An Asymptotic Preserving Maxwell Solver Resulting in the Darwin Limit of Electrodynamics

TL;DR

This paper introduces an asymptotic preserving Maxwell solver using a MOL$^T$ formulation and fast summation, efficiently capturing the Darwin limit in plasma simulations with multi-scale electromagnetic phenomena.

Contribution

The paper presents a novel MOL$^T$ based Maxwell solver with $O(N ext{log}N)$ complexity that asymptotically recovers the Darwin limit in plasma physics.

Findings

Efficient $O(N ext{log}N)$ computational complexity.

The scheme accurately captures the Darwin limit.

The method is asymptotic preserving under scale separation.

Abstract

In plasma simulations, where the speed of light divided by a characteristic length is at a much higher frequency than other relevant parameters in the underlying system, such as the plasma frequency, implicit methods begin to play an important role in generating efficient solutions in these multi-scale problems. Under conditions of scale separation, one can rescale Maxwell's equations in such a way as to give a magneto static limit known as the Darwin approximation of electromagnetics. In this work, we present a new approach to solve Maxwell's equations based on a Method of Lines Transpose (MOL$^T$) formulation, combined with a fast summation method with computational complexity $O(N\log{N})$, where $N$ is the number of grid points (particles). Under appropriate scaling, we show that the proposed schemes result in asymptotic preserving methods that can recover the Darwin limit of electrodynamics.