Grid-based kinetics solvers and AMAR technique

Kinetic equations describe classical transport processes in terms of particle velocity distribution functions. Direct numerical solution of the kinetic equations can be obtained by discretizing phase space (which consists of configuration and velocity spaces). However, substantial challenges exist due to the high dimensionality of phase space (3d3v) in the general case. Two methods to overcome this challenge: a) use problem symmetry to reduce the dimensionality of either configuration or velocity spaces, or b) use splitting procedure to generate a separate mesh in configuration and velocity space and solve transport in these spaces sequentially. In this presentation, we describe grid-based kinetic solvers using both of these techniques.

A whole spectrum of grid-based Vlasov-Poisson solvers has been developed, from an original adaptive scheme using a dynamically refined 6D grid [1] to successive 1d alternate direction stripes in 6D phase space [2]. We have developed kinetic solvers with Adaptive Mesh in Phase Space (AMPS) using a Tree-of-Trees technique [3]. Recently, a hyper.deal library has been developed using a tensor product of two meshes (configuration+velocity) in phase space [4]. Traditional methods have been used for solving kinetic equations for phase-space dimensions less than three. We have developed AMPS kinetic solvers using spherical coordinates in velocity space for several 1d plasma problems [5]. Challenges associated with coupling kinetic solvers in phase space with lower-dimensional Poisson solvers in configurational space have been discussed [6].

The Adaptive Mesh and Algorithm Refinement (AMAR) technique allows dynamically refine the computational mesh and embed kinetic “islands” into the fluid domain to combine the accuracy of kinetic solvers with the efficiency of fluid models [7]. Grid-based kinetic solvers have advantages over particle-based methods for hybrid kinetic-fluid simulations of streamers, shock waves, expanding plasma, etc. The research challenges: identify criteria for selecting appropriate models for (electrons, ions, atoms, and photons), closure of fluid models, coupling kinetic and fluid solvers at interfaces, and practical implementation on modern computing systems.