In this study, we present an optimal implicit algorithm specifically designed to accurately solve the multi-species nonlinear 0D-2V axisymmetric Fokker-Planck-Rosenbluth (FPR) collision equation while preserving mass, momentum, and energy. Our approach relies on the utilization of nonlinear Shkarofsky's formula of FPR (FPRS) collision operator in the spherical-polar coordinate. The key innovation lies in the introduction of a new function named King, with the adoption of the Legendre polynomial expansion for the angular coordinate and King function expansion for the speed coordinate. 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, post-step projection onto manifolds is employed to exactly enforce symmetries of the collision operators. Through solving several typical problems across various nonequilibrium configurations, we demonstrate the high accuracy and superior performance of the presented algorithm for weakly anisotropic plasmas.

翻译：本研究提出了一种最优隐式算法，专门用于精确求解多物种非线性0D-2V轴对称福克-普朗克-罗森布鲁斯碰撞方程，同时保持质量、动量和能量守恒。我们的方法依赖于在球极坐标系中使用非线性Shkarofsky公式的FPR碰撞算子。关键创新在于引入了一个名为King的新函数，并采用勒让德多项式展开处理角坐标，采用King函数展开处理速度坐标。勒让德多项式展开将指数收敛，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，通过后步流形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，我们证明了所提算法对于弱各向异性等离子体具有高精度和优越性能。