Semi-discretization and full-discretization with optimal accuracy for charged-particle dynamics in a strong nonuniform magnetic field

TL;DR

This paper develops semi- and full-discretization methods for charged-particle dynamics in strong nonuniform magnetic fields, achieving optimal accuracy that improves with magnetic field strength, surpassing traditional uniform accuracy methods.

Contribution

It introduces a novel strategy for discretizing charged-particle dynamics with optimal accuracy, incorporating reformulations and two-scale exponential integrators, and extends to a new class of uniformly accurate methods in three dimensions.

Findings

Achieves optimal accuracy that improves with magnetic field strength.

Develops a new class of simple, uniformly accurate methods for 3D CPD.

Numerical tests confirm theoretical accuracy improvements.

Abstract

The aim of this paper is to formulate and analyze numerical discretizations of charged-particle dynamics (CPD) in a strong nonuniform magnetic field. A strategy is firstly performed for the two dimensional CPD to construct the semi-discretization and full-discretization which have optimal accuracy. This accuracy is improved in the position and in the velocity when the strength of the magnetic field becomes stronger. This is a better feature than the usual so called "uniformly accurate methods". To obtain this refined accuracy, some reformulations of the problem and two-scale exponential integrators are incorporated, and the optimal accuracy is derived from this new procedure. Then based on the strategy given for the two dimensional case, a new class of uniformly accurate methods with simple scheme is formulated for the three dimensional CPD in maximal ordering case. All the theoretical results of the accuracy are numerically illustrated by some numerical tests.