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Dynamical low-rank approximation of the Vlasov-Poisson equation with piecewise linear spatial boundary

André Uschmajew, Andreas Zeiser

TL;DR

This work develops a weak, variational projector-splitting framework for dynamical low-rank approximation (DLRA) of the Vlasov--Poisson equation, explicitly addressing inflow boundary conditions on spatial domains with piecewise linear boundaries. By formulating the K-, S-, and L-steps as Friedrichs' systems and coupling them with a rank-adaptive forward S-step, the approach preserves a separated space-velocity representation while respecting boundary fluxes through a boundary penalty structure. Discrete realizations using continuous finite elements and CIP stabilization yield implementable algorithms, including a rank-adaptive variant, and are validated by Landau damping on periodic domains and a boundary-test case with a constant electric field. The results demonstrate principle feasibility and highlight practical considerations such as rank growth, stability, and the potential for flexible discretizations, laying groundwork for conservation properties and nonnegativity in future work.

Abstract

We consider dynamical low-rank approximation (DLRA) for the numerical simulation of Vlasov--Poisson equations based on separation of space and velocity variables, as proposed in several recent works. The standard approach for the time integration in the DLRA model uses a splitting of the tangent space projector for the low-rank manifold according to the separated variables. It can also be modified to allow for rank-adaptivity. A less studied aspect is the incorporation of boundary conditions in the DLRA model. We propose a variational formulation of the projector splitting which allows to handle inflow boundary conditions on spatial domains with piecewise linear boundary. Numerical experiments demonstrate the principle feasibility of this approach.

Dynamical low-rank approximation of the Vlasov-Poisson equation with piecewise linear spatial boundary

TL;DR

This work develops a weak, variational projector-splitting framework for dynamical low-rank approximation (DLRA) of the Vlasov--Poisson equation, explicitly addressing inflow boundary conditions on spatial domains with piecewise linear boundaries. By formulating the K-, S-, and L-steps as Friedrichs' systems and coupling them with a rank-adaptive forward S-step, the approach preserves a separated space-velocity representation while respecting boundary fluxes through a boundary penalty structure. Discrete realizations using continuous finite elements and CIP stabilization yield implementable algorithms, including a rank-adaptive variant, and are validated by Landau damping on periodic domains and a boundary-test case with a constant electric field. The results demonstrate principle feasibility and highlight practical considerations such as rank growth, stability, and the potential for flexible discretizations, laying groundwork for conservation properties and nonnegativity in future work.

Abstract

We consider dynamical low-rank approximation (DLRA) for the numerical simulation of Vlasov--Poisson equations based on separation of space and velocity variables, as proposed in several recent works. The standard approach for the time integration in the DLRA model uses a splitting of the tangent space projector for the low-rank manifold according to the separated variables. It can also be modified to allow for rank-adaptivity. A less studied aspect is the incorporation of boundary conditions in the DLRA model. We propose a variational formulation of the projector splitting which allows to handle inflow boundary conditions on spatial domains with piecewise linear boundary. Numerical experiments demonstrate the principle feasibility of this approach.
Paper Structure (15 sections, 81 equations, 5 figures, 2 algorithms)

This paper contains 15 sections, 81 equations, 5 figures, 2 algorithms.

Table of Contents

  1. Introduction
  2. Weak formulation of the projector splitting integrator
  3. DLRA and projector splitting
  4. Weak formulation of the K-step
  5. Formulation of the S-step
  6. Weak formulation of the L-step
  7. Electrical field
  8. Discrete equations
  9. Discretization
  10. Discrete projector splitting integrator
  11. Rank adaptive algorithm
  12. Numerical experiments
  13. Landau damping
  14. Inflow boundary condition with constant electrical field
  15. Conclusion and outlook

Figures (5)

  • Figure 1: Simulation results of the 2+2-dimensional Landau damping using fixed ranks (Alg. \ref{['alg:lie-trotter']}); electric energy including the analytical decay rate (upper left) and relative error of the invariants \ref{['eq:invariants']}. For the total energy and entropy the lines for all three ranks almost overlap.
  • Figure 2: Simulation results of the 2+2-dimensional Landau damping using the rank adaptive algorithm (Algorithm \ref{['alg:rauc']}) for different tolerances $\epsilon$ and discretizations; electric energy including the analytical decay rate (left) and ranks (right).
  • Figure 3: Spatial basis functions $X_i(\bm x)$ at time $t=50$ for fixed rank ($r=15$, left) and rank adaptive (level 0, $\epsilon=10^{-5}$, right) simulation.
  • Figure 4: Spatial density $\rho^h(t, \bm x)$ (left), see \ref{['eq:rhox']}, and error scaled by a factor of 10 (right) of the numerical solution of \ref{['eq:transport']} with constant electrical field for level $\ell=1$ at times $t = 0.075, \, 0.25, \, 0.375$.
  • Figure 5: Numerical solution of \ref{['eq:transport']} for different levels of discretization. The upper graph shows the $L_2$ error computed with respect to the analytical solution $\bar{f}$, see \ref{['eq:analytical_solution']}, on a uniformly refined grid. The lower graph shows the ranks used by the rank adaptive integrator.