Numerical thermalization in $n$-D particle-in-cell simulations

Abstract

The particle-in-cell (PIC) simulation method is often understood to solve the collisionless Vlasov equation due to the finite shape of its macroparticles. In reality, it can suffer from artificially high collisionality due to the underresolution of particle number; i.e., the use of a large macroparticle weight. The degree to which particle shape effects compensate for a large macroparticle weight in 1D, 2D, and 3D is presented. The collision time is calculated from PIC simulations based on the decay rate of the velocity autocorrelation function and compared directly with the kinetic theory of Okuda, Birdsall, and Langdon. The theory is found to accurately predict the simulated collision time with varied grid spacings, plasma conditions, and simulation dimensionalities. The result is a means to predict the timescale of self-consistent Coulomb interactions in the PIC simulation and thus characterize the relevance and implications of numerical thermalization as a function of grid spacing and macroparticle weight. It is determined that reaching the physical thermalization time, let alone approximating the collisionless Vlasov limit, may often be intractable in 3D for macroparticle sizes that resolve the Debye length.