  A dynamical atomistic chain to simulate mechanical properties of a
one-dimensional material with zero temperature may be modelled by the molecular
dynamics (MD) model. Because the number of particles (atoms) is huge for a MD
model, in practice one often takes a much smaller number of particles to
formulate a coarse-grained approximation. We shall mainly consider the
consistency of the coarse-grained model with respect to the grain (mesh) size
to provide a justification to the goodness of such an approximation. In order
to reduce the characteristic oscillations with very different frequencies in
such a model, we either add a viscous term to the coarse-grained MD model or
apply a space average to the coarse-grained MD solutions for the consistency
study. The coarse-grained solution is also compared with the solution of the
(macroscopic) continuum model (a nonlinear wave equation of mixed type) to show
how well the coarse-grained model can approximate the macroscopic behavior of
the material. We also briefly study the instability of the dynamical atomistic
chain and the solution of the Riemann problem of the continuum model which may
be related to the defect of the atomistic chain under a large deformation in
certain locations.
