  The problem of collisions in a plasma is a wide subject with a huge
historical literature. In fact, the description of realistic plasmas is a tough
problem to attach, both from the theoretical and the numerical point of view,
and which requires in general to approximate the original collisional Landau
integral by simplified differential operators in reduced dimensionality. In
this paper, a Eulerian time-splitting algorithm for the study of the
propagation of electrostatic waves in collisional plasmas is presented.
Collisions are modeled through one-dimensional operators of the Fokker-Planck
type, both in linear and nonlinear form. The accuracy of the numerical code is
discussed by comparing the numerical results to the analytical predictions
obtained in some limit cases when trying to evaluate the effects of collisions
in the phenomenon of wave plasma echo and collisional dissipation of
Bernstein-Greene-Kruskal waves. Particular attention is devoted to the study of
the nonlinear Dougherty collisional operator, recently used to describe the
collisional dissipation of electron plasma waves in a pure electron plasma
column. A receipt to prevent the filamentation problem in Eulerian algorithms
is provided by exploiting the property of velocity diffusion operators to
smooth out small velocity scales.
