Adaptive Methods for Eulerian probability-density-transport equations in turbulent particulate dispersion

本项目旨在发展一种欧拉概率-密度-输运方程的近似方法，该方法涉及大的状态空间维。这些方程包括Liouville方程和Fokker-Planck方程，它们出现在许多涉及统计描述的应用中：从不确定性量化和反问题到湍流混合、化学反应和粒子的分散。在这些问题中，大量独立的状态空间变量使得经典的近似方法由于计算量大而不可行。新技术是一种使用解析求积的Rayleigh-Ritz全局逼近方法。基函数的全局性质将状态空间中昂贵的计算问题转化为有限数量的方程，其维度是支配感兴趣问题的单个实现的确定性方程。该项目中开发的方法将间接帮助改进对一些物理问题的结果的预测，这些问题的边界条件或初始条件只能从统计上获得。这类问题出现在污染物扩散过程中遇到的颗粒流、气溶胶、喷雾和液滴动力学方面，在这些情况下，只有部分统计知识可用。该项目直接关系到涉及携带固体或液体颗粒的流动的物理系统的高性能计算和建模。

This project is to develop an approximation methodology for Eulerian probability-density-transport equations involving a large state-space dimension. These equations include Liouville and Fokker-Planck equations which arise in many applications involving statistical descriptions: from uncertainty quantification and inverse problems to turbulent mixing, chemical reactions and dispersion of particles. The focus is on those problems where the large number of independent state-space variables makes classical approximation methods unfeasible because of their large computation cost. The new technique is a Rayleigh-Ritz global approximation method using analytical quadratures. The global nature of the basis functions transforms an expensive computational problem in state-space into a finite number of equations with the dimensionality of the deterministic equations governing a single realization of the problem of interest. The methods developed in the project will indirectly help improve prediction of the outcome in a number of physical problems where boundary conditions or initial conditions are only available statistically. Such problems arise in particulate flows, aerosols, sprays and droplet dynamics encountered in the dispersion of contaminants, where only partial statistical knowledge of the conditions is available. This project has direct bearing on high-performance computing and modeling of physical systems involving flows carrying solid or liquid particles.