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In this paper, we consider diagonal hyperbolic systems with monotonecontinuous initial data. We propose a natural semi-explicit and upwind firstorder scheme. Under a certain non-negativity condition on the Jacobian matrixof the velocities of the system, there is a gradient entropy estimate for thehyperbolic system. We show that our scheme enjoys a similar gradient entropyestimate at the discrete level. This property allows us to prove theconvergence of the scheme.

Gradient entropy estimate and convergence of a semi-explicit scheme for diagonal hyperbolic systems