A uniformly accurate numerical method for a class of dissipative systems
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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.References
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Additional Information
- Philippe Chartier
- Affiliation: Inria Rennes, IRMAR and ENS Rennes, Campus de Beaulieu, F-35170 Bruz, France
- MR Author ID: 335517
- Email: Philippe.Chartier@inria.fr
- Mohammed Lemou
- Affiliation: CNRS, IRMAR and ENS Rennes, Campus de Beaulieu, F-35170 Bruz, France
- MR Author ID: 355223
- Email: Mohammed.Lemou@univ-rennes1.fr
- Léopold Trémant
- Affiliation: Inria Rennes and IRMAR, Campus de Beaulieu, 35049 Rennes, France
- ORCID: 0000-0002-0512-4286
- Email: Leopold.Tremant@inria.fr
- Received by editor(s): May 25, 2020
- Received by editor(s) in revised form: March 4, 2021, and July 15, 2021
- Published electronically: November 5, 2021
- Additional Notes: This work has been carried out within the framework of the EUROfusion Consortium and has received funding from the Euratom research and training programme 2014-2018 and 2019-2020 under grant agreement No. 633053. The views and opinions expressed herein do not necessarily reflect those of the European Commission.
- © Copyright 2021 American Mathematical Society
- Journal: Math. Comp. 91 (2022), 843-869
- MSC (2020): Primary 65L04, 34E13; Secondary 65L05, 65L20, 65L70
- DOI: https://doi.org/10.1090/mcom/3688
- MathSciNet review: 4379978