Computing Residual Diffusivity by Adaptive Basis Learning via Spectral Method

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.

Computing Effective Diffusivity of Chaotic and Stochastic Flows Using Structure-Preserving Schemes

“…For the Hamiltonian system Eq. (15), the corresponding backward Kolmolgorov equation associated is given by…”

“…where A 1 is a partial differential operator acting on φ 0 (x) that depends on the choice of the numerical method used to solve Eq. (15). If we choose a convergent method to discretize the operator L 0 in Eq.…”

Convergence analysis of a Lagrangian numerical scheme in computing effective diffusivity of 3D time-dependent flows

“…These results include, among others, for time-independent Taylor-Green flows, the authors of [28] proposed a stochastic splitting method and calculated the effective diffusivity in the limit of vanishing molecular diffusion. For time-dependent chaotic flows, an efficient model reduction method based on the spectral method was developed to compute D E using the Eulerian framework [21]. The reader is referred to [22] for an extensive review of many existing mathematical theories and numerical simulations for the passive tracer model with different velocity fields.…”

Computing Residual Diffusivity by Adaptive Basis Learning via Super-Resolution Deep Neural Networks

“…At θ = 1, the flow (1.2) is fully chaotic [21], and has served as a model of chaotic advection for Rayleigh-Bénard experiment [3]. Numerical simulations [2,10,18] suggest that at θ = 1, the effective diffusivity along the x-axis, D E 11 = O(1) as D 0 ↓ 0, the so called residual diffusivity phenomenon emerges. As D 0 ↓ 0, the solutions develop sharp gradients, and render accurate computation costly.…”