Structure-preserving vs. standard particle-in-cell methods: The case of an electron hybrid model

Abstract

We applied two numerical methods both belonging to the class of finite element particle-in-cell methods to a four-dimensional (one dimension in real space and three dimensions in velocity space) hybrid plasma model for electrons in a stationary, neutralizing background of ions. Here, the term hybrid means that (energetic) electrons with velocities close to the phase velocities of the model's characteristic waves are treated kinetically, whereas electrons that are much slower than the phase velocity are treated with fluid equations. The two developed numerical schemes are based on standard finite elements on the one hand and on structure-preserving geometric finite elements on the other hand. We tested and compared the schemes in the linear and in the nonlinear stage. We show that the structure-preserving algorithm leads to better results in both stages. This can be related to the fact that the spatial discretization results in a large system of ordinary differential equations that exhibits a noncanonical Hamiltonian structure. To such systems special time integration schemes with good conservation properties can be applied.