High-Order Low-Order (HOLO) Nonlinear Convergence Accelerator for the Rosenbluth-Fokker-Planck Collision Operator

A fully implicit solver strategy is required for the nonlinear integral-differential Fokker-Planck equation when ε^-1=Δt/τ_col>>1, owing for the need to simultaneously, ensure exact conservation properties (mass, momentum, and energy), as well as the correct asymptotic convergence to the Maxwellian [1]. However, developing an effective solver is challenging, due to the integral-differential nature of the formulation (via the so-called Rosenbluth potentials), leading to a dense linear system. To effectively deal with these numerical challenges, we explore a multiscale iterative strategy based on a HOLO convergence accelerator scheme [2]. HOLO employs an LO (fluid) moment system to accelerate the convergence of the HO (kinetic) system. The LO quantities (n, u, T) inform the Maxwellian component of the potentials, which also contain a perturbation term [of O(ε)<<1] computed from the HO solution. This reformulation shifts the non-local contributions through the potentials from the HO system to the LO one, where they can be dealt with efficiently. Numerical experiments in challenging applications (ICF implosions) demonstrate the enabling capabilities of the HOLO scheme. [1] W.T. Taitano et al., JCP, 297, 357-380 (2015). [2] L. Chacón et al., JCP, 330, 21-45 (2017).