Structure-preserving strategy for conservative simulation of the relativistic nonlinear Landau-Fokker-Planck equation

Mathematical symmetries of the Beliaev-Budker kernel are the most important structure of the relativistic Landau-Fokker-Planck equation. In most numerical simulations, however, one of the symmetries is not preserved in the discrete level resulting in a violation of the energy conservation. Recently, we proposed a charge-momentum-energy-conserving relativistic Vlasov-Maxwell scheme by preserving mathematical formulas in discrete form, and here we apply the concept to the relativistic Landau-Fokker-Planck equation. Through a numerical experiment of relativistic collisional relaxation, a mass-momentum-energy-conserving simulation has been demonstrated without any artificial constraints.

Figure 1

Time development of distribution function initialized by double Maxwell-Jüttner distribution.

Figure 2

Conservation errors for mass, momentum, and energy in a thermal-equilibration verification. (a) Structure-preserving discretization (proposed), (b) Intuitive discretization (conventional), and (c) Comparison of energy conservation property.

Figure 3

Verification of H-theorem. Caution: the proposed scheme does not ensure the H-theorem mathematically.