A semi-Lagrangian adaptive-rank (SLAR) method for linear advection and nonlinear Vlasov-Poisson system

• Published in: Journal of computational physics Vol. 532; p. 113970
• Main Authors: Zheng, Nanyi, Hayes, Daniel, Christlieb, Andrew, Qiu, Jing-Mei
• Format: Journal Article
• Language: English
• Published: Elsevier Inc 01.07.2025
• Subjects: Adaptive rank; Cross approximation; Kinetic Vlasov model; Mass conservation; Semi-Lagrangian; Singular value truncation; Cross approximation; Semi-Lagrangian; Mass conservation; Kinetic Vlasov model; Adaptive rank; Singular value truncation
• ISSN: 0021-9991
• DOI: 10.1016/j.jcp.2025.113970

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 proposed 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. •Developed a semi-Lagrangian method, using adaptive cross approximation and singular value truncation.•Ensured high-order accuracy, mass conservation, and greatly reduced computational complexity.•Introduced a local mass conservation correction for Vlasov-Poisson systems, retaining large time steps.