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.
Xiao et al. (Sun,) studied this question.