  We consider a class of relaxation problems mixing slow and fast variations
which can describe population dynamics models or hyperbolic systems, with
varying stiffness (from non-stiff to strongly dissipative), and develop a
multi-scale method by decomposing this problem into a micro-macro system where
the original stiffness is broken. We show that this new problem can therefore
be simulated with a uniform order of accuracy using standard explicit numerical
schemes. In other words, it is possible to solve the micro-macro problem with a
cost independent of the stiffness (a.k.a. uniform cost), such that the error is
also uniform. This method is successfully applied to two hyperbolic systems
with and without non-linearities, and is shown to circumvent the phenomenon of
order reduction.
