Gradient entropy estimate and convergence of a semi-explicit scheme for   diagonal hyperbolic systems
  In this paper, we consider diagonal hyperbolic systems with monotone continuous initial data. We propose a natural semi-explicit and upwind first order scheme. Under a certain non-negativity condition on the Jacobian matrix of the velocities of the system, there is a gradient entropy estimate for the hyperbolic system. We show that our scheme enjoys a similar gradient entropy estimate at the discrete level. This property allows us to prove the convergence of the scheme. 
Laurent Monasse and R\'egis Monneau
math.NA