A Conservative Discontinuous Galerkin Algorithm for Particle Kinetics on Smooth Manifolds

TL;DR

This work develops a conservative discontinuous Galerkin method for collisionless kinetic evolution on smooth manifolds, anchored in a Hamiltonian formulation with canonical or noncanonical Poisson brackets. A BGK collision operator is incorporated through a moment-matching, implicit scheme that preserves density, momentum, and energy, yielding an asymptotic-preserving discretization that recovers Euler fluid behavior in the collisional limit. The authors prove discrete energy conservation, particle conservation, and, for the canonical case, $L_2$ stability with central fluxes or monotone decay with upwind fluxes, and they demonstrate robustness across flat and curved geometries, including rotating spheres and embedded surfaces. Extensive benchmarks (Sod shocks, Kelvin–Helmholtz instability, and rotating-sphere tests) verify accuracy, alias-free behavior, and convergence to fluid solutions, while normalized momentum coordinates offer a pathway to efficient phase-space representations in complex geometries. The framework lays groundwork for kinetic simulations on curved spacetimes and motivates future general-relativistic extensions using tetrad formalisms and relativistic BGK-like closures.

Abstract

A novel, conservative discontinuous Galerkin algorithm is presented for particle kinetics on manifolds. The motion of particles on the manifold is represented using using both canonical and non-canonical Hamiltonian formulations. Our schemes apply to either formulations, but the canonical formulation results in a particularly efficient scheme that also conserves particle density and energy exactly. The collisionless update is coupled to a Bhatnagar-Gross-Krook (BGK) collision operator that provides a simplified model for relaxation to local thermodynamic equilibrium. An iterative scheme is constructed to ensure collisional invariants (density, momentum and energy) are preserved numerically. Rotation of the manifold is incorporated by modifying the Hamiltonian while ensuring a canonical formulation. Several test problems, including a kinetic version of the classical Sod-shock problem, Kelvin-Helmholtz instability on the surfaces of a sphere and a paraboloid, with and without rotations, is presented. A prospectus for further development of this approach to simulation of kinetic theory in general relativity is presented.

Figures (10)

• Figure 1: Sodshock (1x1v) moment comparison taken at the final timestep between the moments of the canonical bracket solver at various collision frequencies, $\nu$, against the Riemann solution. The collision frequency spans from a nearly collisionless (free-streaming) limit to a highly collisional (fluid-like) limit.
• Figure 2: Annular disk sodshock moments at the final simulation time using the canonical Poisson bracket formalism. From left to right, the columns show density, radial momentum density, and pressure. The top row shows the physical coordinates of the moments of the distribution, and the bottom row of plots show line-outs of the moments compared to the Euler solution.
• Figure 3: Zooming in onto the forward-propagating shock in the annulus sodshock configuration from figure \ref{['fig:can_pb_sodshock_annular_disk']} to examine convergence with increasing spatial resolution. Panel (a) has a gray highlighted region that is zoomed into in panels (b-d) for the density, velocity, and temperature moments respectively.
• Figure 4: Kelvin-Helmholtz instability of the surface of a sphere using the canonical Poisson bracket method to update the distribution function, $f$. Panel (a) is the density moment of the distribution in physical space. Panels (b-e) show the difference between the distribution and local moment-matched Maxwellian, called $df$, at select points indicated by the red-x's on panel (a).
• Figure 5: Kelvin-Helmholtz instability on a hyperbolic surface. To the left, the density moment is mapped onto the physical surface. The panels to the right show the moments on the surface of density, $\theta$ and $z$ momenta, and temperature. The moments in all panels are snapshots taken at time, $t = 4.5$.

Theorems & Definitions (13)

• Proposition 1
• proof
• Lemma 1
• proof
• Lemma 2
• proof
• Proposition 2
• proof
• Proposition 3
• proof