On Numerical Energy Conservation For An Implicit Particle-In-Cell Method Coupled With A Binary Monte-Carlo Algorithm For Coulomb Collisions

Conventional particle-in-cell (PIC) methods suffer from enhanced numerical heating (explicit) or cooling (semi-implicit) when coupled with a Monte-Carlo algorithm for Coulomb collisions. In this work, a fully-implicit θ-PIC scheme (with adjustable time-biasing parameter 1/2 ≤ θ ≤ 1) is considered. The discrete change in energy of a closed system after a time step for this scheme scales with (1/2 - θ)C θ , where C θ >=0 depends on the frequency spectrum of the energy in the fields. Collisions lead to additional energy in the field fluctuations associated with high-frequency light waves produced by a numerical Bremsstrahlung process, which can result in a large increase in the numerical cooling rate for θ > 1/2. However, for θ = 1/2, energy is exactly conserved. In this case, the energy stored as field fluctuations (on long time scales) agrees with that calculated using the equipartition theorem for a classical system in thermodynamic equilibrium. Results from 1D simulations of a magnetic-piston driven shock are presented. It is shown that the fully-implicit PIC scheme can accurately model shock physics in collisional plasmas with far fewer simulation particles than other PIC methods.