High Order Methods for Kinetic Transport Models

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Accurate, robust and efficient simulations of kinetic transport models are of fundamental importance to many applications in physics and engineering, such as rarefied gas dynamics, plasma physics, nuclear and biomedical engineering, and network dynamics. While fluid models often provide lower dimensional approximations, they are valid only when the systems are closed to their equilibria, and therefore kinetic transport models are necessary to account for a wider range of phenomena, including those displaying multiple scales in space and/or time or those involving particles from multiple energy groups. This project aims at advancing computational tools of high order accuracy for time-dependent multi-scale kinetic transport models, both algorithmically and mathematically.Kinetic transport equations are mathematical descriptions of the transport of particles such as neutrons, photons, molecules as well as their interaction with a host medium or among themselves, and they arise in a broad range of applications. The numerical challenges lie in the high dimensionality of the phase space, small scales, multiple scales in space (and even in time), nonlinear interactions, collision operators with multi-fold integrals, as well as the conservation or positivity property of the solutions. The objective of this project is therefore to take important steps in the understanding and simulations of the time-dependent multi-scale kinetic transport models, with the aim of: 1) providing accurate deterministic simulation tools with excellent stability to the kinetic transport community, and 2) developing novel analytical and numerical techniques to address some challenges related to accurate multi-scale kinetic transport simulations. The longer term goal is to develop algorithms robustly simulating more realistic kinetic models with high resolution and good efficiency.This award reflects NSF's statutory mission and has been deemed worthy of support through evaluation using the Foundation's intellectual merit and broader impacts review criteria.