Beyond multiscale and multiphysics: Multimaths for model coupling
Xavier Blanc, ,CEA, DAM, DIF, F-91297, Arpajon, Claude Le Bris, Frédéric Legoll, Tony Lelièvre, ,Université Paris-Est, CERMICS, Ecole des Ponts ParisTech, 6 & 8, avenue Blaise Pascal, 77455 Marne-La-Vallée Cedex 2, ,Université Paris-Est, Institut Navier, LAMI, Ecole Nationale des Ponts et Chaussées, 6 et 8 avenue Blaise Pascal, 77455 Marne-la-Vallée Cedex 2 · Networks and Heterogeneous Media · 2010
The purpose of this article is to present a unified view of somemultiscale models that have appeared in the past decades in computationalmaterials science. Although very different in nature at first sight,since they are employed to simulate complex fluids on the one hand and crystallinesolids on the other hand, the models presented actually share a lot ofsimilarities, many of those being in fact also present inmost multiscale strategies. The mathematical and numericaldifficulties that these models generate, the way in which they areutilized (in particular as numerical strategies coupling different models indifferent 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 andtechniques from different areas in Mathematics: theory of partialdifferential equations, of ordinary differential equations, ofstochastic differential equations, and all the related numericaltechniques appropriate for the simulation of these equations. Webelieve this is a general trend of modern computational modelling.