The Heterogeneous Multiscale Methods with Application to Combustion

The framework of the heterogeneous multiscale methods (HMM) is briefly reviewed. Both the original HMM and the seamless HMM are discussed. Applications to interface capturing and flame front tracking are presented. 18.1 The Heterogeneous Multiscale Method The heterogeneous multiscale method (HMM) is a general framework for designing multiscale methods for a wide range of applications [8, 10]. HMM is a top-down strategy: it starts with an incomplete macroscale model, the missing data in the macroscale model is estimated on the fly using the microscale model. As such, it offers a platform for designing numerical algorithms that makes use of the knowledge we have at both the macroand micro-scales. Here we will briefly review the main ideas, its advantages and limitations. We will also discuss possible applications to combustion. Weinan E Department of Mathematics and Program in Applied and Computational Mathematics, Princeton University, e-mail: [email protected] Björn Engquist Department of Mathematics and Institute for Computational Engineering and Sciences, University of Texas, Austin, e-mail: [email protected] Yi Sun Statistical and Applied Mathematical Sciences Institute, e-mail: [email protected] T. Echekki, E. Mastorakos (eds.), Turbulent Combustion Modeling, Fluid Mechanics and Its Applications 95, DOI 10.1007/978-94-007-0412-1 18, © Springer Science+Business Media B.V. 2011 439 440 Weinan E, Björn Engquist and Yi Sun 18.1.1 The Basic Framework The general setting is as follows. At the macroscopic level, we make an assumption about the form of the macroscale model, which can be expressed abstractly as