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Magnetohydrodynamics (MHD) studies the behaviors of ionized gases, such as plasmas, in the presence of a magnetic field. MHD is used in many applications, such as geophysics, space physics, and nuclear fusion.

Despite intensive research in recent decades, many physical and numerical aspects of MHD are not well understood. The challenges inherent in solving MHD stem from the obstacles encountered in ordinary hydrodynamics, such as those described by the compressible Euler/Navier-Stokes equations, along with the intricacies arising from electromagnetism. A characteristic of compressible flows is their tendency to develop shocks/discontinuities over time. This often leads to unphysical traits in numerical approximations if the capturing scheme is not constructed properly. By physical laws, the magnetic field is solenoidal. However, in practice, numerical schemes seldom ensure this property precisely, which may lead to instability and convergence to wrong solutions. In numerical simulation of many applications, positive physical quantities such as density and pressure can easily become negative. On the whole, preserving the physical relevance of the numerical solutions poses a significant challenge in MHD.

This thesis presents several numerical schemes based on Galerkin approximations to solve MHD. The schemes rely on viscous regularization, a technique to remove mathematical singularities by adding a vanishing viscosity term to the MHD equations. At the continuous level, we propose several choices of viscous regularization and rigorously show that they are consistent with thermodynamics. Based on these choices, we construct numerical schemes of which robustness is confirmed through many challenging benchmarks. Finally, we propose a nonconventional algorithm that simultaneously preserves many desirable physical properties, including positivity of density and internal energy, conservation of total energy, minimum entropy principle, and zero magnetic divergence.