An extension to second-order of a new strategy based on in-cell discontinuous reconstructions to deal with this challenging topic is presented . Several numerical tests are proposed to validate the methods for the Coupled-Burgers system, Gasdynamics equations in Lagrangian coordinates and the modified shallow watersystem . The first-order strategy allowed in particular to capture exactly the isolatedshocks and this new extension keeps this property . We also set thebasis of an extension to high-order methods following this in-Cell methodology following this .

Author(s) : Ernesto Pimentel-García, Manuel J. Castro, Christophe Chalons, Tomás Morales de Luna, Carlos Parés

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Keywords : order - extension - cell - methods - strategy -

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