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

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.

💡 Research Summary

The paper introduces a novel filtered two‑step variational integrator (FVI) for the numerical solution of charged‑particle dynamics (CPD) in both moderate (ε = 1) and strong (0 < ε ≪ 1) magnetic fields. The governing equation is ẍ = ẋ × B(x) + F(x) with B(x) = ε⁻¹B₀ + B₁(x), where ε⁻¹ measures the magnetic field strength. Classical methods such as the Boris scheme preserve certain invariants but suffer from deteriorating accuracy when ε becomes small.

Construction of the integrator
Starting from the Lagrangian L(x,ẋ) = ½|ẋ|² + A(x)·ẋ − U(x), the authors discretize the action over each time step using the midpoint rule, yielding a discrete Lagrangian L_h(xₙ, xₙ₊₁). Applying the discrete Hamilton principle produces the discrete Euler–Lagrange equations (2.1), which form a symmetric, symplectic two‑step scheme.

Filtering for strong fields
To handle the rapid gyromotion induced by a strong magnetic field, two filter functions are introduced:
 ψ(ζ) = tanh(ζ/2)/(ζ/2) and φ(ζ) = sinh(ζ/2)/(ζ/2).
These are turned into matrices Ψ = ψ(−h ε B̃₀) and Φ = φ(−h ε B̃₀), where B̃₀ is the skew‑symmetric matrix defined by −B̃₀v = v × B₀. ψ is applied to the position update, suppressing high‑frequency components and enabling a uniform second‑order error bound in position. φ is used in the velocity averaging formula and is crucial for the long‑time near‑conservation of energy and magnetic moment.

Algorithm
The final scheme consists of a filtered two‑step position recurrence (2.5) and a filtered velocity approximation (2.6). The first step requires an initial value x₁, which is obtained by solving an implicit equation (2.7) via a fixed‑point iteration. The method is fully implicit but the ε⁻¹‑dependent terms are isolated on the left‑hand side to improve stability.

Theoretical results
Moderate magnetic field (ε = 1): Theorem 3.1 proves global errors |xₙ − x(tₙ)|, |vₙ − v(tₙ)| ≤ C h² for step sizes satisfying h ≤ h₀ and |cos(h/2)| > c > 0. Backward error analysis shows that the modified differential equation associated with FVI possesses a Hamiltonian that exactly conserves the discrete energy and momentum, yielding long‑time near‑conservation results.

Strong magnetic field (0 < ε ≪ 1): Using modulated Fourier expansions, the authors establish uniform second‑order accuracy in position and parallel velocity for step sizes that are large relative to ε, while the error with respect to ε is O(ε h). Theorems 3.5–3.7 demonstrate that energy and the magnetic moment are conserved up to O(h²) over time intervals of length O(ε⁻¹), i.e., many gyro‑periods.

Numerical experiments
Four test problems are presented: (i) uniform magnetic field with an electric field, (ii) mildly non‑uniform magnetic field, (iii) strong magnetic field with long‑time integration, and (iv) varying step sizes and ε values. In all cases, FVI matches the predicted convergence rates and exhibits superior long‑time preservation of invariants compared with the Boris method and other filtered schemes. Notably, even for ε = 10⁻³ the method retains second‑order accuracy with relatively large steps.

Conclusion
The filtered two‑step variational integrator combines the structural advantages of variational (symplectic, symmetric) methods with specially designed filters that tame the fast gyromotion. It delivers uniform second‑order accuracy and long‑time near‑conservation of energy, momentum, and magnetic moment across both moderate and strong magnetic field regimes. The work opens the way for extensions to more complex, non‑uniform electromagnetic fields and to kinetic plasma models such as Vlasov‑Poisson systems.