Numerical Analysis of Multiscale Computations (09w5109)


Bjorn Engquist (University of Texas at Austin)

Olof Runborg (KTH)

Steve Ruuth (Simon Fraser University)

Yen-Hsi Tsai (University of Texas at Austin)


The Banff International Research Station will host the "Numerical Analysis of Multiscale Computations" workshop next week, December 6 - December 11, 2009.

The ever increasing power of modern computers and the
accompanying development of numerical algorithms is
pushing the use of large scale computing in science and
engineering into new territories. Where one used to
simulate single physics models, like fluid flow,
and electromagnetic waves, one is now focusing on more
difficult, and computationally expensive, problems
where many physics models are coupled together. These
are problems where the single physics models are not
accurate enough. For instance, in simulation of complex fluids,
such as polymers in a solvent, the flow can reasonably
be described by the classical equations of Navier-Stokes
with certain well-tuned parameters
obtained through empirical arguments based on physical insights.
The accuracy hinges critically on the quality of
the heuristics, which is often not satisfactory. In
the emerging methods, one tries instead to go back to first
principles, and the values of the parameters
would instead be obtained by direct numerical
simulation of the detailed interaction of the polymers and
the background fluid.

The resulting method typically requires solving
problems involving vast differences in spatial and temporal scales;
the polymers are for instance typically
many magnitudes smaller than the domain over which we
want to compute the flow, while at the same time
they are magnitudes bigger than the size of the
fluid particles. We call such problems where a coarse,
macroscale, physical model (Navier-Stokes) is coupled
to a fine, microscale, model (polymer simulation)
multiscale problems. It is the difference in scales
which makes such problems very computationally demanding.
Solving the fine scale
equations accurately over the length and time scales of the
macroscopic quantities, is usually an impossible task.

In this workshop we will consider the mathematics of
multiscale problems and in particular a
new class of numerical methods which
aim to make the coupling between the coarse and
fine scale models more efficient.
This new kind of multiscale approach makes it
feasible to treat problems that were previously
out of reach, and to obtain higher accuracy
when simulating important physical phenomena
in the applied sciences including, materials science,
chemistry, fluid dynamics, and biology.

The Banff International Research Station for Mathematical Innovation and Discovery (BIRS) is a collaborative Canada-US-Mexico venture that provides an environment for creative interaction as well as the exchange of ideas, knowledge, and methods within the Mathematical Sciences, with related disciplines and with industry. The research station is located at The Banff Centre in Alberta and is supported by Canada's Natural Science and Engineering Research Council (NSERC), the US National Science Foundation (NSF), Alberta's Advanced Education and Technology, and Mexico's Consejo Nacional de Ciencia y Tecnologí­a (CONACYT).