Skip to main content Accessibility help

Computing Residual Diffusivity by Adaptive Basis Learning via Spectral Method


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.


Corresponding author

*Corresponding author. Email addresses: (J. C. Lyu), (J. Xin), (Y. F. Yu)


Hide All
[1] Bensoussan, A., Lions, J. L., and Papanicolaou, G., Asymptotic Analysis for Periodic Structures, AMS Chelsea Publishing, 2011.
[2] Biferale, L., Cristini, A., Vergassola, M. and Vulpiani, A., Eddy diffusivities in scalar transport, Phys. Fluids, 7(11) (1995), pp. 27252734.
[3] Camassa, R. and Wiggins, S., Chaotic advection in a Rayleigh-Bénard flow, Phys. Rev. A, 43(2) (1990), pp. 774797.
[4] Childress, S. and Gilbert, A., Stretch, Twist, Fold: The Fast Dynamo, Lecture Notes in Physics Monographs, No. 37, 1995, Springer.
[5] Fannjiang, A. and Papanicolaou, G., Convection enhanced diffusion for periodic flows, SIAM J. Appl. Math., 54(2) (1994), pp. 333408.
[6] Heinze, S., Diffusion-advection in cellular flows with large Peclet numbers, Archive Rational Mech. Anal., 168(4) (2003), pp. 329342.
[7] Holmes, P., Lumley, J. and Berkooz, G., Turbulence, coherent structures, dynamical systems and symmetry, AIAA J., 36(3) (1998), pp. 496496.
[8] Lumley, J., Coherent structures in turbulence, Transition and Turbulence, (1981), pp. 215241.
[9] Majda, A. and Kramer, P., Simplified models for turbulent diffusion: theory, numerical modelling, and physical phenomena, Phys. Reports, 314 (1999), pp. 237574.
[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. 366, CAM Report, pp. 15–63, UCLA, 2016.
[11] Novikov, A. and Ryzhik, L., Boundary layers and KPP fronts in a cellular flow, Arch. Rational Mech. Anal. 184(1) (2007), pp. 2348.
[12] Quarteroni, A. and Rozza, G., Reduced Order Methods for Modeling and Computational Reduction, MS&A, Vol. 9, Springer, 2014.
[13] Taylor, G., Diffusion by continous movements, Proc. London Math. Soc., 20 (1922).
[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. 10491068.
[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. 134.
[16] Zlatoš, A., Sharp asymptotics for KPP pulsating front speed-up and diffusion enhancement by flows, Arch. Ration. Mech. Anal., 195 (2010), pp. 441453.
[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. 3744.


MSC classification

Computing Residual Diffusivity by Adaptive Basis Learning via Spectral Method


Full text views

Total number of HTML views: 0
Total number of PDF views: 0 *
Loading metrics...

Abstract views

Total abstract views: 0 *
Loading metrics...

* Views captured on Cambridge Core between <date>. This data will be updated every 24 hours.

Usage data cannot currently be displayed