Uniformly higher order accurate schemes for dynamics of charged particles under fast oscillating magnetic fields

TL;DR

This paper develops and analyzes uniformly accurate numerical schemes for simulating charged particle dynamics under fast oscillating magnetic fields, effectively handling multiscale oscillations and preserving energy in the averaged limit.

Contribution

It introduces new time integrators that are uniformly accurate regardless of the oscillation scale and extend to energy-preserving schemes for the averaged model.

Findings

Proved uniform accuracy of the proposed schemes.

Constructed energy-preserving extensions for the averaged model.

Numerical results demonstrate the effectiveness of the methods.

Abstract

This work deals with the numerical approximation of plasmas which are confined by the effect of a fast oscillating magnetic field (see \cite{Bostan2012}) in the Vlasov model. The presence of this magnetic field induces oscillations (in time) to the solution of the characteristic equations. Due to its multiscale character, a standard time discretization would lead to an inefficient solver. In this work, time integrators are derived and analyzed for a class of highly oscillatory differential systems. We prove the uniform accuracy property of these time integrators, meaning that the accuracy does not depend on the small parameter $\varepsilon$. Moreover, we construct an extension of the scheme which degenerates towards an energy preserving numerical scheme for the averaged model, when $\varepsilon\to 0$. Several numerical results illustrate the capabilities of the method.