Error estimates of some splitting schemes for charged-particle dynamics under strong magnetic field

TL;DR

This paper analyzes error estimates for splitting schemes used in simulating charged-particle dynamics under strong magnetic fields, introducing a novel energy-preserving scheme with uniform error bounds.

Contribution

A new energy-preserving splitting scheme is proposed that maintains accuracy regardless of magnetic field strength, with rigorous error bounds established.

Findings

The scheme has a computational cost independent of magnetic field strength.

Uniform and optimal error bounds are proven for the scheme and related Lie-Trotter schemes.

Numerical experiments confirm the theoretical error and energy behavior.

Abstract

In this work, we consider the error estimates of some splitting schemes for the charged-particle dynamics under a strong magnetic field. We first propose a novel energy-preserving splitting scheme with computational cost per step independent from the strength of the magnetic field. Then under the maximal ordering scaling case, we establish for the scheme and in fact for a class of Lie-Trotter type splitting schemes, a uniform (in the strength of the magnetic field) and optimal error bound in the position and in the velocity parallel to the magnetic field. For the general strong magnetic field case, the modulated Fourier expansions of the exact and the numerical solutions are constructed to obtain a convergence result. Numerical experiments are presented to illustrate the error and energy behaviour of the splitting schemes.