Beyond multiscale and multiphysics: Multimaths for model coupling
Résumé
The purpose of this article is to present a unified view of some multiscale models that have appeared in the past decades in computational materials science. Although very different in nature at first sight, since they are employed to simulate complex fluids on the one hand and crystalline solids on the other hand, the models presented actually share a lot of similarities, many of those being in fact also present in most multiscale strategies. The mathematical and numerical difficulties that these models generate, the way in which they are utilized (in particular as numerical strategies coupling different models in different regions of the computational domain), the computational load they imply, are all very similar in nature. In particular, a common feature of these models is that they require knowledge and techniques from different areas in Mathematics: theory of partial differential equations, of ordinary differential equations, of stochastic differential equations, and all the related numerical techniques appropriate for the simulation of these equations. We believe this is a general trend of modern computational modelling.