  We are interested in the numerical approximation of discontinuous solutions
in non conservative hyperbolic systems. An extension to second-order of a new
strategy based on in-cell discontinuous reconstructions to deal with this
challenging topic is presented. This extension is based on the combination of
the first-order in-cell reconstruction and the MUSCL-Hancock reconstruction.
The first-order strategy allowed in particular to capture exactly the isolated
shocks and this new second-order extension keep this property. We also set the
basis of an extension to high-order methods following this in-cell methodology
and a way for capturing exactly more than one shock. Several numerical tests
are proposed to validate the methods for the Coupled-Burgers system, Gas
dynamics equations in Lagrangian coordinates and the modified shallow water
system.
