A novel class of high-order uniformly accurate exponential integrators with local linear extension for the charged-particle dynamics under strong magnetic field

Abstract

In this paper, we develop a novel class of high-order uniformly accurate exponential integrators for charged-particle dynamics under a strong magnetic field. The small parameter $0<\varepsilon\ll 1$ induces rapid temporal oscillations, rendering traditional numerical methods prohibitively expensive due to severe step-size restrictions. To address this issue, a linearization technology that introduces auxiliary polynomial variables is employed to recast the original charged-particle dynamics as a higher-dimensional system. Classical exponential integrators are subsequently applied to this augmented formulation, which inherently carries richer structural information, thereby yielding a family of uniformly accurate exponential integrators that can reach arbitrarily high order without requiring any order conditions. For the maximal ordering scaling strong magnetic field, we rigorously demonstrate via algebraic techniques that the proposed schemes with auxiliary polynomial variables of degree $k(k\geq 2)$ achieve an $\mathcal{O}(\varepsilon h^{k+1})$ improved error estimate for the position and a uniform $\mathcal{O}(h^{k+1})$ error estimate for the velocity. Numerical experiments validate the advantages of the methods. The theoretical and numerical in vestigation is finally extended to relativistic charged-particle dynamics in a four-dimensional framework with maximal ordering scaling strong magnetic field.