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A Semi-Lagrangian Adaptive-Rank (SLAR) Method for Linear Advection and Nonlinear Vlasov-Poisson System

Nanyi Zheng, Daniel Hayes, Andrew Christlieb, Jing-Mei Qiu

Abstract

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 $\mathcal{O}(N^2)$ 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.

A Semi-Lagrangian Adaptive-Rank (SLAR) Method for Linear Advection and Nonlinear Vlasov-Poisson System

Abstract

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 per dimension, compared to the 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.

Paper Structure

This paper contains 14 sections, 2 theorems, 41 equations, 15 figures, 1 table, 3 algorithms.

Table of Contents

  1. Introduction
  2. SLAR Methods
  3. SLAR method for linear advection equations
  4. Local non-splitting SL-FD method
  5. ACA of matrices
  6. SLAR algorithm with SVD truncation
  7. LoMaC SLAR method for the nonlinear VP system
  8. RK exponential integrators for nonlinear characteristics tracing
  9. LoMaC SLAR method
  10. Numerical tests
  11. Linear advection equations
  12. Nonlinear VP system
  13. Conclusion
  14. Acknowledgements

Key Result

Proposition 2.1

Assume we have a rank - $(k-1)$ cross approximation $A_{k-1} = C_{k-1}U_{k-1}R_{k-1}$ for row and column indices $\mathcal{I}$ and $\mathcal{J}$, then the cross approximation $A_k$ for rows $\mathcal{I}\cup \{i\}$ and columns $\mathcal{J}\cup \{j\}$ is given by

Figures (15)

  • Figure 1.1: The SLAR method for linear advection equations.
  • Figure 2.2: Schematic illustration of tracing characteristics.
  • Figure 2.3: Visual representation of the CUR decomposition used in SLAR.
  • Figure 3.4: (Rigid body rotation). Log-log plots of CFL numbers versus $L^1$ and $L^{\infty}$ errors with two sets of fixed meshes, $128\times128$ and $256\times256$ at $t = 2\pi$.
  • Figure 3.5: (Rigid body rotation). Left: semi-log plot of CFL numbers versus average ranks of the simulations in \ref{['fig:CFL_RBR']}. Right: semi-log plot of CFL numbers versus compression ratios of DOFs of the simulations in \ref{['fig:CFL_RBR']}.
  • ...and 10 more figures

Theorems & Definitions (7)

  • Proposition 2.1: Recursive update of cross shi2024distributed
  • Corollary 2.2
  • Example 3.1
  • Example 3.2
  • Example 3.3
  • Example 3.4
  • Example 3.5