Hostname: page-component-848d4c4894-5nwft Total loading time: 0 Render date: 2024-05-17T23:50:50.126Z Has data issue: false hasContentIssue false
  • English
  • Français

Computing Residual Diffusivity by Adaptive Basis Learning via Spectral Method

Part of: Partial differential equations Partial differential equations, initial value and time-dependent initial-boundary value problems Numerical methods in Fourier analysis Diffusion and convection

Published online by Cambridge University Press:  09 May 2017

Jiancheng Lyu ,
Jack Xin  and
Yifeng Yu
Jiancheng Lyu*
Affiliation:
Department of Mathematics, University of California at Irvine, Irvine, CA 92697, USA
Jack Xin*
Affiliation:
Department of Mathematics, University of California at Irvine, Irvine, CA 92697, USA
Yifeng Yu*
Affiliation:
Department of Mathematics, University of California at Irvine, Irvine, CA 92697, USA
*
*Corresponding author. Email addresses:jianchel@uci.edu (J. C. Lyu), jxin@math.uci.edu (J. Xin), yyu1@math.uci.edu (Y. F. Yu)
*Corresponding author. Email addresses:jianchel@uci.edu (J. C. Lyu), jxin@math.uci.edu (J. Xin), yyu1@math.uci.edu (Y. F. Yu)
*Corresponding author. Email addresses:jianchel@uci.edu (J. C. Lyu), jxin@math.uci.edu (J. Xin), yyu1@math.uci.edu (Y. F. Yu)
Get access

Abstract

We study the residual diffusion phenomenon in chaotic advection computationally via adaptive orthogonal basis. The chaotic advection is generated by a class of time periodic cellular flows arising in modeling transition to turbulence in Rayleigh-Bénard experiments. The residual diffusion refers to the non-zero effective (homogenized) diffusion in the limit of zero molecular diffusion as a result of chaotic mixing of the streamlines. In this limit, the solutions of the advection-diffusion equation develop sharp gradients, and demand a large number of Fourier modes to resolve, rendering computation expensive. We construct adaptive orthogonal basis (training) with built-in sharp gradient structures from fully resolved spectral solutions at few sampled molecular diffusivities. This is done by taking snapshots of solutions in time, and performing singular value decomposition of the matrix consisting of these snapshots as column vectors. The singular values decay rapidly and allow us to extract a small percentage of left singular vectors corresponding to the top singular values as adaptive basis vectors. The trained orthogonal adaptive basis makes possible low cost computation of the effective diffusivities at smaller molecular diffusivities (testing). The testing errors decrease as the training occurs at smaller molecular diffusivities. We make use of the Poincaré map of the advection-diffusion equation to bypass long time simulation and gain accuracy in computing effective diffusivity and learning adaptive basis. We observe a non-monotone relationship between residual diffusivity and the amount of chaos in the advection, though the overall trend is that sufficient chaos leads to higher residual diffusivity.

Keywords

Advection-diffusionchaotic flowseffective diffusionadaptive basis learningresidual diffusion

MSC classification

Secondary: 76R99: None of the above, but in this section 35B27: Homogenization; equations in media with periodic structure 65T40: Trigonometric approximation and interpolation 65M70: Spectral, collocation and related methods
Type
Research Article
Information
Numerical Mathematics: Theory, Methods and Applications , Volume 10 , Issue 2 , May 2017 , pp. 351 - 372
Copyright
Copyright © Global-Science Press 2017 

Access options

Get access to the full version of this content by using one of the access options below. (Log in options will check for institutional or personal access. Content may require purchase if you do not have access.)

References

[1] Bensoussan, A., Lions, J. L., and Papanicolaou, G., Asymptotic Analysis for Periodic Structures, AMS Chelsea Publishing, 2011.Google Scholar
[2] Biferale, L., Cristini, A., Vergassola, M. and Vulpiani, A., Eddy diffusivities in scalar transport, Phys. Fluids, 7(11) (1995), pp. 2725–2734.Google Scholar
[3] Camassa, R. and Wiggins, S., Chaotic advection in a Rayleigh-Bénard flow, Phys. Rev. A, 43(2) (1990), pp. 774–797.Google Scholar
[4] Childress, S. and Gilbert, A., Stretch, Twist, Fold: The Fast Dynamo, Lecture Notes in Physics Monographs, No. 37, 1995, Springer.Google Scholar
[5] Fannjiang, A. and Papanicolaou, G., Convection enhanced diffusion for periodic flows, SIAM J. Appl. Math., 54(2) (1994), pp. 333–408.Google Scholar
[6] Heinze, S., Diffusion-advection in cellular flows with large Peclet numbers, Archive Rational Mech. Anal., 168(4) (2003), pp. 329–342.Google Scholar
[7] Holmes, P., Lumley, J. and Berkooz, G., Turbulence, coherent structures, dynamical systems and symmetry, AIAA J., 36(3) (1998), pp. 496–496.Google Scholar
[8] Lumley, J., Coherent structures in turbulence, Transition and Turbulence, (1981), pp. 215–241.Google Scholar
[9] Majda, A. and Kramer, P., Simplified models for turbulent diffusion: theory, numerical modelling, and physical phenomena, Phys. Reports, 314 (1999), pp. 237–574.Google Scholar
[10] Murphy, B., Cherkaev, E., Xin, J., Zhu, J. and Golden, K., Spectral analysis and computation of effective diffusivity in space-time periodic incompressible flows, Ann. Math. Sci. Appl., 2(1) (2017), pp. 3–66, CAM Report, pp. 15–63, UCLA, 2016.Google Scholar
[11] Novikov, A. and Ryzhik, L., Boundary layers and KPP fronts in a cellular flow, Arch. Rational Mech. Anal. 184(1) (2007), pp. 23–48.Google Scholar
[12] Quarteroni, A. and Rozza, G., Reduced Order Methods for Modeling and Computational Reduction, MS&A, Vol. 9, Springer, 2014.Google Scholar
[13] Taylor, G., Diffusion by continous movements, Proc. London Math. Soc., 20 (1922).Google Scholar
[14] Xin, J. and Yu, Y., Sharp asymptotic growth laws of turbulent flame speeds in cellular flows by inviscid Hamilton-Jacobi models, Annales I’Institut H. Poincaré Analyse Non Linéaire, 30(6) (2013), pp. 1049–1068.CrossRefGoogle Scholar
[15] Xin, J. and Yu, Y., Front quenching in G-equation model induced by straining of cellular flow, Arch. Rational Mech. Anal., 214 (2014), pp. 1–34.Google Scholar
[16] Zlatoš, A., Sharp asymptotics for KPP pulsating front speed-up and diffusion enhancement by flows, Arch. Ration. Mech. Anal., 195 (2010), pp. 441–453.Google Scholar
[17] Zu, P., Chen, L. and Xin, J., A computational study of residual KPP front speeds in time-periodic cellular flows in the small diffusion limit, Phys. D, 311–312 (2015), pp. 37–44.Google Scholar