A direct Vlasov code to study the non-stationary current collection by a cylindrical Langmuir probe

The time-dependent current collection by a cylindrical Langmuir probe, whose bias is suddenly changed from zero to a positive or negative finite value, is studied with a novel direct Vlasov code. The numerical algorithm is based on finite-difference formulas to approximate spatial and velocity derivatives and the time integration is carried out with an explicit Runge-Kutta method, or in the case of probe radius small compared with the Debye length, by using the unconditionally stable backward Euler scheme. Both electrons and ions are treated kinetically by the code, which implements initial and boundary conditions that are consistent with the presence of the probe. Within the considered parameter range, the plasma sheath around the probe exhibited an overshoot and it later recovered a steady state. Phase space diagrams of the particle trajectories revealed the presence of a trapped population of particles. The dependence of this population as a function of the probe radius is presented as well as a comparison with the stationary theory. The performance of the code and a comparison with previously used particle-in-cell algorithms are discussed.