  This paper presents entropy symmetrization and high-order accurate entropy
stable schemes for the relativistic magnetohydrodynamic (RMHD) equations. It is
shown that the conservative RMHD equations are not symmetrizable and do not
admit a thermodynamic entropy pair. To address this issue, a symmetrizable RMHD
system, equipped with a convex thermodynamic entropy pair, is proposed by
adding a source term into the equations, providing an analogue to the
non-relativistic Godunov--Powell system. Arbitrarily high-order accurate
entropy stable finite difference schemes are developed on Cartesian meshes
based on the symmetrizable RMHD system. The crucial ingredients of these
schemes include (i) affordable explicit entropy conservative fluxes which are
technically derived through carefully selected parameter variables, (ii) a
special high-order discretization of the source term in the symmetrizable RMHD
system, and (iii) suitable high-order dissipative operators based on
essentially non-oscillatory reconstruction to ensure the entropy stability.
Several numerical tests demonstrate the accuracy and robustness of the proposed
entropy stable schemes.
