A Structure- and Pressure-Positivity-Preserving Semi-implicit IMEX Finite Volume Scheme for Ideal MHD at All Acoustic Mach and Alfvén Mach Numbers with Generic Equation of State
We present a conservative, structure-preserving, finite-volume scheme for ideal MHD that ensures pressure positivity, remains applicable across all Mach and Alfven regimes, and handles general nonlinear equations of state. The scheme splits the MHD system into three sub-systems according to characteristic wave scales: an advective part for hydrodynamic transport, a magnetic part for velocity-field coupling, and a pressure part for pressure-velocity coupling. Nonlinear advective terms are explicit, while the other two sub-systems are implicit, yielding a mild, velocity-based CFL condition that is supported by extensive numerical evidence. This makes the scheme suitable for gas-pressure or magnetic-pressure dominated regimes and the incompressible limit. The implicit discretisation gives a pressure equation that reduces to an elliptic form in the low-Mach limit for ideal gases and general thermodynamics. Pressure positivity is ensured via a local conservation-preserving modification of the pressure-internal-energy relation, avoiding a posteriori clipping while preserving conservation. The divergence-free constraint is enforced exactly via constrained transport. Second-order accuracy is achieved with an IMEX Runge-Kutta time integration, TVD reconstruction for explicit fluxes, and central discretisation for implicit terms. The scheme is validated against numerous benchmarks, including high- and low-Mach regimes, strongly magnetised flows, and standard MHD shock problems in 1D and 2D, demonstrating accuracy, stability, and excellent shock-capturing.