**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

email: shlomo@andrew.cmu.edu