Dynamical Low Rank Methods for Multiscale Kinetic Plasma Simulations

Plasmas consist of many charged particles, such as electrons and ions. The Boltzmann equation is often regarded as the first-principle model for plasmas; however, its numerical simulation is prohibitively expensive even on today’s most powerful supercomputers. The challenges manifest as: 1) High-dimensionality. The Boltzmann equation resides in six-dimensional phase space. Hence, full 6D deterministic simulation requires excessive computing effort and memory. 2) Collision operator. Collisions between particles are described by nonlinear, nonlocal integral operators that are extremely difficult to approximate. Yet, they play a critical role in driving the system towards local thermodynamic equilibrium and must be included in the simulation, especially in transition and fluid regimes. 3) Multiple scales. Plasmas inherently exhibit multiscale physics. Different scaling can lead to different asymptotic models. How to conduct efficient kinetic simulations such that multiscale behaviors are properly captured is a long-standing problem. The overall objective of this project is to develop a set of ultra-efficient deterministic numerical methods for multiscale kinetic plasma simulations. The algorithms to be developed in this project have the potential to provide high-fidelity kinetic plasma simulations across a range of regimes at a manageable computational cost. The basic framework we will employ is the dynamical low-rank method (DLRM), a robust dimension reduction technique for solving high-dimensional partial differential equations. In essence, DLRM can be viewed as a time-dependent singular value decomposition; instead of solving the 6D equation, it tracks the dynamics of low-rank factors of the solution, which depend on either the three-dimensional position variable or the three-dimensional velocity variable, thus drastically reducing the computational cost and memory footprint. Our focus will be on the nonlinear collisional kinetic equations for plasmas, allowing us to address a broader range of regimes beyond the collisionless ones. We will design an efficient low-rank ansatz inspired by various asymptotic limits of plasma kinetic equations such that the method only requires a few ranks in the limiting regime and is as efficient as solving the reduced fluid models. We will also study the uniform stability and longtime behavior of DLRM rigorously, justifying the method's robustness for treating multiscale problems. 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.