Computing Residual Diffusivity by Adaptive Basis Learning via Spectral Method
Computing Residual Diffusivity by Adaptive Basis Learning via Spectral Method
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DOI：
10.4208/nmtma.2017.s08
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发表时间：
2017-05
影响因子：
1.3
通讯作者：
J. Lyu;J. Xin;Yifeng Yu
中科院分区：
文献类型：
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作者：
J. Lyu;J. Xin;Yifeng Yu
抽象的。利用自适应正交基对混沌平流中的残余扩散现象进行了计算研究。这种混沌平流是由Rayleigh-Bénard实验中模拟向湍流转变过程中产生的一类时间周期元胞流产生的。剩余扩散是指由于流线的混沌混合，在零分子扩散极限内的非零有效(均匀)扩散。在这个极限下，对流扩散方程的解会出现陡峭的梯度，并且需要大量的傅立叶模式来求解，这使得计算代价很高。我们构造自适应的正交基(训练)，具有内置的尖锐梯度结构，在很少的采样分子扩散系数下从完全分辨的光谱解中得到。这是通过及时获取解决方案的快照，并对由这些快照组成的作为列矢量的矩阵执行奇异值分解来实现的。奇异值迅速衰减，并允许我们提取与顶部奇异值相对应的一小部分左奇异向量作为自适应基向量。经过训练的正交自适应基使在较小的分子扩散系数(测试)下低成本地计算有效扩散系数成为可能。当训练发生在较小的分子扩散系数时，测试误差减小。我们利用对流扩散方程的Poincaré映射绕过了长时间的模拟，在计算有效扩散系数和学习自适应基时获得了精度。我们观察到平流中混沌的数量与剩余扩散系数之间的非单调关系，尽管总体趋势是足够的混沌会导致更高的剩余扩散系数。
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.