Variational discretizations of ideal magnetohydrodynamics in smooth regime using structure-preserving finite elements

We propose a new class of finite element approximations to ideal compressible magnetohydrodynamic equations in smooth regime. Following variational approaches developed for fluid models in the last decade, our discretizations are built via a discrete variational principle mimicking the continuous Euler-Poincaré 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 using spline elements to verify the accuracy of our approach and its ability to preserve the invariants for several test problems.