Uniformly accurate numerical schemes for a class of dissipative systems

TL;DR

This paper introduces a multi-scale numerical method for dissipative systems that maintains uniform accuracy regardless of stiffness, effectively addressing challenges in simulating population dynamics and hyperbolic systems.

Contribution

The paper develops a micro-macro decomposition approach enabling explicit schemes to achieve uniform accuracy across varying stiffness levels.

Findings

Successfully applied to hyperbolic systems with nonlinearities

Achieves uniform order of accuracy independent of stiffness

Circumvents order reduction phenomena

Abstract

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.