A Semi-Lagrangian Adaptive Rank (SLAR) Method for High-Dimensional Vlasov Dynamics

TL;DR

This work develops a high-dimensional semi-Lagrangian adaptive-rank (SLAR) method for Vlasov–Poisson dynamics by integrating a third-order, $d$-dimensional polynomial reconstruction with Hierarchical Tucker Adaptive Cross Approximation (HTACA) and a FFT-based Poisson solver in HTD form. The HTACA framework extends ACA to high-order tensors via a recursive dimension-tree structure, enabling scalable, memory-efficient tensor representations and localized entry access. The combination yields a worst-case complexity of $O(d^{4} N r^{3+\lceil \log_{2} d \rceil})$, with practical linear scaling in $N$ and substantial rank-based compression across up to six dimensions. Numerical tests on Landau damping and two-stream instabilities demonstrate effective rank control, dimensional-tree design choices that minimize growth, and accurate physics in weakly nonlinear regimes, highlighting the approach’s potential for scalable high-dimensional kinetic simulations.

Abstract

We extend our previous work on a semi-Lagrangian adaptive rank (SLAR) integrator, in the finite difference framework for nonlinear Vlasov-Poisson systems, to the general high-order tensor setting. The proposed scheme retains the high-order accuracy of semi-Lagrangian methods, ensuring stability for large time steps and avoiding dimensional splitting errors. The primary contribution of this paper is the novel extension of the algorithm from the matrix to the high-dimensional tensor setting, which enables the simulation of Vlasov models in up to six dimensions. The key technical components include (1) a third-order high-dimensional polynomial reconstruction that scales as $O(d^2)$, providing a point-wise approximation of the solution at the foot of characteristics in a semi-Lagrangian scheme; (2) a recursive hierarchical adaptive cross approximation of high-order tensors in a hierarchical Tucker format, characterized by a tensor tree; (3) a low-complexity Poisson solver in the hierarchical Tucker format that leverages the FFT for efficiency. The computed adaptive rank kinetic solutions exhibit low-rank structures within branches of the tensor tree resulting in substantial computational savings in both storage and time. The resulting algorithm achieves a computational complexity of $O(d^4 N r^{3+\lceil\log_2d\rceil})$, where $N$ is the number of grid points per dimension, $d$ is the problem dimension, and $r$ is the maximum rank in the tensor tree, overcoming the curse of dimensionality. Through extensive numerical tests, we demonstrate the efficiency of the proposed algorithm and highlight its ability to capture complex solution structures while maintaining a computational complexity that scales linearly with $N$.

Figures (10)

• Figure 1: Two candidate dimension trees for 4D (left) and 6D (right) tensors.
• Figure 2: Schematic illustration of the ACA algorithm for a matrix.
• Figure 3: Schematic illustration of 4-D HTACA.
• Figure 4: (2D2V strong Landau damping). Rank history of the SLAR method with the dimension tree in \ref{['fig:balanced_tree_4d']}: Left $-$ modes ordered as {$x$, $y$, $v_x$, $v_y$}; Right $-$ modes ordered as {$x$, $v_x$, $y$, $v_y$}. Simulation settings: $v_{\max}=2\pi$, mesh $256^4$, CFL $=5$, $\epsilon_{\text{Base}} = 10^{-3}$, and no rank limitations.
• Figure 5: (2D2V strong Landau damping). Left: log-log plot of grid points per dimension versus $L^2$ error at $T=0.5$ with $\mathrm{CFL}=1$ and truncation tolerances $\varepsilon_{\text{Base}}=10^{-4},10^{-5},10^{-6},10^{-7}$ for $N=32,64,128,256$, respectively. Right: log-log plot of $\mathrm{CFL}$ versus $L^2$ error at $T=0.5$ with fixed tolerance $10^{-4}$ and mesh $256^4$ for all simulations. No rank limitation is imposed in either test.

Theorems & Definitions (18)

• Proposition 2.1
• proof
• Proposition 3.1
• Definition 3.2
• Definition 3.3
• Lemma 3.4
• Definition 3.5
• Definition 3.6
• Proposition 3.7
• Proposition 3.8