# A low-rank projector-splitting integrator for the Vlasov--Maxwell equations with divergence correction

## TL;DR

This paper introduces a low-rank integrator for the Vlasov--Maxwell equations that reduces computational cost by approximating the solution within a low-rank manifold, effectively capturing plasma dynamics and enforcing divergence constraints.

## Contribution

It develops a dynamic low-rank integrator for the Vlasov--Maxwell system with divergence correction, enabling efficient and accurate plasma simulations.

## Key findings

- The scheme accurately captures key plasma features in numerical tests.
- It maintains divergence constraints with machine precision.
- The low-rank approach reduces computational complexity.

## Abstract

The Vlasov--Maxwell equations are used for the kinetic description of magnetized plasmas. As they are posed in an up to 3+3 dimensional phase space, solving this problem is extremely expensive from a computational point of view. In this paper, we exploit the low-rank structure in the solution of the Vlasov equation. More specifically, we consider the Vlasov--Maxwell system and propose a dynamic low-rank integrator. The key idea is to approximate the dynamics of the system by constraining it to a low-rank manifold. This is accomplished by a projection onto the tangent space. There, the dynamics is represented by the low-rank factors, which are determined by solving lower-dimensional partial differential equations. The proposed scheme performs well in numerical experiments and succeeds in capturing the main features of the plasma dynamics. We demonstrate this good behavior for a range of test problems. The coupling of the Vlasov equation with the Maxwell system, however, introduces additional challenges. In particular, the divergence of the electric field resulting from Maxwell's equations is not consistent with the charge density computed from the Vlasov equation. We propose a correction based on Lagrange multipliers which enforces Gauss' law up to machine precision.