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Path-conservative in-cell discontinuous reconstruction schemes for non conservative hyperbolic systems

We are interested in the numerical approximation of discontinuous solutions in non conservative hyperbolic systems. We introduce the basics of a new strategy based on in-cell discontinuous reconstructions to deal with this challenging topic, and apply it to a 2x2 non conservative toy model, and a 3x3 gas dynamics system in Lagrangian coordinates. The strategy allows in particular to compute exactly isolated shocks. Numerical evidences are proposed.