AI 中文总结
研究垂直强磁场中带电粒子动力学问题，通过升维线性化常微分方程开发任意高阶指数积分器，短时模拟有两个误差界，大步长模拟有均匀收敛率，数值实验验证了理论结果。
AI 中文摘要
本文考虑一类带电粒子动力学问题，粒子受与小参数\(0<\varepsilon\ll1\)成反比的磁通密度的磁力和非线性电力作用，其高振荡行为给数值计算带来挑战。为提高指数积分器性能，采用升维方法线性化常微分方程，开发了新的任意高阶指数积分器族。对于\([0,T]\)上的短时模拟，证明该方法（采用\(k\)次辅助多项式和时间步长\(\Delta t\)）满足\(O(\varepsilon \Delta t^{k + 1})\)和\(O(\varepsilon^{k + 2})\)两个误差界，后者保证步长为\(O(1)\)时算法仍精确。此外，用大步长\(\varepsilon^{-1}\Delta t\)模拟\([0,\varepsilon^{-1}T]\)上的长期动力学时，数值格式达到\(O(\Delta t^{k + 1})\)的均匀收敛率，数值实验证实了这些理论结果。
英文摘要
This paper considers a class of charged-particle dynamics problems in which the particle is subjected to a magnetic force, with a magnetic flux density inversely proportional to a small parameter $0<\varepsilon\ll 1$, and a nonlinear electric force. The resulting highly oscillatory behavior poses significant challenges for numerical computation. To enhance the performance of exponential integrators (EIs), this paper employs a technique that linearizes the ordinary differential equation through a dimension-raising approach. Based on this technique, a new family of EIs is developed that achieves arbitrarily high order. For short-time simulations on the interval $[0,T]$, it is rigorously proved that the proposed method--which employs auxiliary polynomials of degree $k$ and a time step $Δt$--satisfies two distinct error bounds: $O(\varepsilon Δt^{k+1})$ and $O(\varepsilon^{k+2})$. The latter bound guarantees that the algorithm stays accurate even when the step size is of order $O(1)$. Furthermore, when a large step size $\varepsilon^{-1}Δt$ is used to simulate the long-term dynamics over $[0,\varepsilon^{-1}T]$, the numerical scheme attains a uniform convergence rate of $O(Δt^{k+1})$. Several numerical experiments confirm these theoretical results.
