A Conservative Discontinuous Galerkin Algorithm for Particle Kinetics on Smooth Manifolds

We have developed a novel formulation for modeling species in a kinetic continuum scheme with complex geometries. The advantage of our method lies in choosing canonical coordinates to evolve our system, which allows for a simplified evolution equation using the Canonical Poisson bracket. The resulting scheme has no explicit appearance of Christoffel-symbols, and the Poisson bracket is in its simplest, canonical form. Discretizing the Canonical Poisson Bracket in a Discontinuous Galerkin representation results in a high-order scheme for the neutral species. Coupled with an implicit BGK collision term, we can simulate a wide range of collisionality from a collisionless kinetic limit to the fluid limit. We demonstrate this with a transition in collisionality in a sod shock problem. As well, we exemplify the geometric capabilities with Kelvin-Helmholtz Instability on the surface of a sphere. Future application may employ more complex geometries by specifying a metric inverse that encodes the desired geometry. Additionally, from an astrophysical perspective, this formulation provides a pathway towards a first of its kind numeric scheme that can model neutral flows with continuum kinetics around compact objects.