We present a structure-preserving and thermodynamically consistent numerical scheme for classical magnetohydrodynamics, incorporating viscosity, magnetic resistivity, heat transfer, and thermoelectric effect. The governing equations are shown to be derived from a generalized Hamilton's principle, with the resulting weak formulation being mimicked at the discrete level. The resulting numerical method conserves mass and energy, satisfies Gauss' magnetic law and magnetic helicity balance, and adheres to the Second Law of Thermodynamics, all at the fully discrete level. It is shown to perform well on magnetic Rayleigh-B\'enard convection.

翻译：本文提出了一种结构保持且热力学一致的经典磁流体动力学数值格式，该格式包含了粘性、磁电阻、热传导及热电效应。研究表明，控制方程可由广义哈密顿原理导出，其弱形式可在离散层面得到模拟。所得到的数值方法在完全离散层面能够守恒质量和能量，满足高斯磁定律与磁螺旋度平衡，并遵循热力学第二定律。数值实验表明，该方法在磁瑞利-贝纳德对流问题上表现良好。