Solving the Plasma Kinetic Equation Numerically on Smooth Manifolds with Continuum Methods

A major challenge in plasma kinetic simulations is accurately resolving atmospheres around neutron stars and energy dynamics in the ergosphere of Kerr black holes. These regions form extreme density gradients and require fine velocity space resolution; both are not easily captured by particle schemes. Kinetic continuum schemes on the other hand are particularly well suited to handle these cases but have traditionally been considered prohibitively expensive. However, recent developments in applying high-order schemes such as discontinuous Galerkin (DG) have made continuum simulations realizable. We present a kinetic approach using a Hamiltonian-based framework for the stationary spacetimes using a modal DG algorithm. The resulting method conserves energy and density, handles general coordinate systems, and is asymptotic preserving in the limit of high collisionality. Additionally, upwind fluxes result in monotonic entropy growth and L2 decay, increasing simulation stability. We have implemented this scheme in the Gkeyll code base, and benchmarks have shown excellent agreement with existing solvers and analytic results. Current work is aimed at integrating the stationary spacetime general relativistic Vlasov-Maxwell-BGK system aimed at simulating extreme environments around compact objects.