An Envelope Tracking Approach for Particle in Cell Simulations

The state of the art in electromagnetic finite element particle-in-cell (EM-FEMPIC) has advanced significantly in the last few years; these have included understanding function spaces that must be used to represent sources and fields consistently, and how currents should be evolved in space and time. In concert, these achieve satisfaction of Gauss’ laws. All of these were restricted to conditionally stable explicit time stepping. More recently, there have been advances to the state of art: it is now possible to use an implicit EM-FEMPIC method while satisfying Gauss’ law to machine precision. This enables choosing time step sizes dictated by physics as opposed to geometry. In this article, we take this a step further. For devices characterized by a narrowband high-frequency response, choosing a timestep size based on the highest frequency of interest is considerably expensive. In this article, we use methods derived from envelope tracking to construct an EM-FEMPIC method that analytically provides for the high-frequency oscillations of the system, allowing for analysis at considerable coarser timestep sizes even in the presence of non-linear effects from active media such as plasmas. Consequentially, we demonstrate how the point-wise metric used for measuring satisfaction of Gauss’ Laws breaks down when prescribing analytical fast fields and provide a thorough analysis of how charge conservation can be measured. Through a number of examples, we demonstrate that the proposed approach retains the accuracy of the regular scheme while requiring far fewer timesteps in systems where the stepsize is strongly decided by the bandwidth of the fields.