Studies of colliding laser-produced plasmas are of interest in several fields of research, including fusion energy generation, pulsed-laser deposition and potentially as intense short wavelength (X-ray and EUV) light sources. Computer simulations of such systems are a valuable tool in reconciling experimental results with theory and designing new experiments while also providing a platform to explore scenarios which are not easy to realise experimentally. A one-dimensional (1D) hydrodynamic simulation of quasineutrality- compliant colliding plasma systems has been developed and is described. The model is based on a multi-fluid theory due to P W Rambo and J Denavit [J. Comput. Phys. 98 (1992) 317] and uses a flux-corrected transport algorithm with an Eulerian grid in order to solve the three Euler equations for hydrodynamic flow. It includes source terms appropriate for high-density, high-temperature plasmas, such as those produced on dense targets at the focus of intense laser beams. Stagnation and interpenetration effects are simulated by means of collisional momentum coupling between species. Results are compared with the case of colliding plasmas in the absence of collisional coupling.