Asymptotic and non-asymptotic model reduction for kinetic descriptions of plasma

Abstract

Plasma dynamics are coupled across microscopic and macroscopic scales by a variety of nonlinear mechanisms. These include repartition of energy due to kinetic microinstabilities, suppression of fluid instabilities by kinetic stabilization effects, and other mechanisms. At the macroscopic scale plasmas are well-described by fluid equations, whose formal validity depends on the long-time regularization of the phase space distribution function by collisions and magnetic gyrotropization. However, accurately capturing multiscale coupling requires multiscale reduced models which are both efficient and accurate in transition regimes. These regimes, where either collisional or magnetic gyrotropic regularization are marginal, are characterized by the ratio of the (collisional or magnetic) mean free path to a characteristic gradient scale length. This work studies two families of reduced plasma models for transition regimes in depth. The first is an asymptotic expansion for Braginskii-type transport coefficients in the so-called drift ordering for low-beta plasmas. The expansion captures leading-order finite Larmor radius effects for arbitrary collisionality. We present a new derivation of this expansion, evaluate its performance numerically, and provide a numerically feasible approximation. The second family of methods is dynamical low-rank (DLR) methods, which are not based on an asymptotic expansion and have the potential to overcome the curse of dimensionality for kinetic equations. We present two novel DLR schemes for plasma kinetic equations with a focus on fluid-kinetic coupling. One is a DLR method that retains low rank in the highly collisional asymptotic limit. The other is a fully locally conservative DLR method for the Vlasov-Dougherty-Fokker-Planck equation which achieves second-order accuracy in time. All discretizations are described in detail and accompanied by numerical results demonstrating the merit of the proposed approach.