Two-scale exponential integrators with uniform accuracy for three-dimensional charged-particle dynamics under strong magnetic field
Bin Wang, Zhen Miao, Yaolin Jiang
TL;DR
This work tackles the challenge of simulating three-dimensional charged-particle dynamics under a strong, inhomogeneous magnetic field by developing two-scale exponential integrators that are uniformly accurate in the small parameter $\varepsilon$. The authors transform and linearize the system, introducing a two-scale PDE in $(t,\tau)$ and filtering variables to segregate fast cyclotron motion from slow dynamics, with the residual nonlinearities kept uniformly bounded. They construct four practical explicit exponential integrators MO$r$-E of orders 1–4 and prove uniform error bounds $|x(nh)-x^n|+\varepsilon|v(nh)-v^n|\le C(h^r+(2\pi/N_{\tau})^{m_0})$, improving to $\mathcal{O}(\varepsilon^r h^r)$ in the maximal ordering case. Numerical tests demonstrate uniform accuracy in $\varepsilon$, expected time convergence, and computational efficiency gains over existing Boris/RK/CN/AP methods, especially for small $\varepsilon$.
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 $0<\varepsilon \ll 1$. 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 $\varepsilon$. Moreover, in the case of maximal ordering strong magnetic field, improved error bound $\mathcal{O}(\varepsilon^r h^r)$ 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.