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Computing Residual Diffusivity by Adaptive Basis Learning via Spectral Method

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

Information

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 

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