Two-Scale Exponential Integrators with Uniform Accuracy for Three-Dimensional Charged-Particle Dynamics Under a Strong Magnetic Field

Abstract. The numerical simulation of three-dimensional charged-particle dynamics (CPD) under a strong magnetic field is a basic and challenging algorithmic task in plasma physics. 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 and small parameter [Formula: see text]. By dealing with the transformed form of three-dimensional CPD, we linearize the magnetic field and put the residual component in a new nonlinear function which is shown to be uniformly bounded. Based on this foundation and the proposed two-scale exponential integrators, a class of novel integrators is formulated and studied. The corresponding uniform accuracy of the proposed [Formula: see text]th order integrator is shown to be [Formula: see text] in the approximation of the position, where [Formula: see text] and the constant symbolized by [Formula: see text], the time stepsize [Formula: see text], and the computation cost are all independent of [Formula: see text]. Moreover, in the case of a maximal ordering strong magnetic field, an improved error bound [Formula: see text] in the approximation of the position is obtained for the proposed [Formula: see text]th order integrator. A rigorous proof of these uniform and improved error bounds is presented, and a numerical test is performed to illustrate the error and efficiency behavior of the proposed integrators.