Many problems of fundamental and practical importance contain
multiple scale solutions. Composite materials, flow and transport in
porous media, and turbulent flow are examples of this type. Direct
numerical simulations of these multiscale problems are extremely
difficult due to the range of length scales in the underlying physical
problems. Here, we introduce a dynamic multiscale method for computing
nonlinear partial differential equations with multiscale solutions.
The main idea is to construct semi-analytic multiscale solutions
local in space and time, and use them to construct the coarse grid
approximation to the global multiscale solution. Such approach overcomes
the common difficulty associated with the memory effect and the
non-unqiueness in deriving the global averaged equations for flow in
heterogeneous porous media. Such methodology also provides an effective
multiscale numerical method for computing incompressible Euler and
Navier-Stokes equations with multiscale solutions. In a related effort,
we introduce a new class of numerical methods to solve the stochastically
forced Navier-Stokes equations. We will demonstrate that our numerical
method can be used to compute accurately high order statitstical
quantites more efficiently than the traditional Monte-Carlo method.
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