The artificial viscosity method of von Neumann and Richtmyer is used to calculate the flow field for a shocked, elastic‐plastic‐relaxing solid; particular material parameters are chosen to represent the rate‐dependent behavior of iron. The constitutive relation is based on the dislocation model for plastic flow. Dislocation velocity is assumed to be a unique function of shear stress as given by Gilman and Johnston. Two models for dislocation multiplication are studied. The first is the multiple‐cross‐glide model discussed by Gilman, and the second is one in which pinned dislocations become sources of new mobile dislocations in the sense of the Frank‐Read multiplication process. Artificial viscosity calculations yield precursor amplitudes which are sometimes considerably different from those obtained from characteristic theory of precursor decay. These differences are believed to be due to the finite width of the elastic wave front provided by the viscosity. With proper choice of mesh size in the artificial viscosity calculations, a decrease in stress can be seen immediately behind the elastic precursor. This agrees qualitatively with experimentally determined shock profiles in iron.