Numerical capturing of some hydrodynamical shocks

Based on von Neumann ideas (Proposal and analysis of a numerical method for the treatment of hydrodynamical shock problems, in: A.H. Taub, General Ed., Collected Works, Vol. VI, Pergamon Press, Oxford, 1963, pp. 361–379), later developed by Lax (T.Y. Hou, P.D. Lax, Dispersive approximations in fluid dynamics, Commun. Pure Appl. Math. 44 (1991) 1–40) and Greenberg (Commun. Pure Appl. Math. XLV (1992) 1125–1139; J.M. Greenberg, G. Hedstrom, A note on a model for quasilinear wave equation, Arch. Rat. Mech. Anal. 40 (1971) 160–165), in this paper we present a method for hydrodynamical shocks generation. We determine weak solutions of one-dimensional nonlinear wave equation which present shocks and we give also a numerical treatment for capturing these shocks.