  A singularly perturbed linear system of second order partial differential
equations of parabolic reaction-diffusion type with given initial and boundary
conditions is considered. The leading term of each equation is multiplied by a
small positive parameter. These singular perturbation parameters are assumed to
be distinct. The components of the solution exhibit overlapping layers.
Shishkin piecewise-uniform meshes are introduced, which are used in conjunction
with a classical finite difference discretisation, to construct a numerical
method for solving this problem. It is proved that the numerical approximations
obtained with this method are first order convergent in time and essentially
second order convergent in the space variable uniformly with respect to all of
the parameters.
