RUI: Development of Fast Scalable Adaptive High Order Methods for Solving the Boltzmann Equation

This project's goal is to advance one's ability to use computer simulations to address scientific and technological challenges by employing modeling at microscopic scales using the kinetic Boltzmann equation. Applications of this proposal span the dynamics of gas, plasma, self-organizing systems, networks, and bacterial dynamics. The project will focus on a bottleneck issue in kinetic modeling --- the development of fast methods for high fidelity simulations of particle interactions in rarefied gases. The project's most immediate impact is in the development of novel aerospace technologies and in important U.S. initiatives in the development of clean energy, biotechnology, and new materials. This will be through its applications to computer simulation of devices that either operate in rarefied gas or are manufactured in vacuum. The project will provide training for the STEM workforce by engaging students in research.

Despite of being studied intensely in the last decades, deterministic numerical solutions of the Boltzmann equation continue to be evasive. To achieve a full three-dimensional solution suitable for use in applications, fast scalable adaptive numerical approaches for evaluating the five-fold Boltzmann collision integral need to be devised. This proposal will address these shortcomings by developing convolution formulations of the Boltzmann collision integral based on nodal discontinuous Galerkin (nodal-DG) discretizations in the velocity variable, by developing adaptable nodal-DG wavelet discretizations of the collision operator on octree meshes, and by developing fast algorithms for evaluating the convolution form of the collision integral based on an application of the Fourier transform. The new methods will require at most O(n^6) operations for a fully deterministic evaluation of the Boltzmann collision integral, and will require O(n^5) memory units to store the pre-computed collision kernels, where n is the number of discretization points in one dimension in the velocity space. The new methods will be implemented on parallel architectures and will be scalable. Implementation of this proposal will result in the development of capabilities for producing high-fidelity solutions to the Boltzmann equation, capabilities for producing benchmark solutions and methods for validation of kinetic models. The research activities will result in a new application of nodal-DG wavelets to the approximation of the Boltzmann collision integral.