  This work presents an extension of discretely entropy stable discontinuous
Galerkin (DG) methods to the resistive magnetohydrodynamics (MHD) equations.
Although similar to the compressible Navier-Stokes equations at first sight,
there are some important differences concerning the resistive MHD equations
that need special focus. The continuous entropy analysis of the ideal MHD
equations, which are the advective parts of the resistive MHD equations, shows
that the divergence-free constraint on the magnetic field components must be
incorporated as a non-conservative term in a form either proposed by Powell or
Janhunen. Consequently, this non-conservative term needs to be discretized,
such that the approximation is consistent with the entropy. As an extension of
the ideal MHD system, we address in this work the continuous analysis of the
resistive MHD equations and show that the entropy inequality holds. Thus, our
first contribution is the proof that the resistive terms are symmetric and
positive semi-definite when formulated in entropy space as gradients of the
entropy variables. Moreover, this enables the construction of an entropy stable
DG discretization for the resistive MHD equations. However, the resulting
method suffers from large errors in the divergence-free constraint, since no
particular treatment of divergence errors is included in the standard resistive
MHD model. Hence, our second contribution is the extension of the resistive MHD
equations with proper divergence cleaning based on a generalized Lagrange
multiplier (GLM) strategy. We construct and analyze a DG method that is entropy
stable for the resistive MHD equations and has a built-in GLM divergence
cleaning mechanism. The theoretical derivations and proofs are then verified by
several numerical examples...
