High-Order Exponential Integrators with Improved Uniform Accuracy for Charged-Particle in a Perpendicular Strong Magnetic Field

Abstract

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 $0<\epsilon \ll 1$, 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 $\hat{I}”t$--satisfies two distinct error bounds: $O(\epsilon \hat{I}”t^{k+1})$ and $O({\epsilon }^{k+2})$. 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 ${\epsilon }^{-1}\hat{I}”t$ is used to simulate the long-term dynamics over $[0,{\epsilon }^{-1}T]$, the numerical scheme attains a uniform convergence rate of $O(\hat{I}”t^{k+1})$. Several numerical experiments confirm these theoretical results.