Iterative and Analytic Poisson Solving Schemes for the Rosenbluth Potentials of the Fokker-Planck Collision Operator

Abstract

Continuum-kinetic models represent collisional plasma dynamics through distribution functions in phase space, capturing microscopic behavior statistically while avoiding the numerical noise of individual particle-tracking methods. The choice ofcollision operator plays a vital role here in determining what collisional mechanisms are included in the model. The nonlinear Rosenbluth/Fokker-Planck collision operator (FPO) provides a highly accurate treatment of small-angle Coulomb collisions, including a velocity-dependent effective collision frequency that correctly models the high-energy tails of the distribution. Its numerical implementation requires solving two coupled Poisson equations for the Rosenbluth potentials H and G over a three-dimensional velocity space per timestep and species, resulting in a highly-dimensional formulation that is both computationally expensive and difficult to scale, making predictive simulation of collisions challenging within reasonable wall-clock time. In this work, we employ the GKEYLL plasma simulation framework, which applies a discontinuous Galerkin (DG) finite element discretization to the full Vlasov-Maxwell-Fokker-Planck system, and focus specifically on extending the current DG-FPO implementation (MA-DG-FPO): it solves for the Rosenbluth potentials using a Maxwellian approximation, which fails to accurately simulate collisions for non-Maxwellian distributions arising in space, astrophysical, and laboratory plasmas. We study novel numerical and analytical techniques to solve for the Rosenbluth potentials, tested for accuracy, scalability, performance, and integrability with the existing FPO framework. We establish Algebraic Multigrid (AMG), implemented via the HYPRE preconditioner library, as a competitive benchmark, and develop a second-order Richardson iteration scheme which is tested against it, with Richardson as the more suitable numerical engine for our problem. To complement this, an analytic approach is studied using multipole expansions with symmetric trace-free (STF) tensors as a boundary correction to the far-field velocity domain, while Richardson addresses the potential solves in the interior. Together, they form the Extended FPO (X-DG-FPO), integrated into the GKEYLL Vlasov layer to accurately and efficiently simulate collisions across timesteps. The performance and accuracy of THE X-DG-FPO is tested against established collision operators, namely the Bhatnagar-Gross-Krook operator (BGK), the Dougherty- Lenard-Bernstein operator (LBO), and the MA-DG-FPO, using relaxation and thermalization tests on non-Maxwellian distributions. The headline result is that the MA-DG-FPO produces severe spurious oscillations for distributions with heavy tails, such as kappa distributions, while the X-DG-FPO eliminates them completely — demonstrating the importance of accurate handling of the velocity-dependent collision frequency for non-Maxwellian populations. This work motivates continued development towards an efficient, fully conservative, multispecies generalization of the non-linear Rosenbluth/Fokker-Planck collision operator within the continuum-kinetic framework.