Two-scale exponential integrators with uniform accuracy for three-dimensional charged-particle dynamics under strong magnetic field

Abstract

The numerical simulation of three-dimensional charged-particle dynamics (CPD) under 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 $0<\epsilon \ll 1$. 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 $r$-th order integrator is shown to be $\mathcal{O}(h^r)$, where $r=1,2,3,4$ and the constant symbolized by $\mathcal{O}$, the time stepsize $h$ and the computation cost are all independent of $\epsilon$. Moreover, in the case of maximal ordering strong magnetic field, improved error bound $\mathcal{O}({\epsilon }^r{h}^r)$ is obtained for the proposed $r$-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 behaviour of the proposed integrators.