Abstract
We study the limit of the hyperbolic–parabolic approximation
The function \({\tilde {\ss}}\) is defined in such a way as to guarantee that the initial boundary value problem is well posed even if \({\tilde {B}}\) is not invertible. The data \({\bar {g}}\) and \({\bar {v}_{0}}\) are constant. When \({\tilde {B}}\) is invertible, the previous problem takes the simpler form
Again, the data \({\bar {v}_b}\) and \({\bar {v}_0}\) are constant. The conservative case is included in the previous formulations. Convergence of the \({v^{\varepsilon}}\) , smallness of the total variation and other technical hypotheses are assumed, and a complete characterization of the limit is provided. The most interesting points are the following: First, the boundary characteristic case is considered, that is, one eigenvalue of \({\tilde {A}}\) can be 0. Second, as pointed out before, we take into account the possibility that \({\tilde {B}}\) 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 this condition is not satisfied, then pathological behaviors may occur.
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Bianchini, S., Spinolo, L.V. The Boundary Riemann Solver Coming from the Real Vanishing Viscosity Approximation. Arch Rational Mech Anal 191, 1–96 (2009). https://doi.org/10.1007/s00205-008-0177-6
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DOI: https://doi.org/10.1007/s00205-008-0177-6