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

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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.