The second-order Van-Leer MUSCL schemes are actually one of the most popular high order schemes for fluid dynamic computations. In the frame work of the Euler equations, we introduce a new slope limitation procedure to enforce the scheme to preserve the invariant region: namely the positiveness of both density and pressure as soon as the associated first order scheme does it. In addition, we obtain a second-order minimum principle on the specific entropy and second-order entropy inequalities. This new limitation is developed in the general framework of the MUSCL schemes and the choice of the numerical flux functions remains free. The proposed slope limitation can be applied to any change of variables and we do not impose the use of conservative variables in the piecewise linear reconstruction. Several examples are given in the framework of the primitive variables. Numerical 1D and 2D results are performed using several finite volume methods.
"Stability of the MUSCL Schemes for the Euler Equations." Commun. Math. Sci. 3 (2) 133 - 157, June 2005.