A Conservative Discontinuous Galerkin Algorithm for Particle Kinetics on Smooth Manifolds

TL;DR

This paper introduces a conservative discontinuous Galerkin algorithm for particle kinetics on manifolds, capable of conserving key physical quantities and applicable to various Hamiltonian formulations, with potential for future relativity simulations.

Contribution

It presents a new, efficient, and conservative DG scheme for particle kinetics on manifolds, including collision modeling and rotation handling, with demonstrated test problems.

Findings

The scheme conserves density and energy exactly.

It successfully models kinetic phenomena on curved surfaces.

The approach is adaptable to general relativistic kinetic simulations.

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.