A Low Rank Tensor Approach for Nonlinear Vlasov Simulations
In this work, we present a low-rank tensor approach for approximating solutions to the nonlinear Vlasov equation. Our method takes advantage of the tensor-friendly nature of the differential operators in the Vlasov equation to dynamically and adaptively construct a low-rank solution sis bathrough the discretization of the equation and an SVD-type truncation procedure. We utilize finite difference WENO and discontinuous Galerkin spatial discretizations, along with a second-order strong stability preserving multi-step time discretization. To preserve conservation properties, we develop low-rank schemes with local mass, momentum, and energy conservation for the corresponding macroscopic equations. The mass and momentum are conserved using a conservative SVD truncation, while the energy is conserved by replacing the energy component of the kinetic solution with one obtained from a conservative scheme for the macroscopic energy equation. We employ hierarchical Tucker decomposition for high-dimensional problems, and demonstrate the high-order convergence, efficiency, and local conservation properties of our algorithm through a series of linear and nonlinear Vlasov examples.
Jingmei Qiu is a Professor of Mathematics at the University of Delaware, where she specializes in applied mathematics, computational physics, and computational fluid dynamics. She is known for her work on high-order computational algorithms that can simulate complex systems with multi-scale and multi-physics features. Specifically, Qiu has worked on finite element discontinuous Galerkin (DG) methods, finite volume finite difference weighted essentially non-oscillatory (WENO) methods, and spectral methods with shock capturing and structure preservation in computational fluid dynamics. In recent years, she has also focused on developing moving mesh Eulerian-Lagrangian high-order DG algorithms and low-rank tensor approximations for high-dimensional nonlinear dynamics. Qiu’s research has been applied to a wide range of fields, including astrophysics, fluid dynamics, plasma physics, and engineering. Qiu has received continuous support for her research from the NSF, Air Force Office of Scientific Research, and the Department of Energy. She was the recipient of the Air Force Young Investigator award in 2012.