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

TL;DR

This work tackles the challenging nonlinear 0D-2V axisymmetric Fokker-Planck-Rosenbluth collision problem for multi-species plasmas while guaranteeing mass, momentum and energy conservation and adherence to the H-theorem. It introduces a novel implicit solver that combines Legendre polynomial expansions for angular dependence with a King-function-based expansion for the speed coordinate, and employs a post-step manifold projection to enforce exact conservation. A time-block implicit scheme with nonlinear optimization of King parameters (via L01jd2nh, L01jd2, L01jd2NK schemes) and Romberg-integrated moments yields high accuracy for weakly anisotropic plasmas, even under large mass and temperature disparities. Numerical tests across two-, three-species configurations demonstrate second-order time convergence, strong conservation properties, and efficient handling of multi-scale dynamics, illustrating the method’s potential for robust kinetic plasma simulations. The approach is poised to extend toward fully adaptive schemes for broader anisotropy and more complex collision settings, expanding the applicability to fusion-relevant and space plasmas.

Abstract

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.

Figures (26)

• Figure 1: Illustration of the velocity distribution functions in speed coordinate for disparate thermal velocities in a subsonic plasma system.
• Figure 2: Convergence of SHE for drift-Maxwellian distribution: $l_M$ as a function of $\hat{u}_a$ when $Atol_{df}=10^{-10}$.
• Figure 3: Discretization of speed coordinate: field nodes and the Chebyshev grids in the subinterval.
• Figure 4: Two-species temperature equilibration with fixed timestep, $\Delta t=2^{-5}$: Temperatures (upper) and relative disparity of temperatures (lower), $\Delta T = \left |T_a^k - T_b^k \right| / \left (T_a^k + T_b^k \right)$ as functions of time. The color lines represent the numerical solution while the gray and black lines represent the Braginskii's values.
• Figure 5: Two-species temperature equilibration with fixed timestep, $\Delta t$: Demonstration of second-order convergence of the time discretization scheme.