Boris型解算器的剖析和导出大时间步长磁场积分器的Lie算子形式
The anatomy of Boris type solvers and the Lie operator formalism for deriving large time-step magnetic field integrators
摘要：
本文通过对恒定磁场中轨迹误差的详细研究，给出了各种Boris解算器的Lie算子推导。陀螺中心位置和回转半径的这些误差是鲍里斯解算器存在的基础，独立于任何有限差分格式。这项工作表明，有两种不同的方法来消除这些错误，使带电粒子在恒定磁场的轨迹是准确的回旋轨道。一种方法再现已知的二阶对称鲍里斯求解器。另一个产生一个以前未知的，但也在轨道上的求解器，不能从有限差分格式推导出来。通过回顾一些历史计算，发现许多出版物没有区分性能差的蛙跳Boris求解器和对称二阶Boris算法。这种对称的二阶Boris求解器的轨迹更加精确，并且在大于回旋加速器周期的时间步长处，在组合的$nonuniform$电场和磁场中保持接近精确轨道。最后，这种算子形式主义表明，布尼曼的摆线拟合方案在数学上与鲍里斯的轨道求解器相同，鲍里斯的E-B分裂是不必要的。
英文摘要：
This work gives a Lie operator derivation of various Boris solvers via a detailed study of trajectory errors in a constant magnetic field. These errors in the gyrocenter location and the gyroradius are the foundational basis for why Boris solvers existed, independent of any finite-difference schemes. This work shows that there are two distinct ways of eliminating these errors so that the trajectory of a charged particle in a constant magnetic field is exactly on the cyclotron orbit. One way reproduces the known second-order symmetric Boris solver. The other yields a previously unknown, but also on-orbit solver, not derivable from finite-difference schemes. By revisiting some historical calculations, it is found that many publications do not distinguish the poorly behaved leap-frog Boris solver from the symmetric second-order Boris algorithm. This symmetric second-order Boris solver's trajectory is much more accurate and remains close to the exact orbit in a combined $nonuniform$ electric and magnetic field at time-steps greater than the cyclotron period. Finally, this operator formalism showed that Buneman's cycloid fitting scheme is mathematically identical to Boris' on-orbit solver and that Boris' E-B splitting is unnecessary.
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