Combination of Hermite-Legendre bases for kinetic plasma equations

Kinetic plasma simulations allow for modeling of effects associated with non-equilibrium distribution functions in velocity space. In contrast to fluid models, kinetic approaches have to deal with the higher dimensionality associated with resolving velocity space, which can make it challenging or impossible for numerous physical configurations. Thus, decreasing the number of degrees of freedoms (DOFs) is critical for large-scale simulations with kinetic effects. Spectral, particle, and finite difference/element methods are often used to discretize distribution functions in velocity space.

Spectral methods offer many advantages, such as fast global convergence, i.e., allowing fewer DOFs to represent functions behaviour.

We propose a spectral method that combines two bases, Hermite and Legendre, to split the distribution function in velocity space for efficient representation of equilibrium bulk plasma and non-equilibrium complex structures such as shocks, beams, etc. The Hermite basis with tuned parameters may require only a few degrees of freedom (DOF) to represent near-equilibrium (Maxwellian) plasma, while Legendre basis are parameter-free and generally better suited for non-equilibrium distributions. We have implemented a model that combines the two approaches in velocity space while using the discontinuous-Galerkin method for configuration space, with the main emphasis on velocity space. A dynamical scheme is implemented that projects higher Hermite coefficients into Legendre basis to minimize the total number of DOFs. We have performed numerical tests that are based on the evolution of beam-plasma instability with electron beam evolution into a highly non-Maxwellian state. Our initial tests reveal that the dynamical reprojection allows us to achieve better accuracy for the same number of DOFs in the range of a relatively low number of DOFs.