Abstract: We introduce a wave-front tracking algorithm for $N\times N$ hyperbolic systems of conservation laws, $u_t+F(u)_x=0$, that admits characteristic fields which are neither genuinely nonlinear nor linearly degenerate in the sense of Lax. Instead we assume that, for any non genuinely nonlinear $i$-th characteristic family, the derivative of the $i$-th eigenvalue $\lambda_i(u)$ of $DF(u)$ in the direction of the $i$-th right eigenvector $r_i(u)$, vanishes on a single $(N-1)$-dimensional hypersurface in the $u$-space, transversal to the field $r_i(u).$ Systems that fulfill this type of assumptions are of particular interest in studying elastodynamic or rigid heat conductors at low temperature. The first proof of the existence of weak solutions for non genuinely nonlinear systems was given by T.P. Liu, based on a Glimm scheme. Our construction here provides an alternative method to establish the global existence of weak solutions for such systems. Moreover, relying on a stability analysis, we show that these solutions are entropy admissible in the sense of Lax.
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