Hybrid particle-spectral method for kinetic plasma simulations

A hybrid model for numerical solutions of the Vlasov-Maxwell equations is presented which blends spectral and particle approaches.

The model splits the distribution function in the velocity space for plasma species in both spectral and particle representations with the goal of taking advantage of both.

The spectral approach leverages asymmetrically-weighted Hermite basis, and the particle-in-cell (PIC) method is used for the particle method. Configuration phase space is decomposed with the Fourier method which is well suited for periodic problems. The method's conservation properties for mass, momentum, and energy are derived. It is shown that the coupling error between spectral and particle representation is absent in the semi-discrete setting (not taking into account time discretization). Finally, numerical test cases are presented simulating a weak electron beam interaction with plasma, leading to beam-plasma instability. The initially localized electron beam evolved into a highly non-equilibrium distribution function in the velocity space. A small growth rate and resonance nature of instability makes it difficult to obtain accurate solutions for purely particle methods due to present noise which falls as ∽1/√N_p with a number of particles. At the same time, purely spectral methods may require a large number of modes to capture the highly non-equilibrium state of the evolved beam. We show that the hybrid method is well suited for such problems, it reproduces the linear stage with sufficient accuracy as well as nonlinear dynamics with a highly non-equilibrium distribution function.