A MUSCL-scheme on staggered unstructured grids for the Euler equations

We introduce muscl reconstructions for schemes designed on staggered grids for the numerical resolution of the Euler system. The proposed scheme uses principles based on the Discrete Duality Finite Volume framework for which the structure of the unknowns makes the gradient reconstruction naturally amenable. Dealing with general unstructured meshes, we develop a multislope approach where gradients and limiters are constructed face-by-face of the control volumes. Since the numerical unknowns are stored on different grids, several reconstruction techniques have to be combined. We study the stability conditions induced by the preservation of the positivity of the density and internal energy. The scheme is challenged against several numerical test cases.