This paper discusses energy-conserving time-discretizations for finite element particle-in-cell discretizations of the Vlasov–Maxwell system. A geometric spatially discrete system can be obtained using a standard particle-in-cell discretization of the particle distribution and compatible finite element spaces for the fields to discretize the Poisson bracket of the Vlasov–Maxwell model (see Kraus et al. (2017) [1]). In this paper, we derive energy-conserving time-discretizations based on the discrete gradient method applied to an antisymmetric splitting of the Poisson matrix. Firstly, we propose a semi-implicit method based on a splitting that yields constant Poisson matrices in each substep. Moreover, we devise an alternative discrete gradient that yields a time discretization that can additionally conserve Gauss' law. Finally, we explain how substepping for fast species dynamics can be incorporated.

本文讨论了Vlasov-Maxwell系统的有限元单元内离散离散的节能时间离散。几何空间离散系统可以通过使用标准的单元格内离散化粒子分布和兼容的有限元空间获得，以离散化Vlasov-Maxwell模型的泊松括号（请参见Kraus等人（2017）[ 1]）。在本文中，我们基于应用于Poisson矩阵的反对称分裂的离散梯度方法，导出了节能时间离散。首先，我们提出了一种基于分裂的半隐式方法，该方法在每个子步骤中产生恒定的泊松矩阵。此外，我们设计了一种替代的离散梯度，可以产生时间离散化，从而可以进一步保留高斯定律。最后，