Semi-Implicit Continuum Kinetic Modeling of Weakly Collisional Parallel Transport in a Magnetic Mirror

TL;DR

The paper advances continuum kinetic modeling of weakly collisional parallel transport in magnetic mirrors by integrating implicit-explicit time stepping with Jacobian-free Newton–Krylov solvers and AIR-based multigrid preconditioning in COGENT. It validates a bounce-averaged BASM model against analytic loss rates and explores high-energy beam relaxation under FP and LBD collisions, then demonstrates the feasibility and performance of fully kinetic 1D–2V simulations with a self-consistent E-field. The results show time-step and computational speedups up to several thousand-fold over explicit schemes, while highlighting the benefits of fifth-order Vlasov discretization and the limitations of fixed-field preconditioners when collective modes are active. The study lays groundwork for integrated, reactor-scale mirror modeling by outlining future extensions to radial nonuniformities, kinetic electrons, and more complex magnetic geometries.

Abstract

We present implicit-explicit (IMEX) kinetic simulations of weakly collisional parallel plasma transport in magnetic mirror configurations using the continuum code \textsc{COGENT}. The numerical scheme employs a Jacobian-free Newton--Krylov method with algebraic multigrid preconditioning to overcome the severe time-step limitations imposed by strong mirror forces in fully explicit schemes. Applied to parameters relevant to the WHAM mirror experiment, the IMEX approach enables time steps up to $2.5 \times 10^4$ times larger than those permitted by explicit methods, resulting in a 2500x speedup in 1D--2V simulations of parallel transport with kinetic ions and Boltzmann electrons. Additionally, a reduced bounce-averaged model for a square mirror is implemented to support the computationally intensive fully kinetic simulations. The bounce-averaged formulation is used to evaluate the numerical convergence of the velocity-space discretization algorithms and to assess the role of the collision model by comparing simulations employing the nonlinear Fokker--Planck and the simplified Lenard--Bernstein--Dougherty collision operators.

Figures (10)

• Figure 1: Bounce-averaged (0D-2V) simulations. (a) Schematic of the bounce-averaged square-mirror (BASM) model. The blue curve illustrates the loss-cone boundary for $\Phi_m=0$. For the case where the loss-cone boundary intersects a cell $(i,j)$, the shaded area, $dS^{loss}_{i, j}$, is used in Eq. (\ref{['SinkTermDiscr']}). (b) Collisional losses of the electrostatically-confined electron species for $R_m = 32$, $n=e19\m^{-3}$, and $T_e = 940\eV$. The simulation results for the electron confinement time, $\tau_{c,e} = n_e |dn_e/dt |^{-1}$, obtained using the FP (blue) and LBD (orange) e-e collision models are compared with the analytical prediction in Eq. (\ref{['LossesAnalytic']}). (c) Time history of the electron confinement time obtained using the full nonlinear FP model and the fixed-background FP model for $e\Phi_m/T_e =6$.
• Figure 2: High-energy beam relaxation obtained from 0D–2V BASM model simulations. The numerical results using the Fokker–Planck collision model [frames (a)–(c)] are compared with corresponding simulations using the LBD collision model [frames (d)–(f)]. Plotted is the normalized ion distribution function, ${\hat{f}}_i = (2T_0/m_i)^{3/2} \pi f_i /n_0$.
• Figure 3: Comparison of the Fokker-Planck (solid curves) and LBD (dashed curves) collisional models in the 0D-2V BASM-model simulations. Shown are (a) time history of the ion density; and (b) the $\mathrm{v}_{\parallel}$-lineouts extracted from the normalized ion distribution function, ${\hat{f}}_i = (2T_0/m_i)^{3/2} \pi f_i /n_0$, for $\widehat{\mu} = 1$ and $\widehat{\mu} = 6$, at 1000 ms.
• Figure 4: Numerical convergence of the 0D-2V BASM model solution at 84. Shown are (a) the $\mathrm{v}_{\parallel}$-lineouts extracted from the normalized ion distribution function, ${\hat{f}}_i = (2T_0/m_i)^{3/2} \pi f_i /n_0$, for $\widehat{\mu} = 6$; and (b) Richardson extrapolation analysis of the numerical convergence for different values of the trap length, $L_\parallel$. Grid levels $m=(2,3,4,5)$ correspond to the successful refinements of the coarse grid $(N_{\mathrm{v}_{\parallel}},N_\mu)=(128,192)$ by factors of 2, 4, 8, and 16 in both velocity dimensions, respectively.
• Figure 5: Magnetic geometry used in 1D-2V fully kinetic simulations.