We present a discontinuous Galerkin (DG) scheme with
suitable quadrature rules \cite{Tchen2017entropy} for
ideal compressible magnetohydrodynamic (MHD) equations on
structural meshes. The semi-discrete scheme is analyzed to
be entropy stable by using the symmetrizable version of the
equations as introduced by Godunov \cite{godunov1972symmetric},
the entropy stable DG framework with
suitable quadrature rules \cite{Tchen2017entropy},
the entropy conservative flux in
\cite{chandrashekar2016entropy} inside
each cell and the entropy dissipative approximate Godunov type
numerical flux at cell interfaces
to make the scheme entropy stable.
{\color{blue}{
The main difficulty in the generalization of the results in
\cite{Tchen2017entropy} is the appearance
of the non-conservative ``source terms'' added in the
modified MHD model introduced by Godunov \cite{godunov1972symmetric},
which do not exist in the general hyperbolic system studied in
\cite{Tchen2017entropy}. Special care must be taken to discretize these
``source terms'' adequately so that the resulting DG scheme
satisfies entropy stability.
}}
Total variation
diminishing / bounded (TVD/TVB) limiters and bound-preserving
limiters are applied to control spurious oscillations.
We demonstrate the accuracy and robustness of this new scheme
on standard MHD examples.