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Efficient Computation of Instantons for Multi-Dimensional Turbulent Flows with Large Scale Forcing

Published online by Cambridge University Press:  14 September 2015

Tobias Grafke ,
Rainer Grauer  and
Stephan Schindel
Tobias Grafke*
Affiliation:
Department of Physics of Complex Systems, Weizmann Institute of Science, Rehovot 76100, Israel
Rainer Grauer
Affiliation:
Theoretische Physik I, Ruhr-Universität Bochum, Universitätsstr. 150, D44780 Bochum, Germany
Stephan Schindel
Affiliation:
Theoretische Physik I, Ruhr-Universität Bochum, Universitätsstr. 150, D44780 Bochum, Germany
*
*Corresponding author. Email addresses: [email protected] (T. Grafke), [email protected] (R. Grauer), [email protected] (S. Schindel)
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Abstract

Extreme events play a crucial role in fluid turbulence. Inspired by methods from field theory, these extreme events, their evolution and probability can be computed with help of the instanton formalism as minimizers of a suitable action functional. Due to the high number of degrees of freedom in multi-dimensional fluid flows, traditional global minimization techniques quickly become prohibitive in their memory requirements. We outline a novel method for finding the minimizing trajectory in a wide class of problems that typically occurs in turbulence setups, where the underlying dynamical system is a non-gradient, non-linear partial differential equation, and the forcing is restricted to a limited length scale. We demonstrate the efficiency of the algorithm in terms of performance and memory by computing high resolution instanton field configurations corresponding to viscous shocks for 1D and 2D compressible flows.

Keywords

47.11.-j47.27.ef05.10.-a47.40.-xInstanton formalismminimizing trajectorynumerical minimizationGPU programmingCUDA
Type
Research Article
Information
Communications in Computational Physics , Volume 18 , Issue 3 , September 2015 , pp. 577 - 592
Copyright
Copyright © Global-Science Press 2015 

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References

[1]CUDA toolkit documentation, 2014. http://docs.nvidia.com/cuda/index.html.Google Scholar
[2]Assaf, Michael, Roberts, Elijah, and Luthey-Schulten, Zaida. Determining the stability of genetic switches: Explicitly accounting for mrna noise. Phys. Rev. Lett., 106:248102, 2011.Google Scholar
[3]Backus, David, Chernov, Mikhail, and Martin, Ian. Disasters implied by equity index options. The Journal of Finance, 66:1969–2012, 2011.Google Scholar
[4]Balkovsky, E., Falkovich, G., Kolokolov, I., and Lebedev, V.. Intermittency of Burgers’ turbulence. Phys. Rev. Lett., 78:1452, 1997.Google Scholar
[5]Barro, Robert J.. Rare disasters and asset markets in the twentieth century. Quarterly Journal of Economics, 2006.Google Scholar
[6]Celani, Antonio, Cencini, Massimo, and Noullez, Alain. Going forth and back in time: a fast and parsimonious algorithm for mixed initial/final-value problems. Physica D: Nonlinear Phenomena, 195(3):283–291, 2004.Google Scholar
[7]Chernykh, A. I. and Stepanov, M. G.. Large negative velocity gradients in Burgers turbulence. Phys. Rev. E, 64:026306, 2001.Google Scholar
[8]Dominicis, C. de. Techniques de renormalisation de la théorie des champs et dynamique des phénomènes critiques. J. Phys. C, 1:247, 1976.Google Scholar
[9]Dembo, Amir and Zeitouni, Ofer. Large deviations techniques and applications. Springer-Verlag, Berlin, 2010.CrossRefGoogle Scholar
[10]E, W., Ren, W., and Vanden-Eijnden, E.. Minimum action method for the study of rare events. Commun. Pure Appl. Math., 57:1–20, 2004.Google Scholar
[11]E, Weinan, Ren, Weiqing, and Vanden-Eijnden, Eric. String method for the study of rare events. Phys. Rev. B, 66(5):052301, 2002.Google Scholar
[12]Elgart, Vlad and Kamenev, Alex. Classification of phase transitions in reaction-diffusion models. Phys. Rev. E, 74:041101, 2006.Google Scholar
[13]Fogedby, H. C. and Ren, W.. Minimum action method for the Kardar-Parisi-Zhang equation. Phys. Rev. E, 80:041116, 2009.Google Scholar
[14]Freidlin, Mark I and Wentzell, Alexander D. Random perturbations of dynamical systems, volume 260. Springer, 2012.CrossRefGoogle Scholar
[15]Grafke, T., Grauer, R., and Schäfer, T.. Instanton filtering for the stochastic Burgers equation. J. Phys. A, 46(6):62002, 2013.Google Scholar
[16]Grafke, T., Grauer, R., Schäfer, T., and Vanden-Eijnden, E.. Relevance of instantons in Burgers turbulence. arXiv preprint arXiv:1412.0255, December 2014.Google Scholar
[17]Grafke, T., Grauer, R., Schäfer, T., and Vanden-Eijnden, Eric. Arclength parametrized Hamilton’s equations for the calculation of instantons. Multiscale Modeling & Simulation, 12(2):566–580, 2014.Google Scholar
[18]Gurarie, V. and Migdal, A.. Instantons in the Burgers equation. Phys. Rev. E, 54:4908, 1996.Google Scholar
[19]Henkelman, Graeme and Jónsson, Hannes. Improved tangent estimate in the nudged elastic band method for finding minimum energy paths and saddle points. The Journal of Chemical Physics, 113:9978, 2000.Google Scholar
[20]Heymann, M. and Vanden-Eijnden, E.. Pathways of maximum likelihood for rare events in nonequilibrium systems: application to nucleation in the presence of shear. Phys. Rev. Lett., 100(14):140601, 2008.Google Scholar
[21]Heymann, M. and Vanden-Ejnden, E.. The geometric minimum action method: A least action principle on the space of curves. Commun. Pure Appl. Math., 61:1053, 2008.Google Scholar
[22]Janssen, H.K.. On a Lagrangian for classical field dynamics and renormalization group calculations of dynamical critical properties. Z. Physik B, 23:377, 1976.Google Scholar
[23]Kramers, H. A.. Brownian motion in a field of force and the diffusion model of chemical reactions. Physica, 7(4):284–304, April 1940.Google Scholar
[24]Laurie, Jason and Bouchet, Freddy. Computation of rare transitions in the barotropic quasi-geostrophic equations. arXiv preprint arXiv:1409.3219, 2014.Google Scholar
[25]Martin, P. C., Siggia, E. D., and Rose, H. A.. Statistical dynamics of classical systems. Phys. Rev. A, 8:423, 1973.Google Scholar
[26]Onorato, M., Residori, S., Bortolozzo, U., Montina, A., and Arecchi, F.T.. Rogue waves and their generating mechanisms in different physical contexts. Physics Reports, 528(2):47–89, 2013.Google Scholar
[27]Schmeits, Maurice J. and Dijkstra, Henk A.. Bimodal behavior of the Kuroshio and the Gulf Stream. J. Phys. Ocean., 31:3435, 2001.Google Scholar
[28]Vanden-Eijnden, Eric and Heymann, Matthias. The geometric minimum action method for computing minimum energy paths. The Journal of Chemical Physics, 128:061103, 2008.Google Scholar
[29]Vanden-Ejnden, E. and Heymann, M.. The geometric minimum action method for computing minimum energy paths. Jour. Chem. Phys., 128:061103, 2008.Google Scholar
[30]Zhou, Xiang, Ren, Weiqing, and E., WeinanAdaptive minimum action method for the study of rare events. J. Chem. Phys., 128:104111, 2008.Google Scholar
[31]Wang, Q., Moin, P., and Iaccarino, G.. Minimal repetition dynamic checkpointing algorithm for unsteady adjoint calculation. SIAM Journal on Scientific Computing, 31(4):2549–2567, 2009.Google Scholar