Here we reproduce the complete text of Los Alamos Scientific Laboratory Report LA-671 authored by Robert Richtmyer in 1948. This report, though unpublished, is arguably one of the most important and influential monographs in computational fluid dynamics. It will be recognized that this report is the source of much of the material in the highly cited 1950 paper of von Neumann and Richtmyer. However, ideas in this report go beyond that later paper. In particular, we find here a motivation and a derivation of the quadratic form of artificial viscosity, an enduring concept in the simulation of flows with shocks, with turbulence models, and more recently, with theory in fluid dynamics. We reproduce the report in its entirety and then append a brief commentary to familiarize the reader with the environment in which Richtmyer worked, the motivation for his work, and some ensuing research. Artificial viscosity remains essential in Lagrangian simulations of fusion technologies such as inertial confinement fusion.