A Structure-Preserving Penalization Method for the Single-species Rosenbluth-Fokker-Planck Equation

TL;DR

This paper introduces a novel structure-preserving penalization scheme for the stiff single-species Rosenbluth-Fokker-Planck equation, ensuring conservation, positivity, and stability in numerical simulations of plasma dynamics.

Contribution

It develops a new discretization method that generalizes Chang-Cooper, incorporates an isotropic penalization operator, and uses adaptive timestepping to improve numerical stability and physical property preservation.

Findings

Conserves mass, momentum, and energy exactly.

Unconditionally stable and positivity-preserving scheme.

Successfully applied to complex anisotropic diffusion examples.

Abstract

The Rosenbluth-Fokker-Planck (RFP) equation describes Coulomb collisional dynamics within and across species in plasmas. It belongs to the broader class of anisotropic-diffusion-advection equations, whose numerical approximation is highly-nontrivial due to its nonlinearity, stiffness, and structural properties such as conservation and entropy dissipation (hence with the Maxwellian distribution as the equilibrium state). In this paper, we propose a structure-preserving penalization scheme for the stiff, single-species RFP equation. The scheme features three novel components: 1) a novel generalization of the well-known Chang-Cooper discretization for the RFP equation that is equilibrium-preserving and enables positivity while preserving mass, momentum, and energy; 2) an easy-to-invert isotropic variable-coefficient penalization operator to deal with the temporal stiffness without resorting to a fully implicit scheme, borrowing ideas from explicit-implicit-null (EIN) methods, and 3) an adaptive timestepping strategy that preserves the positivity of the full penalized scheme. The resulting scheme conserves mass, momentum, and energy strictly, is unconditionally stable, and robustly positivity preserving. The scheme is demonstrated with linear and nonlinear anisotropic diffusion examples of increasing complexity, including several single-species RFP examples.