A hybrid-Vlasov model based on the current advance method for the simulation of collisionless magnetized plasma

We present a numerical scheme for the integration of the Vlasov–Maxwell system of equations for a non-relativistic plasma, in the hybrid approximation, where the Vlasov equation is solved for the ion distribution function and the electrons are treated as a fluid. In the Ohm equation for the electric field, effects of electron inertia have been retained, in order to include the small scale dynamics up to characteristic lengths of the order of the electron skin depth. The low frequency approximation is used by neglecting the time derivative of the electric field, i.e. the displacement current in the Ampere equation.The numerical algorithm consists in coupling the splitting method proposed by Cheng and Knorr in 1976 [C.Z. Cheng, G. Knorr, J. Comput. Phys. 22 (1976) 330–351.] and the current advance method (CAM) introduced by Matthews in 1994 [A.P. Matthews, J. Comput. Phys. 112 (1994) 102–116.] In its present version, the code solves the Vlasov–Maxwell equations in a five-dimensional phase space (2-D in the physical space and 3-D in the velocity space) and it is implemented in a parallel version to exploit the computational power of the modern massively parallel supercomputers. The structure of the algorithm and the coupling between the splitting method and the CAM method (extended to the hybrid case) is discussed in detail. Furthermore, in order to test the hybrid-Vlasov code, the numerical results on propagation and damping of linear ion-acoustic modes and time evolution of linear elliptically polarized Alfvén waves (including the so-called whistler regime) are compared to the analytical solutions. Finally, the numerical results of the hybrid-Vlasov code on the parametric instability of Alfvén waves are compared with those obtained using a two-fluid approach.