ABSTRACT. 

The purpose of this talk is to provide an update of several
recent results regarding the stability of a well-known family of finite
difference schemes for the initial value problem associated with the
Petrowski well-posed, multispace-dimensional parabolic system

\[\frac{\partial{\bf u}({\bf x},t)}{\partial t} = \sum_{1\leq p\leq q\leq s}
A_{pq}\frac{\partial2{\bf u}({\bf x},t)}{\partial x_p\partial x_q} +
\sum_{1\leq p\leq s}
B_p\frac{\partial{\bf u}({\bf x},t)}{\partial x_p} +C{\bf u}({\bf x},t),\]

where $A_{pq},B_p$ and $C$ are constant matrices, $A_{pq}$ being Hermitian.