IN GENERAL, SOLUTIONS TO THE ONE-DIMENSIONAL, unisteady hydrodynamic equations can only be obtained by numerical integration. Several methods of performing such an integration are possible [1-6]. Thus a choice of method must be made, and unless this decision is made with care, considerable labor can be expended in arriving at the optimum method for the problem in hand. It is the purpose of this paper to discuss two typical but rather divergent methods from the viewpoint of the user, who has access to a fair sized digital computer. Consideration is restricted to motion depending on only one space variable. Unsteady problems involving two or more space variables are quite complicated, and adequate comparison of the various methods of numerical solution does not seem possible, at present. The hydrodynamic equations can be the mathematical model for a number of physical problems, depending on the assumptions one feels justified in using to obtain them. An example of such a physical problem is the use of the hydrodynamic equations to represent a strictly one-dimensional motion of a deformable solid, strictly one-dimensional meaning the only space variable is a linear cartesian coordinate. The set of equations under consideration is comprised of differential statements of the conservation of mass, momentum, and energy (equations (1), (2) and (3) below respectively) together with a constitutive equation for the medium, of which equation (4) is a typical example. If the stress is a function of the state of strain and internal energy only, the constitutive equation is an equation of state in the usual thermodynamic sense. In the following this assumption is implicit, and hence the medium may be considered a fluid. In Lagrangian form we then have: