Abstract

Combining the Lorentz force equations with Newton's law gives a secondorder dierential equation in space for the motion of a charged particle in a magneticeld. The most natural and widely used numerical discretization is the Boris algorithm,which is explicit, symmetric, volume-preserving, and of order 2.In a rst part we discuss geometric properties (long-time behaviour, and in particularnear energy conservation) of the Boris algorithm. This is achieved by applying standardbackward error analysis. Near energy conservation can be obtained also in situations,where the method is not symplectic.In a second part we consider the motion of a charged particle in a strong magnetic eld.Backward error analysis can no longer be applied, and the accuracy (order 2) breaksdown. To improve accuracy we modify the Boris algorithm in the spirit of exponentialintegrators. Theoretical estimates are obtained with the help of modulated Fourierexpansions of the exact and numerical solutions.This talk is based on joint work with Christian Lubich, and Bin Wang.Related publications (2017{2019) can be downloaded from