Relativistic Magnetohydrodynamics: Renormalized eigenvectors and full
  wave decomposition Riemann solver
  We obtain renormalized sets of right and left eigenvectors of the flux vector
Jacobians of the relativistic MHD equations, which are regular and span a
complete basis in any physical state including degenerate ones. The
renormalization procedure relies on the characterization of the degeneracy
types in terms of the normal and tangential components of the magnetic field to
the wavefront in the fluid rest frame. Proper expressions of the renormalized
eigenvectors in conserved variables are obtained through the corresponding
matrix transformations. Our work completes previous analysis that present
different sets of right eigenvectors for non-degenerate and degenerate states,
and can be seen as a relativistic generalization of earlier work performed in
classical MHD. Based on the full wave decomposition (FWD) provided by the the
renormalized set of eigenvectors in conserved variables, we have also developed
a linearized (Roe-type) Riemann solver. Extensive testing against one- and
two-dimensional standard numerical problems allows us to conclude that our
solver is very robust. When compared with a family of simpler solvers that
avoid the knowledge of the full characteristic structure of the equations in
the computation of the numerical fluxes, our solver turns out to be less
diffusive than HLL and HLLC, and comparable in accuracy to the HLLD solver. The
amount of operations needed by the FWD solver makes it less efficient
computationally than those of the HLL family in one-dimensional problems.
However its relative efficiency increases in multidimensional simulations.
