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.

翻译：本研究提出了一种最优隐式算法，旨在精确求解多物种非线性0D-2V轴对称Fokker-Planck-Rosenbluth（FRP）碰撞方程，同时保持质量、动量和能量守恒。我们采用基于勒让德多项式展开的非线性Shkarofsky FRP（FRPS）碰撞算子公式。我们无网格方法的关键在于：角向采用勒让德多项式展开，速度轴向采用King函数（Eq.\EQ{King}）展开。勒让德多项式展开将呈指数级收敛，而King方法作为一种矩收敛算法，能够在离散形式下高精度地保证守恒性。此外，我们还采用了流形后步投影来精确强制碰撞算子的对称性。通过求解多种非平衡构型下的若干典型问题，我们证明了该算法具有优越的性能和高精度。