A pseudospectral implicit particle-in-cell method with exact energy and charge conservation

The standard particle-in-cell (PIC) method employs explicit finite-difference (FD) schemes, such as the leap-frog scheme, for spatial and temporal integrations. In this study, we introduce a novel approach by employing a pseudospectral method for solving the Poisson equation and adopting a fully implicit time integration technique to achieve exact energy conservation. The pseudospectral field solver offers the advantage of spectral accuracy in obtaining field solutions. Previous investigations on implicit time integration of PIC FD equations have demonstrated the capability of enforcing precise energy exchange between the field and particles, resulting in energy-conserving schemes. We prove that the exact energy conservation property can be extended to the pseudospectral scheme. Additionally, we provide a solution to ensure a pseudospectral charge continuity equation. To validate our methodology, we implement the new scheme in a 2D electrostatic PIC code and present numerical examples that confirm the theoretical findings.