A numerical scheme for the integration of the Vlasov–Poisson system of equations, in the magnetized case

We present a numerical algorithm for the solution of the Vlasov–Poisson system of equations, in the magnetized case. The numerical integration is performed using the well-known “splitting” method in the electrostatic approximation, coupled with a finite difference upwind scheme; finally the algorithm provides second order accuracy in space and time. The cylindrical geometry is used in the velocity space, in order to describe the rotation of the particles around the direction of the external uniform magnetic field. polar coordinates, the integration of the Vlasov equation is very simplified in the velocity space with respect to the cartesian geometry, because the rotation in the velocity cartesian space corresponds to a translation along the azimuthal angle in the cylindrical reference frame. The scheme is intrinsically symplectic and significatively simpler to implement, with respect to a cartesian one. The numerical integration is shown in detail and several conservation tests are presented, in order to control the numerical accuracy of the code and the time evolution of the entropy, strictly related to the filamentation problem for a kinetic model, is discussed.