We propose a new class of finite element approximations to ideal compressible magnetohydrodynamic equations in smooth regime. Following variational approximations developed for fluid models in the last decade, our discretizations are built via a discrete variational principle mimicking the continuous Euler-Poincar\'e principle, and to further exploit the geometrical structure of the problem, vector fields are represented by their action as Lie derivatives on differential forms of any degree. The resulting semi-discrete approximations are shown to conserve the total mass, entropy and energy of the solutions for a wide class of finite element approximations. In addition, the divergence-free nature of the magnetic field is preserved in a pointwise sense and a time discretization is proposed, preserving those invariants and giving a reversible scheme at the fully discrete level. Numerical simulations are conducted to verify the accuracy of our approach and its ability to preserve the invariants for several test problems.

翻译：本文针对理想可压缩磁流体动力学方程在光滑区域中的求解，提出了一类新型有限元逼近方法。该方法沿袭过去十年流体模型变分逼近的研究思路，通过离散变分原理构建离散格式，该原理模拟了连续欧拉-庞加莱原理。为进一步利用问题的几何结构，向量场通过其作为任意阶微分形式李导数的动作进行表示。研究表明，对于广泛类别的有限元逼近，所得到的半离散格式能够保持解的总质量、熵和能量。此外，磁场无散特性在逐点意义上得到保持，同时本文还提出了一种时间离散化方法，该方法在完全离散水平上保持上述守恒量并形成可逆格式。通过数值模拟验证了该方法的精度及其在多个测试问题中保持守恒量的能力。