The numerical simulation of three-dimensional charged-particle dynamics (CPD) under strong magnetic field is challenging. In this paper, we introduce a new methodology to design two-scale exponential integrators for three-dimensional CPD whose magnetic field's strength is inversely proportional to a dimensionless parameter $0<\varepsilon \ll 1$. By dealing with the transformed form of three-dimensional CPD, we linearize the magnetic field and put the rest part in a nonlinear function which can be shown to be small. Based on which and the proposed two-scale exponential integrators, a class of novel integrators is formulated. The corresponding uniform accuracy over $\mathcal{O}(1/\varepsilon^{\beta})$ time interval is $\mathcal{O}(\varepsilon^{r\beta} h^r)$ for the $r$-th order integrator with the time stepsize $h$, $r=1,2,3,4$ and $0<\beta<1$. A rigorous proof of this error bound is presented and a numerical test is performed to illustrate the error behaviour of the proposed integrators.

翻译：强磁场下的三维带电粒子动力学数值模拟具有挑战性。本文针对磁场强度与无量纲参数$0<\varepsilon \ll 1$成反比的三维带电粒子动力学问题，提出了一种设计双尺度指数积分器的新方法。通过对三维带电粒子动力学变换形式的处理，我们将磁场线性化，并将剩余部分归入可证明为小量的非线性函数中。基于此分析及所提出的双尺度指数积分器，构建了一类新型积分器。对于$r$阶积分器（$r=1,2,3,4$，时间步长为$h$，$0<\beta<1$），在$\mathcal{O}(1/\varepsilon^{\beta})$时间区间上的均匀精度为$\mathcal{O}(\varepsilon^{r\beta} h^r)$。本文给出了该误差界的严格证明，并通过数值实验验证了所提出积分器的误差特性。