Continuous-stage symplectic adapted exponential methods for charged-particle dynamics with arbitrary electromagnetic fields

TL;DR

This paper develops new symplectic numerical methods using continuous-stage and exponential integrators for charged-particle dynamics in arbitrary electromagnetic fields, achieving uniform accuracy and improved stability.

Contribution

It introduces a general class of symplectic methods for CPD with arbitrary fields, including practical schemes up to order four and extensions to non-homogeneous magnetic fields.

Findings

Second order scheme has uniform accuracy in position relative to magnetic field strength

Proposed methods demonstrate high accuracy and stability in numerical experiments

Error estimates confirm the theoretical uniform accuracy of the methods

Abstract

This paper is devoted to the numerical symplectic approximation of the charged-particle dynamics (CPD) with arbitrary electromagnetic fields. By utilizing continuous-stage methods and exponential integrators, a general class of symplectic methods is formulated for CPD under a homogeneous magnetic field. Based on the derived symplectic conditions, two practical symplectic methods up to order four are constructed where the error estimates show that the proposed second order scheme has a uniform accuracy in the position w.r.t. the strength of the magnetic field. Moreover, the symplectic methods are extended to CPD under non-homogeneous magnetic fields and three algorithms are formulated. Rigorous error estimates are investigated for the proposed methods and one method is proved to have a uniform accuracy in the position w.r.t. the strength of the magnetic field. Numerical experiments are provided for CPD under homogeneous and non-homogeneous magnetic fields, and the numerical results support the theoretical analysis and demonstrate the remarkable numerical behavior of our methods.