Multiscale Discontinuous Galerkin Methods for Kinetic Models of Gas and Plasma

Variants of the celebrated Boltzmann equation can be used to model the dynamics of rarefied gases (i.e., collections of molecules that move around in space and interact through collisions), as well as plasma (i.e., collections of positively and negatively charged ions that move around in space and interact through collisions and electromagnetic forces). As such, solutions of the Boltzmann equation can be used to describe and predict the dynamics in various applications, such as flow in microfluidic devices, hypersonic and space vehicle aerodynamics, flow in magnetically confined fusion reactors, and particle acceleration in laser-plasma systems.

A critical challenge is that computing solutions to the Boltzmann equation in realistic scenarios is prohibitively expensive, even on modern massively parallel computers. An important goal of this research is to develop reduced-order models that can capture important flow features but that can be more readily solved on modern computer architectures.

The approach pursued in this research is to decompose the solution into a macroscopic portion that describes large-scale features and a microscopic portion that describes smaller-scale features; macroscopic features can be computed relatively cheaply and accurately, while microscopic features are expensive to compute. Various adaptive strategies are explored to reduce the expense of the microscopic portions.

The primary objective of this research is to develop accurate and efficient computational methods for solving the kinetic Boltzmann and Vlasov equation for modeling rarefied gases and plasma. The main challenge in solving kinetic models is that solutions live in high-dimensional phase space and contain information over wide-ranging spatial and temporal scales.

An important goal is to develop reduced models that capture the important physics and can be more readily solved on modern computer architectures. The approach pursued in this research is based on decomposing the kinetic particle density function into macroscopic and microscopic pieces, allowing for different computational techniques on each portion.

This research leverages several key innovations, including (1) high-order discontinuous Galerkin finite element methods for spatial discretization, (2) novel explicit and semi-implicit time-stepping techniques, (3) adaptive refinement strategies to reduce the computational expense of the microscopic portion of the update, and (4) implementation of the resulting algorithms on massively parallel computer architectures. Verification and validation will be performed on several test cases relevant to the simulation of rarefied gases and plasma.

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.