The Heterogeneous Multiscale Method and the “Equation-free” Approach to Multiscale Modeling

In many problems of multiscale modeling, we are interested in capturing the macroscale behavior of the system with the help of some accurate microscale models, bypassing the need of using empirical macroscale models. This paper gives an overview of the recent efforts on establishing general strategies for designing such algorithms. After reviewing some important classical examples, the Car-Parrinello molecular dynamics, the quasicontinuum method for modeling the deformation of solids and the kinetic schemes for gas dynamics, we discuss three attempts that have been made for designing general strategies: Brandt’s renormalization multi-grid method (RMG), the heterogeneous multiscale method (HMM) and the “equation-free” approach. We will discuss the relative merits and difficulties with each strategy and we will make an attempt to clarify their similarities and differences. We end with some perspectives about this kind of approach.