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We study the limit of the hyperbolic-parabolic approximation 8 < " 2 RN v"(t, 0)  ¯ vb v"(0, x)  ¯ v0. Again, the datavb and ¯ v0 are constant. The conservative case is included in the previous formulations. It is assumed convergence of the v", 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, as pointed out before 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.

The Boundary Riemann Solver Coming from the Real Vanishing Viscosity Approximation