Exact moment methods for conservation laws in phase space

We construct exact moment models for a class of conservation laws based on a parametrization of the distribution functions by its moments as proposed in [Burby, Sci. Rep. 13 (2023)]. In particular, we derive moment models for the Vlasov--Maxwell and the relativistic Vlasov--Maxwell systems. The new models can be viewed as a higher degree extension of the Direct Quadrature Method of Moments (DQMOM). We propose slicing moment method for the representation of the distribution function. For simulations with strong kinetic effects, we utilize a hybrid formulation that transforms adaptively a fraction of fluid nodes into particles in order to capture multi-streaming. We demonstrate convergence and performance of our strategies on the weak and strong Landau damping and two-stream instability experiments in non-relativistic and relativistic limits.