A conservative, implicit solver for 0D-2V multi-species nonlinear Fokker-Planck collision equations

In this study, we present an optimal implicit algorithm designed to accurately solve the multi-species nonlinear 0D-2V axisymmetric Fokker-Planck-Rosenbluth (FRP) collision equation while preserving mass, momentum, and energy. We rely on the nonlinear Shkarofsky's formula of FRP (FRPS) collision operator in terms of Legendre polynomial expansions. The key to our meshfree approach is the adoption of the Legendre polynomial expansion for the angular direction and King function (Eq.\EQ{King}) expansion for the velocity axis direction. The Legendre polynomial expansion will converge exponentially and the King method, a moment convergence algorithm, could ensure the conservation with high precision in discrete form. Additionally, a post-step projection to manifolds is employed to exactly enforce symmetries of the collision operators. Through solving several typical problems across various nonequilibrium configurations, we demonstrate the superior performance and high accuracy of our algorithm.