Why the MUSCL-Hancock Scheme is L1-stable

The finite volume methods are one of the most popular numerical procedure to approximate the weak solutions of hyperbolic systems of conservation laws. They are developed in the framework of first-order numerical schemes. Several approaches are proposed to increase the order of accuracy. The van Leer methods are interesting ways. One of them, namely the MUSCL-Hancock scheme, is full time and space second-order accuracy. In the present work, we exhibit relevant conditions to ensure the L1-stability of the method. A CFL like condition is established, and a suitable limitation procedure for the gradient reconstruction is developed in order to satisfy the stability criterion. In addition, we show that the conservative variables are not useful within the gradient reconstruction and the procedure is extended in the framework of the primitive variables. Numerical experiments are performed to show the interest and the robustness of the method.