The state of art of charge-conserving electromagnetic finite element particle-in-cell has grown by leaps and bounds in the past few years. These advances have primarily been achieved for leap-frog time stepping schemes for Maxwell solvers, in large part, due to the method strictly following the proper space for representing fields, charges, and measuring currents. Unfortunately, leap-frog based solvers (and their other incarnations) are only conditionally stable. Recent advances have made Electromagnetic Finite Element Particle-in-Cell (EM-FEMPIC) methods built around unconditionally stable time stepping schemes were shown to conserve charge. Together with the use of a quasi-Helmholtz decomposition, these methods were both unconditionally stable and satisfied Gauss' Laws to machine precision. However, this architecture was developed for systems with explicit particle integrators where fields and velocities were off by a time step. While completely self-consistent methods exist in the literature, they follow the classic rubric: collect a system of first order differential equations (Maxwell and Newton equations) and use an integrator to solve the combined system. These methods suffer from the same side-effect as earlier--they are conditionally stable. Here we propose a different approach; we pair an unconditionally stable Maxwell solver to an exponential predictor-corrector method for Newton's equations. As we will show via numerical experiments, the proposed method conserves energy within a PIC scheme, has an unconditionally stable EM solve, solves Newton's equations to much higher accuracy than a traditional Boris solver and conserves charge to machine precision. We further demonstrate benefits compared to other polynomial methods to solve Newton's equations, like the well known Boris push.

翻译：近年来，基于电荷守恒的电磁有限元粒子网格（FEMPIC）方法取得了突飞猛进的发展。这些进展主要针对麦克斯韦求解器的蛙跳时间步进方案实现，很大程度上得益于该方法严格遵循了场、电荷表示及电流测量的正确空间。然而，基于蛙跳的求解器（及其衍生形式）仅具有条件稳定性。近期研究表明，基于无条件稳定时间步进方案的电磁有限元粒子网格方法能够实现电荷守恒。通过结合准亥姆霍兹分解，这些方法既满足无条件稳定性，又能以机器精度满足高斯定律。但该架构是为显式粒子积分器系统设计的，其中场与速度存在一个时间步的偏差。尽管文献中存在完全自洽的方法，但它们遵循经典范式：建立一阶微分方程组（麦克斯韦方程组与牛顿方程）并利用积分器求解组合系统。这些方法存在与先前方法相同的缺陷——即条件稳定性。本文提出一种新方法：将无条件稳定的麦克斯韦求解器与牛顿方程的指数预测-校正方法相结合。数值实验表明，所提方法能在粒子网格方案中保持能量守恒，实现电磁求解的无条件稳定性，以远高于传统Boris求解器的精度求解牛顿方程，并以机器精度实现电荷守恒。我们进一步展示了该方法相较于其他多项式方法（如著名的Boris推进算法）在求解牛顿方程方面的优势。