A Dynamical Systems Approach to Turbulence -- Challenges for
High-Performance Computing

Bruce Boghosian
Department of Mathematics, Tufts University

Turbulence has been called the "last unsolved problem of classical
mechanics."  While it has long been understood that the details of
turbulent flow are essentially unpredictable beyond a certain number
of Lyapunov times due to the so-called "Butterfly effect," hope
remains for a comprehensive statistical description of turbulence.
Two developments in dynamical systems theory over the past twenty
years provide solid foundation for that hope.  The first is the
observation, placed on firm foundation in the 1980s, that
Navier-Stokes flow has a finite-dimensional set of attracting states.
The second is the development, especially that in the 1980s and 1990s,
of the dynamical zeta function formalism, enabling statistical
descriptions of chaotic dynamical systems, given knowledge of their
unstable periodic orbits (UPOs).  For this reason, the efficient
numerical computation of UPOs has gained great importance over the
past decade, in both the dynamical systems and turbulence literature.
Periodic orbits for high-dimensional state spaces are devilishly
difficult to calculate, requiring high-performance computing and
placing new demands on algorithms, accuracy and hardware.  This talk
will discuss a general software framework for the computation of
periodic orbits of high-dimensional dynamical systems, that is based
on a matrix-free implementation of a Newton-Krylov optimization
algorithm suggested by Viswanath.  This talk will describe some of the
computational challenges involved in computing UPOs, discuss the
possible future implementation of this software on heterogeneous
architectures, and demonstrate the successful computation of UPOs of
driven Navier-Stokes turbulence in two and three spatial dimensions.