  A combination of implicit and explicit timestepping is analyzed for a system
of ODEs motivated by ones arising from spatial discretizations of evolutionary
partial differential equations. Loosely speaking, the method we consider is
implicit in local and stabilizing terms in the underlying PDE and explicit in
nonlocal and unstabilizing terms. Unconditional stability and convergence of
the numerical scheme are proven by the energy method and by algebraic
techniques. This stability result is surprising because usually when different
methods are combined, the stability properties of the least stable method plays
a determining role in the combination.
