CNA 2001 Summer School




Numerical Simulations and Coarse Graining

Shlomo Ta'asan

ABSTRACT: Numerical solution of multiscale problems requires new approaches, as direct simulation techniques are infeasible. Models may be given at atomistic/molecular level while scientific questions of interest are at higher scales, such as the mesoscale or the macroscopic scale. Coarse graining procedures are essential for efficient simulation as well as for the interpretation of the data. As the complexity of the models increase, numerical methods become indispensable. The common use of numerical methods is to compute solutions of equations or processes. Here we will focus on the use of simulations to construct models and equations, that hold at larger temporal and/or spatial scales. The main issues are: 1) the choice of variables at different levels, 2) the representation (stochastic/deterministic, continuous/discrete), 3) the interaction between scales. Examples from different fields will be discussed. These will include passage from deterministic Hamiltonian systems to stochastic models, from atomistic models of fluids to macroscopic dynamics (partial differential equations), and atomistic models for polycrystals to grain boundary evolution equations, and possible more.

Shlomo Ta'asan
Carnegie Mellon University
Department of Mathematical Sciences
Pittsburgh, PA 15213