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We are interested in the numerical approximation of discontinuous solutionsin non conservative hyperbolic systems. An extension to second-order of a newstrategy based on in-cell discontinuous reconstructions to deal with thischallenging topic is presented. This extension is based on the combination ofthe first-order in-cell reconstruction and the MUSCL-Hancock reconstruction.The first-order strategy allowed in particular to capture exactly the isolatedshocks and this new second-order extension keep this property. We also set thebasis of an extension to high-order methods following this in-cell methodologyand a way for capturing exactly more than one shock. Several numerical testsare proposed to validate the methods for the Coupled-Burgers system, Gasdynamics equations in Lagrangian coordinates and the modified shallow watersystem.

In-cell Discontinuous Reconstruction path-conservative methods for non conservative hyperbolic systems -- Second-order extension