Axisymmetric Gyrokinetic Simulation of ASDEX-Upgrade Scrape-off Layer Using a Conservative Implicit BGK Collision Operator

TL;DR

The paper addresses the time-step bottleneck in high-collisionality SOL gyrokinetic simulations by introducing an implicit, conservative BGK collision operator within the DG-based full-f gyrokinetic solver Gkeyll. The method preserves density, momentum, and energy through a moment-correcting Maxwellian projection and supports multispecies collisions via Greene's cross-Maxwellian formulation, all within an IMEX time-stepping framework. Benchmarks confirm moment conservation and accurate relaxation behavior, while axisymmetric ASDEX-Upgrade SOL simulations demonstrate that the implicit BGK achieves significant speedups (up to about 56×) and robust convergence at moderate resolutions, with results in good agreement with the more accurate but costlier LBD operator. This approach offers a practical path to efficient, kinetic SOL modeling and lays groundwork for future 3D turbulence studies and shoulder-formation investigations in reactor-relevant plasmas.

Abstract

Collisions play an important role in turbulence and transport of fusion plasmas. For kinetic simulations, as the collisionality increases in the domain of interest, the size of the time step to resolve the collisional physics can become overly restrictive in an explicit time integration scheme, leading to high computational cost. With the aim of overcoming such restriction, we have implemented an implicit Bhatnagar-Gross-Krook (BGK) collision operator for use in the discontinuous Galerkin (DG) full-f gyrokinetic solver within the Gkeyll framework, which, when combined with Gkeyll's traditional explicit time integrator for collisionless advection, can significantly increase the time step in gyrokinetic simulations of highly collisional regimes. To ensure conservation of density, momentum, and energy, we utilize an iterative scheme to correct the discretized approximation to the equilibrium Maxwellian distribution to which the BGK collision operator relaxes. We have further generalized the BGK infrastructure, both the implicit scheme and the correction routine, to handle cross species collisions. This improved implicit and conservative BGK operator is benchmarked against the more accurate but more computationally expensive Lenard-Bernstein-Dougherty (LBD) operator which has been utilized in prior studies with Gkeyll. The implicit BGK operator enables 2D axisymmetric simulations of the ASDEX-Upgrade scrape-off layer to run 56 times faster to completion than the simulations with the LBD operator, because the BGK operator is more robust and converges at a lower resolution than is required by the LBD operator. Additionally, in this more collisional limit, we demonstrate that the results of our simulations utilizing the implicit BGK operator agreed well with simulations utilizing the more computationally expensive LBD operator.

Figures (8)

• Figure 1: Relaxation of a bump-on-tail distribution. The distribution along the parallel velocity is shown at $\mu=0.0$ at four time slices.
• Figure 2: The maximum relative errors of parallel velocity and error of thermal velocity squared in configuration space drop to near the machine precision within $9$ iterations.
• Figure 3: 1D Sod shock propagation. (a) Comparison of solutions to Euler equations and to Gyrokinetic equation with implicit BGK. (b) Comparison of solutions to Gyrokinetic equation with implicit BGK and explicit LBD. The dotted lines represent initial density.
• Figure 4: Mean parallel velocity and temperature relaxation of an anisotropic deuterium plasma. The solid curves represent results of the test with the implicit BGK, and the dotted curves represent results of the test with the explicit LBD.
• Figure 5: Poloidal projection of the flux tube. The black dots represent the nodes; the blue lines represent the boundary of the cells. The divertor plates are horizontally placed for convenience. This plot only shows 4 cells in the x direction for clarity.