[en] A singularities tracking version of the GRP scheme for the integration of the Euler equations for compressible duct flow is presented. Flow singularities corresponding to contact (material), shock, or gradient discontinuities are represented by special grid points that move through the fixed grid at the appropriate speed of propagation. The primary modification to the Eulerian GRP scheme is the evaluation of fluxes at singular points in a unified grid containing the union of regular Euler grid points and singular moving points. Interactions between singular points (shock-shock or shock-contact interactions) are treated accurately by solving the appropriate generalized Riemann problem. The new GRP/ST scheme combines some merits of computation by characteristics, with a robustness approaching that of a finite difference conservation laws scheme. Shock wave phenomena illustrating the capabilities of the tracking method are presented, demonstrating improved resolution and accuracy at a given grid