Choosing good modelling method for solving Boltzmann equation

I’m writing a solver for Boltzmann Equations (BE) including a force term in rarefied plasma, for my PhD. The aim is to see if an instability occurs inside an electric streamer (theoretically it should, basing on the analysis of the dispersion relation).

I used finite differences method with MPI parallelization, but the model is unstable before the instability shows. Unfortunately, the methods that I used are slow. I’m thinking of changing the paradigm and turning to Particle-In-Cell (PIC) model. Anyway there still are other methods like Lattice Boltzmann Method (LBM) that might be used without reducing particles to macro-particles as in PIC while improving the speed.

Have you got any recommendations for me where to turn with my investigations? Maybe it would be useful to go few steps higher, trying to see if hydrodynamic approach would do? However, the analysis linked is based purely on kinetic theory. I note that I got the access to huge computing resources.

The set of BE being solved is:

On the right hand side there is a BGK collision term ( for $alpha$ type of particles, $Phi_alpha$ is a Maxwellian distribution function shifted by the velocity of a streamer), r is displacement vector, v is velocity vector, E is a constant field that acts on the whole fluid, but E’ and B’ are locally-variable fields that are related to the small wavy plasma disturbance calculated from Maxwell equations in each step.