## Section: New Results

### Envelope following methods

One difficulty when solving problems in plasma physics is the behaviour at several scales in time and space of the solutions of equations. For example, central equations in this domain of application are highly oscillatory in time. The multiscale aspect makes the models difficult to tackle when we aim at avoiding a high computational cost. A solution to this problem is to solve the models by designing adapted numerical methods with a low computational cost and which are able to deal efficiently with rapid and slow scales in time. In this direction, we worked on envelope following methods, which have been efficiently applied in the community of oscillators in RF circuits. The method has (at least) two variants: in a first place, it is based on the concept of using extra variables to represent the changing rapid period and the cumulative effect of changing periods and then, use of Newton iterations allows to find these unknowns. In a second place, we adopt a similar strategy except that the rapid period is not an extra variable but a direct outcome of the numerical integration by the use of the Poincaré map. We implemented and tested both approaches for equations of interest in plasma physics and we observed that these methods didn't perform accurate results.