Convergence of an implicit scheme for diagonal non-conservative hyperbolic systems

In this paper, we consider diagonal non-conservative hyperbolic systems in one space dimension with monotone and large Lipschitz continuous data. Under a certain nonnegativity condition on the Jacobian matrix of the velocity of the system, global existence and uniqueness results of a Lipschitz solution for this system, which is not necessarily strictly hyperbolic, was proved in El Hajj and Monneau (J. Hyperbolic Differ. Equ. 10 (2013) 461-494). We propose a natural implicit scheme satisfiying a similar Lipschitz estimate at the discrete level. This property allows us to prove the convergence of the scheme without assuming it strictly hyperbolic.