  The paper presents two contributions in the context of the numerical
simulation of magnetized fluid dynamics. First, we show how to extend the ideal
magnetohydrodynamics (MHD) equations with an inbuilt magnetic field divergence
cleaning mechanism in such a way that the resulting model is consistent with
the second law of thermodynamics. As a byproduct of these derivations, we show
that not all of the commonly used divergence cleaning extensions of the ideal
MHD equations are thermodynamically consistent. Secondly, we present a
numerical scheme obtained by constructing a specific finite volume
discretization that is consistent with the discrete thermodynamic entropy. It
includes a mechanism to control the discrete divergence error of the magnetic
field by construction and is Galilean invariant. We implement the new
high-order MHD solver in the adaptive mesh refinement code FLASH where we
compare the divergence cleaning efficiency to the constrained transport solver
available in FLASH (unsplit staggered mesh scheme).
