Development of a moment-conditioned, asymptotic preserving and mass, momentum and energy-conserving Particle-in-Cell solver for the Vlasov-Ampére system*

The coupled Vlasov-Ampére (V-A) set of equations describes the temporal evolution of a collisionless electrostatic plasma, and the accurate solution of these equations is key to the computational study of plasmas. However, the V-A system has a number of properties which are difficult to satisfy discretely including: conservation of  mass, momentum, and energy, preservation of the asymptotic quasi-neutral limit, satisfaction of the Gauss's law, and maintaining positivity of the velocity distribution function (VDF). A numerical scheme which robustly preserves all of the discrete requirements have not yet been developed.

In this work, we discuss a new formulation of the kinetic equation and implement it using PIC. In this formulation, the VDF is transformed through a change of variables and coordinates to separate particle motion from the evolution of the conserved hydrodynamic moments, which are evolved using the fluid moment equations and self-consistently closed from the kinetic solution. In this way, the constraints on the numerical method set above are divorced from the particle kinetics and instead are segregated into the moment equations, where the constraints are easier to satisfy. We will demonstrate the properties and advantages of this formulation on classic electrostatic test cases, including nonlinear Landau damping and a multi-scale ion acoustic shock.