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A filtered two-step variational integrator for charged-particle dynamics in a moderate or strong magnetic field

Ting Li, Bin Wang

Abstract

This article is concerned with a new filtered two-step variational integrator for solving the charged-particle dynamics in a mildly non-uniform moderate or strong magnetic field with a dimensionless parameter $\varepsilon$ inversely proportional to the strength of the magnetic field. In the case of a moderate magnetic field ($\varepsilon=1$), second-order error bounds and long-time near-conservation of energy and momentum are obtained. Moreover, the proof of the long-term analysis is accomplished by the backward error analysis. For $0<\varepsilon \ll 1$, the proposed integrator achieves uniform second-order accuracy in the position and the parallel velocity for large step sizes, while attaining first-order accuracy with respect to the small parameter $\varepsilon$ for smaller step sizes. The error bounds are derived from a comparison of the modulated Fourier expansions of the exact and numerical solutions. Moreover, long-time near-conservation of the energy and the magnetic moment is established using modulated Fourier expansion and backward error analysis. All the theoretical results of the error behavior and long-time near-conservation are numerically demonstrated by four numerical experiments.

A filtered two-step variational integrator for charged-particle dynamics in a moderate or strong magnetic field

Abstract

This article is concerned with a new filtered two-step variational integrator for solving the charged-particle dynamics in a mildly non-uniform moderate or strong magnetic field with a dimensionless parameter inversely proportional to the strength of the magnetic field. In the case of a moderate magnetic field (), second-order error bounds and long-time near-conservation of energy and momentum are obtained. Moreover, the proof of the long-term analysis is accomplished by the backward error analysis. For , the proposed integrator achieves uniform second-order accuracy in the position and the parallel velocity for large step sizes, while attaining first-order accuracy with respect to the small parameter for smaller step sizes. The error bounds are derived from a comparison of the modulated Fourier expansions of the exact and numerical solutions. Moreover, long-time near-conservation of the energy and the magnetic moment is established using modulated Fourier expansion and backward error analysis. All the theoretical results of the error behavior and long-time near-conservation are numerically demonstrated by four numerical experiments.

Paper Structure

This paper contains 23 sections, 11 theorems, 147 equations, 12 figures, 1 algorithm.

Table of Contents

  1. Introduction
  2. Numerical integrator
  3. Main results and numerical experiments
  4. Theoretical and numerical results in a moderate magnetic field ($\varepsilon=1$)
  5. Theoretical results in a moderate magnetic field
  6. Numerical experiments in a moderate magnetic field
  7. Theoretical and numerical results in a strong magnetic field ($0<\varepsilon\ll 1$)
  8. Theoretical results in a strong magnetic field
  9. Numerical experiments in a strong magnetic field
  10. Error bounds and long-term analysis in a moderate magnetic field (Proof of Theorems \ref{['error bound 1']}--\ref{['conservation-M1']})
  11. Error bounds (Proof of Theorem \ref{['error bound 1']})
  12. Conservation properties (Proof of Theorems \ref{['conservation-H1']}--\ref{['conservation-M1']})
  13. Long-time near-conservation of energy (Proof of Theorem \ref{['conservation-H1']})
  14. Long-time near-conservation of momentum (Proof of Theorem \ref{['conservation-M1']})
  15. Error bounds and long-term analysis in a strong magnetic field (Proof of Theorems \ref{['error bound 2']}--\ref{['conservation-M2']})
  16. ...and 8 more sections

Key Result

Theorem 3.1

(Error bounds) It is assumed that charged-particle has sufficiently smooth solutions, and the functions $A(x)$ and $F(x)$ are sufficiently differentiable. Moreover, we assume that $A(x)$ and $F(x)$ are locally Lipschitz continuous with Lipschitz constants $L$. There exists a constant $h_0>0$, such t where $C>0$ is a generic constant independent of $h$ or $n$ but depends on $L$ and $T$.

Figures (12)

  • Figure 1: Problem 1. The global errors $error_x$ and $error_v$ with $t=1$ and $h=1/2^{k}$ for $k=1,\ldots,8$ (the dash-dot line is slope two).
  • Figure 2: Problem 1. Evolution of energy error $e_H$ and momentum error $e_{M}$ with different step sizes $h$.
  • Figure 3: Problem 2. The global errors $error_x$ and $error_v$ with $t=1$ and $h=1/2^{k}$ for $k=1,\ldots,8$ (the dash-dot line is slope two).
  • Figure 4: Problem 2. Evolution of energy error $e_H$ and momentum error $e_{M}$ with different step sizes $h$.
  • Figure 5: Problem 3. The global errors in $x$, $v_{\parallel}$ and $v_{\perp}$ at time $t=\pi/2$ vs. $\varepsilon$ ($\varepsilon=1/2^{k}, k=6,\ldots,17$) with different $h$.
  • ...and 7 more figures

Theorems & Definitions (20)

  • Theorem 3.1
  • Theorem 3.2
  • Theorem 3.3
  • Theorem 3.4
  • proof
  • Remark 3.1
  • Theorem 3.5
  • Remark 3.2
  • Remark 3.3
  • Theorem 3.6
  • ...and 10 more