We propose a new class of finite element approximations to ideal compressible magnetohydrody- namic 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-Poincare principle, and to further exploit the geometrical structure of the prob- lem, 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 discretiza- tion 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：我们提出了一类新的有限元近似方法，用于处理光滑区域中可压缩理想磁流体动力学方程。遵循过去十年为流体模型开发的变分近似方法，我们通过模拟连续Euler-Poincaré原理的离散变分原理构建离散化方案，并为进一步利用该问题的几何结构，将向量场表示为它们作为李导数对任意阶微分形式的作用。所得到的半离散近似方法被证明能保持广类有限元近似解的总质量、熵和能量。此外，磁场无散特性在逐点意义上得到保持，并提出了一种时间离散化方法，该方法保留了这些不变量，并在完全离散层面提供了可逆方案。通过数值模拟验证了我们方法的准确性及其在多个测试问题中保持不变量的能力。