@Article{NMTMA-10-351,
author = {Jiancheng Lyu, Jack Xin and Yifeng Yu},
title = {Computing Residual Diffusivity by Adaptive Basis Learning via Spectral Method},
journal = {Numerical Mathematics: Theory, Methods and Applications},
year = {2017},
volume = {10},
number = {2},
pages = {351--372},
abstract = {<p style="text-align: justify;">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&nbsp;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.</p>},
issn = {2079-7338},
doi = {https://doi.org/10.4208/nmtma.2017.s08},
url = {http://global-sci.org/intro/article_detail/nmtma/12350.html}
}