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Networks and Heterogeneous Media

2010, Volume 5, Issue 3: 423-460. doi: 10.3934/nhm.2010.5.423
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Beyond multiscale and multiphysics: Multimaths for model coupling

  • 1.
    CEA, DAM, DIF, F-91297, Arpajon
  • 2.
    Université Paris-Est, CERMICS, Ecole des Ponts ParisTech, 6 & 8, avenue Blaise Pascal, 77455 Marne-La-Vallée Cedex 2
  • 3.
    Université Paris-Est, Institut Navier, LAMI, Ecole Nationale des Ponts et Chaussées, 6 et 8 avenue Blaise Pascal, 77455 Marne-la-Vallée Cedex 2
  • Received: 01 January 2010 Revised: 01 April 2010

  • 35Q30, 35Q35, 35Q74, 60H35, 65C05, 65C30, 65K10, 65M12, 65M60, 65N12, 65N15, 70-08, 70C20, 70G75, 74B20, 74G15, 74G65, 74G70, 76A05, 76A10, 76M10, 82C31, 82C80, 82D60, 49M25.

  • 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.

    Citation: Xavier Blanc, Claude Le Bris, Frédéric Legoll, Tony Lelièvre. Beyond multiscale and multiphysics: Multimaths for model coupling[J]. Networks and Heterogeneous Media, 2010, 5(3): 423-460. doi: 10.3934/nhm.2010.5.423

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  • 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.


  • This article has been cited by:

    1. Claude Le Bris, Tony Lelièvre, Micro-macro models for viscoelastic fluids: modelling, mathematics and numerics, 2012, 55, 1674-7283, 353, 10.1007/s11425-011-4354-y
    2. Kristian Debrabant, Giovanni Samaey, Przemysław Zieliński, Study of micro–macro acceleration schemes for linear slow-fast stochastic differential equations with additive noise, 2020, 60, 0006-3835, 959, 10.1007/s10543-020-00804-5
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  • © 2010 the Author(s), licensee AIMS Press. This is an open access article distributed under the terms of the Creative Commons Attribution License (http://creativecommons.org/licenses/by/4.0)
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