We have solved the coupled time-dependent cosmic ray transport equation and gas equations numerically for a number of one-dimensional cases. Although we have kept the physics as simple as possible, we have included injection of particles at a gas shock and the momentum dependence of the cosmic ray diffusion coefficient. As well as confirming the existence of strictly steady solutions for low-Mach shocks, we have found that there exist quasi-steady solutions in which the post-shock cosmic ray pressure remains constant while the particle spectrum evolves continuously. For high Mach numbers we show that it is possible to set up cosmic-ray-dominated shocks with no current source of particles in which the pressure is provided by an ever-decreasing number of particles at higher and higher energies. Apart from this we have been able to confirm a number of previous conjectures based partly on test-particle theory and partly on the two-fluid approximation.