Robustness of MUSCL schemes for 2D unstructured meshes

We consider second-order accuracy MUSCL schemes to approximate the solutions of hyperbolic system of conservation laws. In the context of the 2D unstructured grids, we propose a limitation procedure on the gradient reconstruction to enforce several stability properties. We establish that the MUSCL scheme preserves the invariant domains and satisfy a set of entropy inequalities. A conservation assumption is not useful in the present work to define the piecewise linear approximations and the proposed limitation can be understood as a systematic correction of the standard gradient reconstruction procedure. The numerical method is applied to the compressible Euler equations. The gradient reconstruction is performed using the characteristic variables. Several numerical tests exhibit stability and robustness of the scheme.