  Fully compressible magnetohydrodynamic (MHD) simulations are a fundamental
tool for investigating the role of dynamo amplification in the generation of
magnetic fields in deep convective layers of stars. The flows that arise in
such environments are characterized by low (sonic) Mach numbers (M_son < 0.01
). In these regimes, conventional MHD codes typically show excessive
dissipation and tend to be inefficient as the Courant-Friedrichs-Lewy (CFL)
constraint on the time step becomes too strict. In this work we present a new
method for efficiently simulating MHD flows at low Mach numbers in a
space-dependent gravitational potential while still retaining all effects of
compressibility. The proposed scheme is implemented in the finite-volume
Seven-League Hydro (SLH) code, and it makes use of a low-Mach version of the
five-wave Harten-Lax-van Leer discontinuities (HLLD) solver to reduce numerical
dissipation, an implicit-explicit time discretization technique based on Strang
splitting to overcome the overly strict CFL constraint, and a well-balancing
method that dramatically reduces the magnitude of spatial discretization errors
in strongly stratified setups. The solenoidal constraint on the magnetic field
is enforced by using a constrained transport method on a staggered grid. We
carry out five verification tests, including the simulation of a small-scale
dynamo in a star-like environment at M_son ~ 0.001 . We demonstrate that the
proposed scheme can be used to accurately simulate compressible MHD flows in
regimes of low Mach numbers and strongly stratified setups even with moderately
coarse grids.
