用于相对论粒子模拟的电荷守恒指数预测-校正FEMPIC格式
A Charge Conserving Exponential Predictor Corrector FEMPIC Formulation for Relativistic Particle Simulations
AI总结：
提出一种电荷守恒的指数预测-校正FEMPIC格式，结合无条件稳定Maxwell求解器，实现能量守恒、高精度粒子求解及机器精度电荷守恒。
摘要：
电荷守恒电磁有限元粒子网格方法的最新进展在过去几年中突飞猛进。这些进展主要是在Maxwell求解器的蛙跳时间步进方案中实现的，很大程度上是因为该方法严格遵循了表示场、电荷和测量电流的适当空间。不幸的是，基于蛙跳的求解器（及其其他形式）只是条件稳定的。最近的进展表明，围绕无条件稳定时间步进方案构建的电磁有限元粒子网格（EM-FEMPIC）方法能够保持电荷守恒。结合准Helmholtz分解的使用，这些方法既无条件稳定，又能在机器精度内满足Gauss定律。然而，这种架构是为具有显式粒子积分器的系统开发的，其中场和速度相差一个时间步。虽然文献中存在完全自洽的方法，但它们遵循经典规则：收集一阶微分方程组（Maxwell和Newton方程）并使用积分器求解组合系统。这些方法遭受与早期相同的副作用——它们是条件稳定的。在这里，我们提出了一种不同的方法；我们将无条件稳定的Maxwell求解器与用于Newton方程的指数预测-校正方法配对。正如我们将通过数值实验展示的，所提出的方法在PIC方案中保持能量守恒，具有无条件稳定的电磁求解，以比传统Boris求解器高得多的精度求解Newton方程，并在机器精度内保持电荷守恒。我们进一步展示了与其他求解Newton方程的多项式方法（如众所周知的Boris推进）相比的优势。
英文摘要：
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.
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