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Convergence analysis of a Crank-Nicolson scheme for strongly magnetized plasmas

Francis Filbet, L Miguel Rodrigues, Kim Han Trinh

TL;DR

This work delivers a rigorous convergence analysis for an asymptotic-preserving particle scheme (a Crank–Nicolson-type method) used to push particles in a PIC framework for the Vlasov equation under a strong, inhomogeneous magnetic field. By augmenting the formulation and introducing guiding-center variables, the authors derive discrete ε-asymptotics and establish uniform-in-ε error bounds that connect the discrete solution to the slow, large-scale dynamics even when the time step is not small relative to the fast oscillations. The analysis combines discrete stability and consistency to prove a sharp error estimate of the form min{ε+Δt^2, Δt^2/ε^5}, with extensions to the guiding-center variables, showing the scheme remains accurate in the stiff regime. These results underpin the reliability and efficiency of AP PIC schemes for magnetized plasmas and provide a template for analyzing similar implicit–explicit augmentations in oscillatory kinetic systems.

Abstract

The present paper is devoted to the convergence analysis of an asymptotic preserving particle scheme designed to serve as a particle pusher in a Particle-In-Cell (PIC) method for the Vlasov equation with a strong inhomogeneous magnetic field. The asymptotic preserving scheme that we study removes classical strong restrictive stability constraints on discretization steps while capturing the large-scale dynamics, even when the discretization is too coarse to capture fastest scales. Our error bounds are explicit regarding the discretization and stiffness parameters and match sharply numerical tests. The present analysis is expected to be representative of the general analysis of a class of schemes, developed by the authors, conceived as implicit-explicit schemes on augmented formulations.

Convergence analysis of a Crank-Nicolson scheme for strongly magnetized plasmas

TL;DR

This work delivers a rigorous convergence analysis for an asymptotic-preserving particle scheme (a Crank–Nicolson-type method) used to push particles in a PIC framework for the Vlasov equation under a strong, inhomogeneous magnetic field. By augmenting the formulation and introducing guiding-center variables, the authors derive discrete ε-asymptotics and establish uniform-in-ε error bounds that connect the discrete solution to the slow, large-scale dynamics even when the time step is not small relative to the fast oscillations. The analysis combines discrete stability and consistency to prove a sharp error estimate of the form min{ε+Δt^2, Δt^2/ε^5}, with extensions to the guiding-center variables, showing the scheme remains accurate in the stiff regime. These results underpin the reliability and efficiency of AP PIC schemes for magnetized plasmas and provide a template for analyzing similar implicit–explicit augmentations in oscillatory kinetic systems.

Abstract

The present paper is devoted to the convergence analysis of an asymptotic preserving particle scheme designed to serve as a particle pusher in a Particle-In-Cell (PIC) method for the Vlasov equation with a strong inhomogeneous magnetic field. The asymptotic preserving scheme that we study removes classical strong restrictive stability constraints on discretization steps while capturing the large-scale dynamics, even when the discretization is too coarse to capture fastest scales. Our error bounds are explicit regarding the discretization and stiffness parameters and match sharply numerical tests. The present analysis is expected to be representative of the general analysis of a class of schemes, developed by the authors, conceived as implicit-explicit schemes on augmented formulations.

Paper Structure

This paper contains 12 sections, 20 theorems, 159 equations, 3 figures.

Table of Contents

  1. Introduction
  2. Continuous analysis
  3. Original variables
  4. Guiding-center variables
  5. Discrete uniform bounds
  6. Asymptotic analysis
  7. Direct convergence analysis
  8. Stability analysis
  9. Consistency analysis
  10. Convergence
  11. Proof of the main theorem, Theorem \ref{['th:main']}
  12. Implicit solving

Key Result

Theorem 1.1

Let $T>0$, $M>0$ and $\varepsilon _0>0$. There exist $\delta_0>0$ and $C>0$ such that when $0<\varepsilon \leq\varepsilon _0$ and $0<\Delta t\leq\delta_0$ if $({\mathbf x},{\mathbf v})$ solves eq:ODE on $[0,T]$ with initial datum $({\mathbf x}^0,{\mathbf v}^0)$ such that $\|{\mathbf v}^0\|\leq M$, a

Figures (3)

  • Figure 1: Convergence errors, that is, comparison of the solution to the scheme \ref{['scheme:CN']} with the solution to \ref{['eq:ODE']}. (Left) Comparison of the original slow variables $({\mathbf x},e)$. (Right) Comparison of their guiding center modifications $({\mathbf x}_{\rm gc},e_{\rm gc})$.
  • Figure 2: Discrete $\varepsilon$ asymptotic errors, that is, comparison of the solution to the scheme \ref{['scheme:CN']} with solutions to \ref{['scheme:asymp']}. (Left) Comparison of the original slow variables $({\mathbf x},e)$. (Right) Comparison of their guiding center modifications $({\mathbf x}_{\rm gc},e_{\rm gc})$. Note that initial data for \ref{['scheme:asymp']} are set respectively to $({\mathbf x}^0,e^0)$ for the computation of the left panel and to $({\mathbf x}_{\rm gc}^0,e_{\rm gc}^0)$ for the computation of the right one.
  • Figure 3: Continuous $\varepsilon$ asymptotic errors, that is, comparison of the solution to the scheme \ref{['scheme:CN']} with solutions to \ref{['eq:asymp']}. (Left) Comparison of the original slow variables $({\mathbf x},e)$. (Right) Comparison of their guiding center modifications $({\mathbf x}_{\rm gc},e_{\rm gc})$. Note that initial data for \ref{['scheme:asymp']} are set respectively to $({\mathbf x}^0,e^0)$ for the computation of the left panel and to $({\mathbf x}_{\rm gc}^0,e_{\rm gc}^0)$ for the computation of the right one.

Theorems & Definitions (29)

  • Theorem 1.1
  • Lemma 2.1
  • Lemma 2.2
  • proof
  • Proposition 2.3
  • proof
  • Lemma 2.4
  • Proposition 2.5
  • Lemma 3.1
  • proof
  • ...and 19 more