  We present an ``equation-free'' multiscale approach to the simulation of
unsteady diffusion in a random medium. The diffusivity of the medium is modeled
as a random field with short correlation length, and the governing equations
are cast in the form of stochastic differential equations. A detailed
fine-scale computation of such a problem requires discretization and solution
of a large system of equations, and can be prohibitively time-consuming. To
circumvent this difficulty, we propose an equation-free approach, where the
fine-scale computation is conducted only for a (small) fraction of the overall
time. The evolution of a set of appropriately defined coarse-grained variables
(observables) is evaluated during the fine-scale computation, and ``projective
integration'' is used to accelerate the integration. The choice of these coarse
variables is an important part of the approach: they are the coefficients of
pointwise polynomial expansions of the random solutions. Such a choice of
coarse variables allows us to reconstruct representative ensembles of
fine-scale solutions with "correct" correlation structures, which is a key to
algorithm efficiency. Numerical examples demonstrating accuracy and efficiency
of the approach are presented.

Numerical Analysis