Kinetic spectral simulations with local-implicit global-explicit approach*

An implicit-explicit (IMEX) temporal integration approach for kinetic spectral models is presented where fast/stiff operators are treated implicitly. We solve the Boltzmann equation with the Bhatnagar–Gross–Krook (BGK) collision operator for a gas dynamics problem (GD), and the Vlasov-Ampere (VA) system for collisionless plasma physics applications. Kinetic spectral models help to manage the curse of dimensionality present in kinetic problems, however

retain drastic time scale separation which is common for plasma and fluid applications.

Thus, for GD problems, the BGK operator introduces a stiff collisional term, while for the VA system, the stiff parts are the particle acceleration term and current source in Ampere's equation.

The purpose of this presentation is to show speed-up by stepping over fast scale dynamics. This procedure ensures locality in physical space for the implicit part, making it particularly efficient where standard physical space domain decomposition is used.

The local property makes the parallel implementation and its preconditioning significantly easier than for a

fully implicit methods which require global nonlinear iterations.

Both GD and VA systems use a spectral expansion in velocity space that leverages Hermite basis, and finite differences for spatial discretization.

To illustrate the method, we present a Sedov problem, where an initial pressure gradient leads to shock-wave formation and its propagation with IMEX time stepping exceeds the explicit time step by ∼10⁴ times yet recovers the correct dynamics accurately. For plasma applications, the evolution of a large magnetohydrodynamics scale ion-acoustic wave can be evolved with IMEX time stepping of over ∼800 the fastest scale of the problem, which is the electron plasma period.