A numerical scheme for the one‐dimensional pressureless gases system

In this work, we investigate the numerical approximation of the one‐dimensional pressureless gases system. After briefly recalling the mathematical framework of the duality solutions introduced by Bouchut and James (Comm. Partial Differential Equations 24 (1999), 2173–2189), we point out that the upwind scheme for density and momentum does not satisfy the one‐sided Lipschitz (OSL) condition on the expansion rate required for the duality solutions. Then we build a diffusive scheme which allows the OSL condition to be recovered by following the strategy described by Boudin (SIAM J Math Anal 32 (2000), 172–193) for the continuous model.