We present an explicit temporal discretization of particle-in-cell schemes for the Vlasov equation that results in exact energy conservation when combined with an appropriate spatial discretization. The scheme is inspired by a simple, second-order explicit scheme that conserves energy exactly in the Eulerian context. We show that direct translation to particle-in-cell does not result in strict conservation, but derive a simple correction based on an analytically solvable optimization problem that recovers conservation. While this optimization problem is not guaranteed to have a real solution for every particle, we provide a correction that makes imaginary values extremely rare and still admits $\mathcal{O}(10^{-12})$ fractional errors in energy for practical simulation parameters. We present the scheme in both electrostatic -- where we use the Amp\`{e}re formulation -- and electromagnetic contexts. With an electromagnetic field solve, the field update is most naturally linearly implicit, but the more computationally intensive particle update remains fully explicit. We also show how the scheme can be extended to use the fully explicit leapfrog and pseudospectral analytic time-domain (PSATD) field solvers. The scheme is tested on standard kinetic plasma problems, confirming its conservation properties.

翻译：本文提出了一种用于Vlasov方程的粒子网格方案显式时间离散化方法，当与适当的空间离散化结合时，该方法可实现精确的能量守恒。该方案受到欧拉框架下精确守恒能量的简单二阶显式格式的启发。我们证明直接迁移到粒子网格框架无法实现严格守恒，但通过基于解析可解优化问题的简单修正恢复了守恒性。虽然该优化问题不能保证对每个粒子都存在实数解，但我们提供的修正使得虚数值出现概率极低，且在实际模拟参数下仍能保持$\mathcal{O}(10^{-12})$量级的能量相对误差。我们在静电（采用安培表述）和电磁两种情境中阐述了该方案。在电磁场求解中，场更新本质上为线性隐式，但计算量更大的粒子更新仍保持完全显式。我们还展示了如何将该方案扩展至完全显式的蛙跳格式和伪谱解析时域（PSATD）场求解器。通过对标准等离子体动理学问题的测试，验证了该方案的守恒特性。