Boris-type exponential integrators for charged-particle dynamics in strong magnetic fields
Abstract
We study numerical time integration for the motion of a charged particle in a strong magnetic field, focusing on the regime in which the fast gyration is not resolved by the time step. Standard Boris-type methods are structure preserving and accurate when the gyration is resolved, but their error constants deteriorate with increasing field strength; filtered Boris methods improve this behavior but may suffer from singularities and resonance-induced error blow-up. In this work we consider the case of a constant strong magnetic field and derive a broad family of one-step exponential integrators from the variation-of-constants formula, formulated in terms of matrix-valued filter functions. This framework includes existing filtered Boris variants and permits the construction of schemes with uniformly bounded filters. Using a discrete variation-of-constants representation together with summation-by-parts arguments, we establish error bounds whose constants are independent of the magnetic field strength and characterize the filter conditions required for first- and second-order accuracy. In particular, we show that bounded filters avoid resonance blow-up and allow arbitrary time steps, at the cost of a controlled order reduction near resonant frequencies. Numerical experiments confirm the predicted convergence behavior, the different accuracy of position and velocity components, and the role of the leading error terms across a wide range of step-size-field-strength products.
Related
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BibTeX
@misc{hochbruck2026boristype,
title = {{Boris-type exponential integrators for charged-particle dynamics in strong magnetic fields}},
author = {Marlis Hochbruck and Sebastian Merk},
howpublished = {arXiv},
year = {2026},
month = sep,
doi = {10.48550/arxiv.2609.35679},
eprint = {2609.35679},
archivePrefix = {arXiv},
url = {https://arxiv.org/abs/2609.35679},
}
RIS
TY - UNPB
TI - Boris-type exponential integrators for charged-particle dynamics in strong magnetic fields
AU - Marlis Hochbruck
AU - Sebastian Merk
PY - 2026
DA - 2026/09/28
PB - arXiv
DO - 10.48550/arxiv.2609.35679
UR - https://arxiv.org/abs/2609.35679
AB - We study numerical time integration for the motion of a charged particle in a strong magnetic field, focusing on the regime in which the fast gyration is not resolved by the time step. Standard Boris-type methods are structure preserving and accurate when the gyration is resolved, but their error constants deteriorate with increasing field strength; filtered Boris methods improve this behavior but may suffer from singularities and resonance-induced error blow-up. In this work we consider the case of a constant strong magnetic field and derive a broad family of one-step exponential integrators from the variation-of-constants formula, formulated in terms of matrix-valued filter functions. This framework includes existing filtered Boris variants and permits the construction of schemes with uniformly bounded filters. Using a discrete variation-of-constants representation together with summation-by-parts arguments, we establish error bounds whose constants are independent of the magnetic field strength and characterize the filter conditions required for first- and second-order accuracy. In particular, we show that bounded filters avoid resonance blow-up and allow arbitrary time steps, at the cost of a controlled order reduction near resonant frequencies. Numerical experiments confirm the predicted convergence behavior, the different accuracy of position and velocity components, and the role of the leading error terms across a wide range of step-size-field-strength products.
ER -