The space-time evolution of an expanding plasma is studied numerically and analytically. Based on hydrodynamic equations a Lagrangian algorithm is developed both in the collisionless and in the dissipative regime. Several studies are made to determine the dependence of this nonlinear dynamical system on charge separation and electron thermodynamics, on ion viscosity, on the initial conditions, and on the degree of implicitness of the code. Without dissipation ion wavebreaking and an associated ion bunching is found to take place limiting the computational time. Dissipation removes this singularity allowing long time calculations. The advantage of a full implicit scheme is emphasized, and support comes from a conventional stability analysis making use of the concept of pseudo-sound velocity. Analytically, wavebreaking is confirmed by establishing and solving a scalar wave equation. It describes the entire dynamical system and includes charge separation. The latter is found to be essential in describing the first behaviour. Fast ion velocities of three and more ion sound velocities are obtained in agreement with laboratory experiments.