Explicit Uniformly Accurate Integrators for the Relativistic Charged-Particle Dynamics under a Strong Magnetic Field

In this paper, we propose two novel explicit uniformly accurate exponential integrators for solving the relativistic charged-particle dynamics under a strong magnetic field, which is commonly utilized in the guiding center theory. The solutions exhibit highly oscillatory behavior in time, characterized by a small parameter 0<ε≪1. For constant magnetic field intensity, we introduce a transformation to filter out the oscillatory term entirely. Combining the two-scale formulation, two explicit exponential integrators with second-order and fourth-order uniform accuracy are constructed and a rigorous convergence analysis is provided within the 4D motion equation framework. The error estimate demonstrates that the accuracy and computational cost of the algorithms are both independent of the magnetic field strength. By re-scaling the time, the methods are extended to handle the general strong magnetic field with varying intensity and direction. Moreover, the proposed numerical schemes are also applicable to scenarios involving the maximal ordering scaling magnetic field. Numerical experiments are carried out to validate the efficiency of the methods across diverse magnetic fields.