A First-Order Reduced Model for a Highly Oscillating Differential Equation with Application in Penning Traps

.We derive a reduced first-order model from a two-scale asymptotic expansion in a small parameter in order to approximate the solution of a stiff differential equation. The problem of interest is a multiscale Newton–Lorentz equation modeling the dynamics of a charged particle under the influence of a linear electric field and of a perturbed strong magnetic field. First, we show that in short times, the first-order model provides a much better approximation than the zero-order one, since it contains terms evolving at slow time scales. Then, thanks to the source-free property of the equations, we propose a volume-preserving method using a particular splitting technique to solve numerically the first-order model. Finally, it turns out that the first-order model does not systematically provide a satisfactory approximation in long times. To overcome this issue, we implement a recent strategy based on the Parareal algorithm, in which the first-order approximation is used for the coarse solver. This approach allows one to perform efficient and accurate long-time simulations for any small parameter. Numerical results for two realistic Penning traps are provided to support these statements.Keywordsmultiscale modelNewton–Lorentz equationfirst-order approximationvolume-preserving integratorParareal algorithmspeedupMSC codes34Exx34E1365L04