A uniformly accurate numerical method for a class of dissipative systems

Abstract

We consider a class of ordinary differential equations mixing slow and fast variations with varying stiffness (from non-stiff to strongly dissipative). Such models appear for instance in population dynamics or propagation phenomena. We develop a multi-scale approach by splitting the equations into a micro part and a macro part, from which the original stiffness has been removed. We then show that both parts can be simulated numerically with uniform order of accuracy using standard explicit numerical schemes. As a result, solving the problem in its micro-macro formulation can be done with a cost and an accuracy independent of the stiffness. This work is also a preliminary step towards the application of such methods to hyperbolic partial differential equations and we will indeed demonstrate that our approach can be successfully applied to two discretized hyperbolic systems (with and without non-linearities), though with some ad-hoc regularization.