A critical analysis of well-known procedures for the computation of one-dimensional shocked flows is made, in order to show the inconveniences of computing finite differences across a discontinuity and to prove that the use of the equations of motion in conservation form does not make the results any more accurate. A technique is developed to treat one-dimensional inviscid problems and it is applied to the problem of an accelerating piston. Practical and safe ways to predict the formation of a shock and to follow it up in its evolution are given.