Grid-Point Requirements for Direct Kinetic Simulation of the Vlasov and Boltzmann Equations

Vlasov-based numerical methods, or direct kinetic (DK) methods, are increasingly being used to simulate the dynamics of plasmas in a variety of applications ranging from astrophysics and nuclear fusion to space propulsion and materials processing. These methods involve the direct Eulerian simulation of the particle distribution function in the Vlasov or Boltzmann equation, which describes the kinetic transport of this probability density function in configuration (physical) and velocity spaces. The DK method eliminates statistical noise inherent in Lagrangian particle methods and is preferable for multiscale time-varying problems where the tolerable noise floor is low. However, it is generally associated with larger costs due to its higher dimensionality. Viable widespread usage of the DK method requires a detailed understanding of its grid-point requirements so that the underlying mesh is designed efficiently without compromising on solution accuracy. We examine these requirements for a plasma plume expansion process, such as that produced by a laser ablation procedure, which produces a bi-Maxwellian ion distribution due to ambipolar acceleration by fast electrons. In addition to the commonly known Debye-length requirement in physical space, we investigate the corresponding requirement in velocity space, as well as the sensitivity of these requirements to the quantities of interest, such as the species number densities, velocities, and temperatures, as well as their associated electric potential and field.