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 nonnegativity 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.