  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.
