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A Characteristic Mapping Method for Vlasov-Poisson with Extreme Resolution Properties

Philipp Krah, Xi-Yuan Yin, Julius Bergmann, Jean-Christophe Nave, Kai Schneider

TL;DR

The study develops a semi-Lagrangian characteristic mapping method (CMM) for the 1d+1d Vlasov–Poisson system, enabling high-fidelity advection on coarse grids by evolving a backward characteristic map and composing with the initial distribution $f_0$. Through a three-grid discretization, Hermite interpolation, and a Jacobian-based remapping strategy, CMM achieves global third-order convergence in space and time and demonstrates strong conservation properties. Numerical experiments on linear and nonlinear Landau damping and the two-stream instability benchmark the method against a Fourier pseudo-spectral code, revealing significant efficiency gains and the ability to resolve fine-scale structures via map composition and sub-grid transport. The results indicate that CMM can suppress filamentation-related recurrence effects and provides a scalable framework with open-source implementations for high-resolution plasma simulations, with planned extensions to higher dimensions and rigorous conservation proofs.

Abstract

We propose an efficient semi-Lagrangian characteristic mapping method for solving the one+one-dimensional Vlasov-Poisson equations with high precision on a coarse grid. The flow map is evolved numerically and exponential resolution in linear time is obtained. Global third-order convergence in space and time is shown and conservation properties are assessed. For benchmarking, we consider linear and nonlinear Landau damping and the two-stream instability. We compare the results with a Fourier pseudo-spectral method. The extreme fine-scale resolution features are illustrated showing the method's capabilities to efficiently treat filamentation in fusion plasma simulations.

A Characteristic Mapping Method for Vlasov-Poisson with Extreme Resolution Properties

TL;DR

The study develops a semi-Lagrangian characteristic mapping method (CMM) for the 1d+1d Vlasov–Poisson system, enabling high-fidelity advection on coarse grids by evolving a backward characteristic map and composing with the initial distribution . Through a three-grid discretization, Hermite interpolation, and a Jacobian-based remapping strategy, CMM achieves global third-order convergence in space and time and demonstrates strong conservation properties. Numerical experiments on linear and nonlinear Landau damping and the two-stream instability benchmark the method against a Fourier pseudo-spectral code, revealing significant efficiency gains and the ability to resolve fine-scale structures via map composition and sub-grid transport. The results indicate that CMM can suppress filamentation-related recurrence effects and provides a scalable framework with open-source implementations for high-resolution plasma simulations, with planned extensions to higher dimensions and rigorous conservation proofs.

Abstract

We propose an efficient semi-Lagrangian characteristic mapping method for solving the one+one-dimensional Vlasov-Poisson equations with high precision on a coarse grid. The flow map is evolved numerically and exponential resolution in linear time is obtained. Global third-order convergence in space and time is shown and conservation properties are assessed. For benchmarking, we consider linear and nonlinear Landau damping and the two-stream instability. We compare the results with a Fourier pseudo-spectral method. The extreme fine-scale resolution features are illustrated showing the method's capabilities to efficiently treat filamentation in fusion plasma simulations.
Paper Structure (24 sections, 48 equations, 15 figures, 10 tables)

This paper contains 24 sections, 48 equations, 15 figures, 10 tables.

Table of Contents

  1. Introduction
  2. The Vlasov--Poisson Equation
  3. Characteristic Mapping Method for Vlasov--Poisson
  4. Characteristic Mapping Method
  5. Advecting the Particle Distribution Function
  6. Periodic Boundary Conditions in Velocity Space
  7. Numerical Implementation
  8. Spatial Discretization.
  9. Temporal Discretization.
  10. Compositional Structure and Remapping Criteria.
  11. Error Analysis
  12. Pseudo-Spectral Discretization of Vlasov--Poisson
  13. Numerical Studies
  14. Landau Damping
  15. Linear Landau Damping
  16. ...and 9 more sections

Figures (15)

  • Figure 1: Stream function $\psi$ for the initial condition of the two-stream instability.
  • Figure 2: Periodization of the velocity field using a rescaled and shifted smooth Heaviside function $h(v)$. From left to right: resulting periodized stream function, a smooth continuation of the velocity component $u_1$ at different grid locations $v$, and the rescaled and shifted Heaviside function.
  • Figure 3: Convergence studies for the linear Landau damping test case at time $t=10$ with physical parameters listed in \ref{['tabl:landaudamping']}. Shown are the relative errors that correspond to the error measures defined in \ref{['eq:dist_error', 'eq:map_error', 'eq:MconsError', 'eq:PconsError', 'eq:EconsError']}. (a) Spatial convergence of evolutionary quantities $f$, $\textrm{\boldmath${X}$}$ and the conserved quantities $\Delta \mathcal{M},\Delta \mathcal{P} \Delta \mathcal{E}$ using the parameters listed in \ref{['tabl:convergence_space']}. (b) Temporal convergence with parameters stated in \ref{['tabl:convergence_time']}.
  • Figure 4: Spatial convergence of the linear Landau damping test case. Relative error as a function of $N$ of CMM with respect to the Fourier--Galerkin reference solution computed at resolution $512^2$.
  • Figure 5: Comparison of the damping of the potential energy for $k=0.5$. (a). Comparison of CPU-time for CMM and the spectral method for different grids (b).
  • ...and 10 more figures

Theorems & Definitions (2)

  • Definition 2.1: Mass and momentum
  • Definition 2.2: Energy