On the Riemann problem for non-conservative hyperbolic systems

We consider the construction and the properties of the Riemann solver for the hyperbolic system \begin{equation}\label{E:hyp0} u_t + f(u)_x = 0, \end{equation} assuming only that $Df$ is strictly hyperbolic. In the first part we prove a general regularity theorem on the admissible curves $T_i$ of the $i$-family, depending on the number of inflection points of $f$: namely, if there is only one inflection point, $T_i$ is $C^{1,1}$. If the $i$-th eigenvalue of $Df$ is genuinely nonlinear, by it is well known that $T_i$ is $C^{2,1}$. However, we give an example of an admissible curve $T_i$ which is only Lipschitz continuous if $f$ has two inflection points. In the second part, we show a general method for constructing the curves $T_i$, and we prove a stability result for the solution to the Riemann problem. In particular we prove the uniqueness of the admissible curves for \eqref{E:hyp0}. Finally we apply the construction to various approximations to \eqref{E:hyp0}: vanishing viscosity, relaxation schemes and the semidiscrete upwind scheme. In particular, when the system is in conservation form, we obtain the existence of smooth travelling profiles for all small admissible jumps of \eqref{E:hyp0}.