Methods for computing (asynchronously) spatiotemporal solutions of Navier-Stokes

Recent papers on HPC argue that computing directly spatio-temporal solutions of chaotic/turbulent systems is more efficiently parallelized and more robust than the traditional forward-in-time evolution.

References

Wang, Q.; Gomez, S. A.; Blonigan, P. J.; Gregory, A. L. & Qian, E. Y. Towards scalable parallel-in-time turbulent flow simulations Phys. Fluids (2013) 25, 110818

Cvitanovic, P. & Lan, Y. Turbulent fields and their recurrences, in Antoniou, N. (Ed.) Correlations and Fluctuations in QCD : Proceedings of 10th International Workshop on Multiparticle Production, World Scientific (2003) 313-325

A spatiotemporal theory of transitional turbulence

UCSB Mathematics Colloquium Thu, Jan 26 at 3:30-4:30pm in South Hall 6635 (campus map is here)(South Hall floor plan) P. Cvitanović (with M. Gudorf, N.B. Budanur and B. Gutkin)
Center for Nonlinear Science,
Georgia Tech, Atlanta GA 30332-0430, USA predrag@gatech.edu

Recent advances in fluid dynamics reveal that the recurrent flows observed in moderate Reynolds number turbulence result from close passes to unstable invariant solutions of Navier-Stokes equations. By now hundreds of such solutions been computed for a variety of flow geometries, but always confined to small computational domains (minimal cells).

The 2016 Gutkin and Osipov paper on many-particle quantum chaos opens a path to determining such solutions on spatially infinite domains. Flows of interest (pipe, channel flows) often come equipped with D continuous spatial symmetries. If the theory is recast as a (D+1)-dimensional space-time theory, the space-time invariant solutions are (D+1)-tori (and not the 1-dimensional periodic orbits of the traditional periodic orbit theory). The symbolic dynamics is likewise (D+1)-dimensional (rather than a single temporal string of symbols), and the corresponding zeta functions should be sums over tori, rather than 1-dimensional periodic orbits. In this theory there is no time, there is only a repertoire of admissible spatiotemporal patterns.

## Methods for computing (asynchronously) spatiotemporal solutions of Navier-Stokes

Recent papers on HPC argue that computing directly spatio-temporal solutions of chaotic/turbulent systems is more efficiently parallelized and more robust than the traditional forward-in-time evolution.

ReferencesWang, Q.; Gomez, S. A.; Blonigan, P. J.; Gregory, A. L. & Qian, E. Y.

Towards scalable parallel-in-time turbulent flow simulations

Phys. Fluids (2013) 25, 110818

Donzis, D. A. & Aditya, K.

Asynchronous finite-difference schemes for partial differential equations

J. Computational Physics (2014), 274, 370-392

Christian Kuhn's codes: are an extension of NGSolve, see

gitlab.asc.tuwien.ac.at/jschoeberl/ngsolve-docu/wikis/home

Also see

www-m8.ma.tum.de/personen/kuehn/software.htmlDyGluS, here:Lan, Y. & Cvitanovic, P.

Variational method for finding periodic orbits in a general flow

Phys. Rev. E, (2004) 69, 016217

Cvitanovic, P. & Lan, Y.

Turbulent fields and their recurrences, in Antoniou, N.

(Ed.)Correlations and Fluctuations in QCD : Proceedings of 10th International Workshop on Multiparticle Production, World Scientific (2003) 313-325## A spatiotemporal theory of transitional turbulence

UCSB Mathematics Colloquium Thu, Jan 26 at 3:30-4:30pm in South Hall 6635(campus map is here)(South Hall floor plan)

P. Cvitanović(with M. Gudorf, N.B. Budanur and B. Gutkin)Center for Nonlinear Science,

Georgia Tech, Atlanta GA 30332-0430, USA

predrag@gatech.edu

Recent advances in fluid dynamics reveal that the recurrent flows observed in moderate Reynolds number turbulence result from close passes to unstable invariant solutions of Navier-Stokes equations. By now hundreds of such solutions been computed for a variety of flow geometries, but always confined to small computational domains (minimal cells).

The 2016 Gutkin and Osipov paper on many-particle quantum chaos opens a path to determining such solutions on spatially infinite domains. Flows of interest (pipe, channel flows) often come equipped with D continuous spatial symmetries. If the theory is recast as a (D+1)-dimensional space-time theory, the space-time invariant solutions are (D+1)-tori (and not the 1-dimensional periodic orbits of the traditional periodic orbit theory). The symbolic dynamics is likewise (D+1)-dimensional (rather than a single temporal string of symbols), and the corresponding zeta functions should be sums over tori, rather than 1-dimensional periodic orbits. In this theory there is no time, there is only a repertoire of admissible spatiotemporal patterns.

This talk was not recorded. The slides are here.

Matt Gudorf KITP conference poster

Linear encoding of the spatiotemporal cat map

B Gutkin, L Han, R Jafari, A K Saremi, and P Cvitanović

(a VERY preliminary draft of January 3, 2017)

Semiclassical Identification of Periodic Orbits in a Quantum Many-Body System

Maram Akila, Daniel Waltner, Boris Gutkin, Petr Braun, Thomas Guhr