# A semi-implicit, energy- and charge-conserving particle-in-cell algorithm for the relativistic Vlasov-Maxwell equations

## TL;DR

This paper introduces a semi-implicit particle-in-cell algorithm that conserves energy and charge exactly while maintaining accurate light-wave dispersion, improving simulations of relativistic plasma phenomena.

## Contribution

It combines leap-frog and Crank-Nicolson methods to create an energy- and charge-conserving PIC algorithm with accurate light-wave dispersion properties.

## Key findings

- Exact energy and charge conservation achieved.
- Preserves light-wave dispersion properties.
- Validated with relativistic plasma instability simulations.

## Abstract

Conventional explicit electromagnetic particle-in-cell (PIC) algorithms do not conserve discrete energy exactly. Time-centered fully implicit PIC algorithms can conserve discrete energy exactly, but may introduce large dispersion errors in the light-wave modes. This can lead to intolerable simulation errors where accurate light propagation is needed (e.g. in laser-plasma interactions). In this study, we selectively combine the leap-frog and Crank-Nicolson methods to produce an exactly energy- and charge-conserving relativistic electromagnetic PIC algorithm. Specifically, we employ the leap-frog method for Maxwell's equations, and the Crank-Nicolson method for the particle equations. The semi-implicit algorithm admits exact global energy conservation, exact local charge conservation, and preserves the dispersion properties of the leap-frog method for the light wave. The algorithm employs a new particle pusher designed to maximize efficiency and minimize wall-clock-time impact vs. the explicit alternative. It has been implemented in a code named iVPIC, based on the Los Alamos National Laboratory VPIC code (\url{https://github.com/losalamos/vpic}). We present numerical results that demonstrate the properties of the scheme with sample test problems: relativistic two-stream instability, Weibel instability, and laser-plasma instabilities.