Hybrid modeling of plasmas

Here we present the mathematical and numerical details of a general hybrid model for plasmas. All grid quantities are stored at cell centers on the grid. The most common discretization of the fields in PIC solvers is to have the electric and magnetic fields staggered, introduced by Yee. This automatically ensures that div(B)=0, down to round-off errors. Here we instead present a cell centered discretization of the magnetic field. That the standard cell centered second order stencil for rot(E) in Faraday's law will preserve div(B)=0 was noted by Toth. The advantage of a cell centered discretization is ease of implementation, and the possibility to use available solvers that only handle cell centered variables. We also show that the proposed method has very good energy conservation for a simple test problem in three dimensions, when compared to a commonly used algorithm.