Tuan Anh Dao: A structure-preserving numerical method for ideal magnetohydrodynamics

Abstract:

This talk discusses our new structure-preserving numerical method for the compressible ideal magnetohydrodynamics (MHD) equations. The MHD equations model the dynamics of conducting fluids in magnetic fields. In the special case of non-conductivity or absence of a magnetic field, the MHD equations reduce to the compressible Euler equations. The additional challenges in solving MHD equations compared to compressible Euler equations include the preservation of zero magnetic divergence and positivity of different physical properties such as density, internal energy, and pressure.

This technique addresses the MHD equations using a non-divergence formulation, where the contributions of the magnetic field to the momentum and total mechanical energy are treated as source terms. Our approach uses the Marchuk-Strang splitting technique and involves three distinct components: a compressible Euler solver, a source-system solver, and an update procedure for the total mechanical energy. The scheme allows for significant freedom on the choice of Euler’s equation solver, while the magnetic field is discretized using a curl-conforming finite element space, yielding exact preservation of the involution constraints. We prove that the method preserves invariant domain properties, including the positivity of density, the positivity of internal energy, and the minimum principle of the specific entropy. If the scheme used to solve Euler’s equation conserves total energy, then the resulting MHD scheme can be proven to preserve total energy. Similarly, if the scheme used to solve Euler’s equation is entropy-stable, then the resulting MHD scheme is entropy-stable as well. In our approach, the CFL condition does not depend on magnetosonic wave speeds but only on the usual maximum wave speed from Euler’s system. By numerical experiments, we show that the method is capable of delivering high-order accuracy in space for smooth problems, while also offering unconditional robustness in the shock hydrodynamics regime.