Abstract This paper treats the collective behavior of hot plasma as modified by the numerical time integration methods used to integrate the particle equations of motion in computer simulation of plasmas. No approximation, other than ignoring roundoff errors, is made in analyzing the finite-difference algorithms. Our results reduce simply and exactly to the corresponding results of plasma theory in the limit Δt → 0. The possibility of nonphysical instability is considered. The results of this and of previous papers are combined to describe both the spatial and temporal difference algorithms. The theory is generalized to a class of integration schemes, some algorithms are analyzed, and a new example is synthesized. The difficulty of developing algorithms stable at very large time steps is examined. The present analysis may be combined with an earlier rigorous analysis of the spatial grid used for field equations, to develop a kinetic theory of simulation plasmas paralleling that for real plasmas. This theory may be of use in the design and interpretation of computer simulation experiments.