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

This work deals with the numerical approximation of plasmas that are confined by the effect of a fast oscillating magnetic field [Bostan, M. (2012), Transport of charged particles under fast oscillating magnetic fields. SIAM J. Math. Anal., 44, 1415–1447] 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.