Kinetic spectral simulations with implicit-explicit time integration*

Kinetic models for plasma or rarefied gas problems include physics with drastic time scale separation, which we address with the implicit-explicit (IMEX) temporal integration approach. 

We demonstrate the approach using a kinetic plasma  / gas dynamics (GD) solver. For plasmas, we solve the Vlasov-Maxwell (VM) system, where the stiff parts are the particle acceleration term and current source in Maxwell's equation. For GD, we solve the Boltzmann equation with the Bhatnagar-Gross-Krook (BGK) collision operator (stiff term). The spatial dynamics are treated explicitly, thus the procedure ensures locality in physical space for the implicit part, making it particularly efficient where standard physical space domain decomposition is used. Hence, the parallel implementation and preconditioning of implicit solvers are significantly easier than for fully implicit methods which require global nonlinear iterations. 

Both GD and VM systems use a spectral expansion in velocity space that leverages Hermite basis (featuring fluid-kinetic coupling that enables its multi-physics application), and finite differences for spatial discretization.For plasma applications, we present the evolution of a large magnetohydrodynamics scale ion-acoustic and Alfvén waves with IMEX time stepping few orders larger than the electron plasma period (the fastest scale). For GD, we recover the correct dynamics of a shock wave propagation (Sedov problem) with IMEX time stepping exceeding ∼10⁴ times the collision period.