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.
We study the structure-preserving discretizations of a hybrid model with kinetic ions and mass-less electrons. Different from most existing works in the literature, we conduct the discretizations based on two equivalent formulations with vector potentials in different gauges, and the distribution functions depend on canonical momentum (not velocity). Particle-in-cell methods are used for the distribution functions, and vector potentials are discretized by finite element methods in the framework of finite element exterior calculus. Splitting methods are used for time discretizations. For the first formulation, filters are used to reduce the noises from particles and are shown to improve the numerical results significantly. The schemes of the second formulation show good stability and accuracy because of the use of symplectic methods for canonical Hamiltonian systems. Magnetic fields obtained from the vector potentials are divergence-free naturally. Some numerical experiments are conducted to validate and compare the two discretizations.
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