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A novel directly energy-preserving method for charged particle dynamics

Yexin Li, Ping Jiang, Haochen Li

TL;DR

This work reformulates the Lorentz force dynamics as a non-canonical Hamiltonian system with Hamiltonian $H(z)=\tfrac{1}{2}\mathbf{v}^T\mathbf{v}+U(\mathbf{x})$ and develops energy-preserving CIDG integrators. The authors introduce CIDG-I, its adjoint CIDG-II, and the symmetric CIDG-C composition, proving exact energy conservation without quadrature and 2nd-order accuracy for CIDG-C. Numerical experiments show CIDG-C maintains energy exactly and yields bounded invariants over long times, outperforming the Boris method in energy fidelity, while offering competitive CPU performance relative to alternative energy-preserving schemes. The results provide a robust toolkit for accurate, long-time simulations of charged particle dynamics in static and geometrically complex electromagnetic fields, including tokamak configurations.

Abstract

In this paper, we apply the coordinate increment discrete gradient (CIDG) method to solve the Lorentz force system which can be written as a non-canonical Hamiltonian system. Then we can obtain a new energy-preserving CIDG-I method for the system. The CIDG-I method can combine with its adjoint method CIDG-II which is also a energy-preserving method to form a new method, namely CIDG-C method. The CIDG-C method is symmetrical and can conserve the Hamiltonian energy directly and exactly. With comparison to the well-used Boris method, numerical experiments indicate that the CIDG-C method holds advantage over the Boris method in terms of energy-conserving.

A novel directly energy-preserving method for charged particle dynamics

TL;DR

This work reformulates the Lorentz force dynamics as a non-canonical Hamiltonian system with Hamiltonian and develops energy-preserving CIDG integrators. The authors introduce CIDG-I, its adjoint CIDG-II, and the symmetric CIDG-C composition, proving exact energy conservation without quadrature and 2nd-order accuracy for CIDG-C. Numerical experiments show CIDG-C maintains energy exactly and yields bounded invariants over long times, outperforming the Boris method in energy fidelity, while offering competitive CPU performance relative to alternative energy-preserving schemes. The results provide a robust toolkit for accurate, long-time simulations of charged particle dynamics in static and geometrically complex electromagnetic fields, including tokamak configurations.

Abstract

In this paper, we apply the coordinate increment discrete gradient (CIDG) method to solve the Lorentz force system which can be written as a non-canonical Hamiltonian system. Then we can obtain a new energy-preserving CIDG-I method for the system. The CIDG-I method can combine with its adjoint method CIDG-II which is also a energy-preserving method to form a new method, namely CIDG-C method. The CIDG-C method is symmetrical and can conserve the Hamiltonian energy directly and exactly. With comparison to the well-used Boris method, numerical experiments indicate that the CIDG-C method holds advantage over the Boris method in terms of energy-conserving.
Paper Structure (8 sections, 1 theorem, 25 equations, 4 figures, 1 table)

This paper contains 8 sections, 1 theorem, 25 equations, 4 figures, 1 table.

Table of Contents

  1. Introduction
  2. Non-canonical Hamiltonian form of the system
  3. Coordinate increment discrete gradient methods
  4. Numerical experiments
  5. 2D dynamics in a static electromagnetic field
  6. Energy behavior test
  7. 2D dynamics in an axisymmetric tokamak geometry
  8. Conclusions

Key Result

Theorem 3.1

CIDG-I, CIDG-II and CIDG-C methods can all conserve the Hamiltonian energy of the system eq:3s5. Moreover, the CIDG-C method is symmetrical and has order 2.

Figures (4)

  • Figure 1: The Boris method is applied to the simple 2D dynamics with step $h=\pi/10$. (a) The orbit in the 100-st turn; (b) Errors of the angular momentum $p_{\xi}$, the magnetic moment $\mu$ and the energy $H$ for $t\in [0,5\times 10^{5}h]$.
  • Figure 3: The LIM(2,2) method is applied to the simple 2D dynamics with step $h=\pi/10$. (a) The orbit in the 100-st turn; (b) Errors of the angular momentum $p_{\xi}$, the magnetic moment $\mu$ and the energy $H$ for $t\in [0,5\times 10^{5}h]$.
  • Figure 5: Hamiltonian error solving problem\ref{['eq:6s1']}-\ref{['eq:6s3']} with step $h = 10^{-2}$. (a) The Boris method; (b)The CIDG-C method ;(c)The LIM(2,2) method ;(d)The BDLI method.
  • Figure 6: Numerical solution of the CIDG-C method with step $h=\pi/10$ for $t\in [0,5\times 10^{5}h]$. (a) Transit orbit; (b) Banana orbit; (c) Energy preservation.

Theorems & Definitions (5)

  • Theorem 3.1
  • proof
  • Remark 3.1
  • Remark 3.2
  • Remark 4.1