2024/11/27 by Nanyi Zheng, Daniel Hayes, Zheng, Nanyi +5 · 6 citations
Engineering · Mathematics · #65 #Computational Fluid Dynamics and Aerodynamics #FOS: Mathematics #Gas Dynamics and Kinetic Theory #Lattice Boltzmann Simulation Studies #Numerical Analysis (math.NA)
paper · pdf · doi:10.48550/arxiv.2411.17963
openalex publication_date 2024/11/27 · openalex created_date 2025/10/10 · openalex updated_date 2026/08/01
High-order semi-Lagrangian methods for kinetic equations have been under rapid development in the past few decades. In this work, we propose a semi-Lagrangian adaptive rank (SLAR) integrator in the finite difference framework for linear advection and nonlinear Vlasov-Poisson systems without dimensional splitting. The proposed method leverages the semi-Lagrangian approach to allow for significantly larger time steps while also exploiting the low-rank structure of the solution. This is achieved through cross approximation of matrices, also referred to as CUR or pseudo-skeleton approximation, where representative columns and rows are selected using specific strategies. To maintain numerical stability and ensure local mass conservation, we apply singular value truncation and a mass-conservative projection following the cross approximation of the updated solution. The computational complexity of our method scales linearly with the mesh size N per dimension, compared to the O(N2) complexity of traditional full-rank methods per time step. The algorithm is extended to handle nonlinear Vlasov-Poisson systems using a Runge-Kutta exponential integrator. Moreover, we evolve the macroscopic conservation laws for charge densities implicitly, enabling the use of large time steps that align with the semi-Lagrangian solver. We also perform a mass-conservative correction to ensure that the adaptive rank solution preserves macroscopic charge density conservation. To validate the efficiency and effectiveness of our method, we conduct a series of benchmark tests on both linear advection and nonlinear Vlasov-Poisson systems. The propose algorithm will have the potential in overcoming the curse of dimensionality for beyond 2D high dimensional problems, which is the subject of our future work.