In recent years, several gauge-symmetric particle-in-cell (PIC) methods have been developed whose simulations of particles and electromagnetic fields exactly conserve charge. While it is rightly observed that these methods’ gauge symmetry gives rise to their charge conservation, this causal relationship has generally been asserted via ad hoc derivations of the associated conservation laws. In this work, we develop a comprehensive theoretical grounding for charge conservation in gauge-symmetric Lagrangian and Hamiltonian PIC algorithms. For Lagrangian variational PIC methods, we apply Noether’s second theorem to demonstrate that gauge symmetry gives rise to a local charge conservation law as an off-shell identity. For Hamiltonian splitting methods, we show that the momentum map establishes their charge conservation laws. We define a new class of algorithms – gauge-compatible splitting methods – that exactly preserve the momentum map associated with a Hamiltonian system’s gauge symmetry – even after time discretization. This class of algorithms affords splitting schemes a decided advantage over alternative Hamiltonian integrators. We apply this general technique to design a novel, explicit, symplectic, gauge-compatible splitting PIC method, whose momentum map yields an exact local charge conservation law. Our study clarifies the appropriate initial conditions for such schemes and examines their symplectic reduction.

中文翻译：

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细胞内粒子模拟中电荷守恒的几何理论

近年来，已经开发了几种规范对称的细胞内粒子（PIC）方法，其对粒子和电磁场的模拟精确地保存了电荷。虽然正确地观察到这些方法的规范对称性导致了它们的电荷守恒，但这种因果关系通常通过以下方式断言：特设相关守恒定律的推导。在这项工作中，我们为规范对称拉格朗日和哈密顿 PIC 算法中的电荷守恒建立了全面的理论基础。对于拉格朗日变分 PIC 方法，我们应用 Noether 的第二定理来证明规范对称性会产生局部电荷守恒定律作为壳外恒等式。对于哈密顿分裂方法，我们表明动量图建立了它们的电荷守恒定律。我们定义了一类新的算法——规范兼容分裂方法——即使在时间离散化之后，它也能准确地保留与哈密顿系统的规范对称性相关的动量图。这类算法为分裂方案提供了优于替代哈密顿积分器的决定性优势。我们应用这种通用技术来设计一部小说，显式、辛、规范兼容的分裂 PIC 方法，其动量图产生精确的局部电荷守恒定律。我们的研究阐明了此类方案的适当初始条件并检查了它们的辛约简。