高阶时间积分器用于磁四极子中带电粒子运动的模拟
High order time integrators for the simulation of charged particle motion in magnetic quadrupoles
AI总结：
本文研究了如何通过高阶时间积分器提高磁四极子中带电粒子运动模拟的稳定性与效率，分析了不同高阶积分方法的精度和效率，并探讨了矢量势插值对粒子跟踪精度的影响。
摘要：
磁四极子是粒子加速器如大型强子对撞机中的关键组件。为了研究粒子束穿过四极子时的稳定性，必须模拟大量的粒子回旋，这导致需要在长时间间隔内保持运动不变量的数值保真度，且计算成本主要与矢量势的重复评估有关。本文为了减少此成本，首先考虑一种特定的规范变换，可以显著减少矢量势评估次数。然后分析了数值解对计算矢量势数据所需插值过程的敏感性。最后比较了多种高阶积分技术，以评估其在长期模拟中的精度和效率。考虑了显式高阶李方法，以及隐式高阶辛积分器和传统显式龙格-库塔方法。在辛方法中，高阶李积分器在成本/精度比方面表现最佳，但非辛的龙格-库塔方法在非常长的长期模拟中也表现优异。此外，场重建和插值技术的精度被证明是影响粒子跟踪精度的限制因素。
英文摘要：
Magnetic quadrupoles are essential components of particle accelerators like the Large Hadron Collider. In order to study numerically the stability of the particle beam crossing a quadrupole, a large number of particle revolutions in the accelerator must be simulated, thus leading to the necessity to preserve numerically invariants of motion over a long time interval and to a substantial computational cost, mostly related to the repeated evaluation of the magnetic vector potential. In this paper, in order to reduce this cost, we first consider a specific gauge transformation that allows to reduce significantly the number of vector potential evaluations. We then analyze the sensitivity of the numerical solution to the interpolation procedure required to compute magnetic vector potential data from gridded precomputed values at the locations required by high order time integration methods. Finally, we compare several high order integration techniques, in order to assess their accuracy and efficiency for these long term simulations. Explicit high order Lie methods are considered, along with implicit high order symplectic integrators and conventional explicit Runge Kutta methods. Among symplectic methods, high order Lie integrators yield optimal results in terms of cost/accuracy ratios, but non symplectic Runge Kutta methods perform remarkably well even in very long term simulations. Furthermore, the accuracy of the field reconstruction and interpolation techniques are shown to be limiting factors for the accuracy of the particle tracking procedures.
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