The boundary Riemann solver coming from the real vanishing viscosity   approximation
  We study a family of initial boundary value problems associated to mixed hyperbolic-parabolic systems:   v^{\epsilon} _t + A (v^{\epsilon}, \epsilon v^{\epsilon}_x ) v^{\epsilon}_x = \epsilon B (v^{\epsilon} ) v^{\epsilon}_{xx}   The conservative case is, in particular, included in the previous formulation.   We suppose that the solutions $v^{\epsilon}$ to these problems converge to a unique limit. Also, it is assumed smallness of the total variation and other technical hypotheses and it is provided a complete characterization of the limit.   The most interesting points are the following two.   First, the boundary characteristic case is considered, i.e. one eigenvalue of $A$ can be $0$.   Second, we take into account the possibility that $B$ is not invertible. To deal with this case, we take as hypotheses conditions that were introduced by Kawashima and Shizuta relying on physically meaningful examples. We also introduce a new condition of block linear degeneracy. We prove that, if it is not satisfied, then pathological behaviours may occur. 
S. Bianchini and L. V. Spinolo