# Towards a uniformly accurate numerical scheme for stiff ODEs with a central manifold

We look at a simplified model with a fast relaxation towards a central manifold involving a small parameter ε. This kind of behaviour is found in reaction-diffusion systems, population dynamics and some kinetic models. At the moment, numerical simulations of this kind of systems involve error bounds requiring time-step small compared to ε. Recent algebraic methods allow more satisfactory results in the case of very small ε, but the error bound still depends on this small parameter, and computations are heavy.
Using a two-scale expansion, we set out to find a "uniformly accurate" (UA) numerical scheme, meaning a scheme for which the error bound is independent of the small parameter ε. This expansion yields a transport equation which we analyse and which we use to devise a new, UA numerical scheme. Implementing this scheme reveals limits in terms of efficiency which we try to overcome. Analysing the computed functions reveals that the main problem is the representation and the computation of convolution products of functions on large intervals.