  This paper develops high-order accurate entropy stable (ES) adaptive moving
mesh finite difference schemes for the two- and three-dimensional special
relativistic hydrodynamic (RHD) and magnetohydrodynamic (RMHD) equations, which
is the high-order accurate extension of [J.M. Duan and H.Z. Tang, Entropy
stable adaptive moving mesh schemes for 2D and 3D special relativistic
hydrodynamics, J. Comput. Phys., 426(2021), 109949]. The key point is the
derivation of the higher-order accurate entropy conservative (EC) and ES finite
difference schemes in the curvilinear coordinates by carefully dealing with the
discretization of the temporal and spatial metrics and the Jacobian of the
coordinate transformation and constructing the high-order EC and ES fluxes with
the discrete metrics. The spatial derivatives in the source terms of the
symmetrizable RMHD equations and the geometric conservation laws are
discretized by using the linear combinations of the corresponding second-order
case to obtain high-order accuracy. Based on the proposed high-order accurate
EC schemes and the high-order accurate dissipation terms built on the WENO
reconstruction, the high-order accurate ES schemes are obtained for the RHD and
RMHD equations in the curvilinear coordinates. The mesh iteration
redistribution or adaptive moving mesh strategy is built on the minimization of
the mesh adaption functional. Several numerical tests are conducted to validate
the shock-capturing ability and high efficiency of our high-order accurate ES
adaptive moving mesh methods on the parallel computer system with the MPI
communication. The numerical results show that the high-order accurate ES
adaptive moving mesh schemes outperform both their counterparts on the uniform
mesh and the second-order ES adaptive moving mesh schemes.
