A Computational scheme for Quasilinear Diffusion for magnetized fast electrons in a mean field of quasi-particle wave packets.

The space averaged quasi-linear diffusion model for magnetized fast electrons in momentum space results from stimulated emission and absorption of waves packets via wave-particle resonances.  The model consists in solving the dynamics of a coupled system of classical kinetic diffusion processes described by the balance equations for electron probability density functions (electron pdf) coupled to the time dynamics on spectral energy waves (quasi-particles) in a quantum process of their resonant interaction. Such description results in a ‘mean field’ model where diffusion coefficients are determined by the local spectral energy density of excited waves whose perturbations depend on flux averages of the electron pdf.

These non-linear coupled system of equations is approximated by an implicit iteration scheme. Given pdf at a previous time step, the wave spectral density is solved implicitly; and similarly, the pdf is updated with an implicit scheme using wave spectral density at the previous step, under a resonance condition which couples the spectral energy to the electron pdf momentum variable. This resonance condition is modeled as a singular measure (Dirac delta) concentrated on the dispersion relation computed by a local level set approximation on each element.

Our approach not only ensures a semi-discrete exact conservation of mass, preserves total momentum and total energy up to the accuracy of basis function approximation, but also identifies resonant element pairs that can be utilized to optimize the computation domain. Numerical simulations are presented showing a strong anisotropic diffusion.