随机过程动态自适应小波独立网格多尺度模拟

Capturing Allscales of Random Modes on Independent Grids

在随机过程数值仿真中,由多项式混沌展开谱方法得到求解展开系数的确定性偶合方程组。该方程组比相应的确定性仿真时增大许多。并且当多项式展开阶数和随机空间维数提高时,方程维数急剧增加。由于待求未知分量为表征不同尺度波动的混沌展开模,形成节点意义下的的多尺度问题,传统的网格细分自适应逼近不再适用。为此我们采用了小波的多尺度离散,并建立基于空间细化的动态自适应系统,让每个求解点上的多个未知分量有各自独立的小波网格。本文以随机对流扩散方程为例,进行了二个算例的数值实验,论证了此方法的优点。

In stochastic computations, the spectral approach based on the polynomial chaos expansion in random space leads to a coupled system of deterministic equations for the coefficients of the expansion. The size of this system is much larger than the deterministic counterpart and grows quickly with an increasing number of independent random variables and/or order of polynomial chaos expansion. Moreover, the solution components at each computational grid point represent fluctuations of different scales, their coupled grid-wise, multiscale nature, renders the conventional mesh-refinement unsuitable. Here we approach this challenge using a novel approach based on a dynamically adaptive wavelet methodology involving space-refinement oil physical space that allows all scales of each solution component （random mode） to be refined independently of the rest. We exemplify this by focusing on the convection-diffusion model with random input data. We present two numerical examples illustrating the salient features of the proposed method.