Conservative velocity mappings for discontinuous Galerkin kinetics

TL;DR

This work introduces a first-of-its-kind DG discretization of velocity space using mapped coordinates to implement nonuniform grids for continuum kinetic models, specifically demonstrated in a long-wavelength gyrokinetic framework. By mapping the physical velocity coordinates $(v_igparallel,)$ to computational coordinates $(,)$ and applying a conservative DG formulation, the authors achieve exact particle and energy conservation for collisionless dynamics and exact conservation of particles, momentum, and energy for collisional operators. The method employs a continuous-in-$x^3$ potential via a projection operator, carefully designed surface fluxes (Lax-Friedrichs style) and analytic, flux-conserving volume terms, enabling stable explicit time integration (SSP3) on GPU-accelerated architectures. Across 1D, 2D, and 3D test cases (HTS mirror, ASDEX-Upgrade SOL, and LAPD turbulence), the approach reproduces standard results while reducing velocity-space DOF by factors up to 6–60 and achieving speed-ups of 22–60x, depending on geometry and parameters. The framework thus offers substantial computational savings for high-dimensional, velocity-structured kinetic simulations and sets the stage for further extensions, such as position-dependent mappings and multi-block velocity grids, to broaden applicability and robustness in fusion-plasma modeling.

Abstract

Continuum computational kinetic plasma models evolve the distribution function of a plasma species $f_s$ on a phase-space grid over time. In many problems of interest the distribution function has limited extent in velocity space; hence, using a uniform, highly refined mesh would be costly and slow. Nonuniform velocity grids can reduce the computational cost by placing more degrees of freedom where $f_s$ is appreciable and fewer where it is not. In this work we introduce a first-of-its kind discontinuous Galerkin approach to nonuniform velocity-space discretization using mapped velocity coordinates. This new method is presented in the context of a gyrokinetic model used to study magnetized plasmas. We create discretizations of collisionless and collisional terms using mappings in a way that exactly conserves particles and energy. Numerical tests of such properties are presented, and we show that this new discretization can reproduce earlier gyrokinetic simulations using grids with up to 6-60 times fewer cells and 22X-60X speed-ups depending on dimensionality, geometry and plasma parameters.

Figures (14)

• Figure 1: (a) A uniform $v_{\parallel}(\eta)$ map (solid blue), and the map in equation \ref{['eq:linquad_vpar_map']} (dashed orange). (b) Initial ion distribution at $z=0$, as well as the nonuniform velocity grid (grey mesh). Both figures show demarcate the region with a linear $v_{\parallel}(\eta)$ map using dotted green lines.
• Figure 2: (a) Field energy as a function of time (normalized to the ion cyclotron frequency $\omega_{ci}$) of ion acoustic wave simulation exhibiting collisionless Landau damping with uniform and nonuniform velocity mappings for $k_\parallel\rho_i=0.25$. (b) Damping rate as a function of wave number for uniform and nonuniform velocity mappings. Grey dotted lines show a case in which only the $\left| \eta \right|<1/2$ part velocity-space is used.
• Figure 3: Ion distribution function averaged along $\mu$ at $\omega_{ci}t=35$ (a) and $\omega_{ci}t=90$ (b); these look nearly the same in both uniform and nonuniform velocity simulations. (c) Normalized field energy for simulations with uniform (solid blue) and nonuniform velocity maps (dashed orange).
• Figure 4: (a) Initial ion distribution function in velocity space with the nonuniform grid overlaid (grey mesh). (b) Parallel ($T_{\parallel e}$, solid) and perpendicular ($T_{\perp e}$, dashed) temperatures as a function of time (normalized to the electron collision frequency $\nu_{ee}$) for collisional relaxation simulations with uniform (blue) and nonuniform (orange) velocity space grids.
• Figure 5: Relative error in integrated particle number (a), momentum (b) and kinetic energy (c) density as a function of the number of cells along $v_{\parallel}$ ($N_{v_{\parallel}}$) and along $\mu$ ($N_\mu$). The error is of order of the machine precision in all cases.