmholtz free energy density, we systematically derive key thermodynamic variables-chemical potential, entropy density, and internal energy-in a self-consistent manner. Building on this foundation, we construct a temporally discrete numerical scheme that inherits the thermodynamic consistency of the continuous model. The scheme is proven to adhere rigorously to both the first and second laws of thermodynamics. For the implemented two-dimensional degenerate system, we establish comprehensive a priori error estimates in space and time. Numerical simulations validate the model's effectiveness in capturing essential plasma phenomena, demonstrating its applicability to complex physical scenarios.

Thermodynamically consistent modeling and simulation of two-fluid magnetohydrodynamic equations

TL;DR

The paper addresses the challenge of enforcing the first and second laws of thermodynamics in two-fluid MHD by formulating the model from a Helmholtz free energy functional. It derives thermodynamic quantities such as the chemical potential , entropy , and internal energy self-consistently from the free energy, with a representative density–temperature free energy . A semi-implicit time-stepping scheme, built on the convex–concave structure of the free energy, preserves discrete energy conservation and entropy non-decrease, and a fully discrete mixed finite element method is developed for a 2D degenerate formulation with rigorous and error estimates. Numerical experiments demonstrate convergence and physically relevant phenomena such as magnetic reconnection, current-sheet dynamics, and entropy production, validating the framework and its numerical implementation. The results provide a principled pathway toward thermodynamically sound plasma simulations and motivate extensions to 2.5D and full 3D configurations across complex magnetic and thermal couplings.

Abstract

Based on a rigorous thermodynamic framework, this work develops a two-fluid magnetohydrodynamic model grounded in the Helmholtz free energy formalism. The model maintains full thermodynamic consistency by simultaneously satisfying energy conservation and entropy production laws in two-fluid systems. By analyzing the convex-concave structure of the Helmholtz free energy density, we systematically derive key thermodynamic variables-chemical potential, entropy density, and internal energy-in a self-consistent manner. Building on this foundation, we construct a temporally discrete numerical scheme that inherits the thermodynamic consistency of the continuous model. The scheme is proven to adhere rigorously to both the first and second laws of thermodynamics. For the implemented two-dimensional degenerate system, we establish comprehensive a priori error estimates in space and time. Numerical simulations validate the model's effectiveness in capturing essential plasma phenomena, demonstrating its applicability to complex physical scenarios.

Paper Structure

This paper contains 27 sections, 11 theorems, 165 equations, 10 figures, 2 tables.

Table of Contents

  1. Introduction
  2. Mathematical model
  3. Formulations of Thermodynamical Variables
  4. The original model
  5. New formulations and thermodynamical consistency
  6. New formulations
  7. Thermodynamical consistency
  8. Thermodynamically consistent numerical method
  9. Fully Discrete Error Analysis for 2D Degenerate Equations
  10. Finite element space
  11. Variational Formulation of 2D Equations
  12. Well-posedness of the EBJ System
  13. Error estimation
  14. Step1(Error estimation of all variables in $\mathbf{x}^{k+1}_h$ but excluding $E^{k+1}_h$)
  15. Proof of \ref{['eq:error_vartheta']}
  16. ...and 12 more sections

Key Result

Theorem 3.1

The new 2F-MHD system satisfies the first law of thermodynamics, namely the conservation of total energy: where the total energy functional is defined by $\mathcal{E} = \mathcal{H} + \mathcal{U} + \mathcal{B}$, $\mathcal{H} = \frac{1}{2} \int_{\Omega}\rho|\mathbf{u}|^{2} dx$, $\mathcal{U} = \int_{\Omega} \epsilon d x$ and $\mathcal{B} = \frac{1}{2 \mu_{0}} \int_{\Omega} |\mathbf{B}|^{2} d x$ wit

Figures (10)

  • Figure 6.1: Energy variance over time.
  • Figure 6.2: Evolution of ion entropy (a), electron entropy (b), and iteration numbers (c).
  • Figure 6.3: Plasma number density $n$ at selected times.
  • Figure 6.4: Current density $J$ at selected times.
  • Figure 6.5: Evolution of magnetic field streamlines.
  • ...and 5 more figures

Theorems & Definitions (23)

  • Theorem 3.1
  • proof
  • Theorem 3.2
  • proof
  • Lemma 4.1
  • proof
  • Theorem 4.2
  • proof
  • Lemma 4.3: Discrete Helmholtz free energy inequality
  • proof
  • ...and 13 more