# An \mathcal{O}(N) Maxwell solver with improved numerical dispersion properties

## TL;DR

This paper introduces an (N) Maxwell solver based on finite element methods that enhances numerical dispersion properties in relativistic PIC simulations, suitable for parallel computing and reducing instabilities.

## Contribution

It presents a novel finite element Maxwell solver with (N) complexity that improves dispersion accuracy and suppresses numerical instabilities in high-speed plasma simulations.

## Key findings

- Reduces Numerical Cherenkov instability growth rate
- Achieves (N) computational complexity
- Enhances dispersion properties in relativistic PIC simulations

## Abstract

A Maxwell solver derived from finite element method with \mathcal{O}(N) computing cost is developed to improve the numerical dispersion properties in relativistic particle-in-cell (PIC) simulations. The correction of the dispersion relation of the electromagnetic wave is achieved using the neighboring cells via an iteration scheme without decomposing into Fourier modes. The local nature of the communication is ideally suited to massively parallel computer architectures. This Maxwell solver constrains the Numerical Cherenkov instability (NCI) for the ultra-relativistic drifting pair plasma in x direction to large wave vectors for two dimensional grid. The growth rate of NCI is suppressed by using the low pass filtering.