UC Berkeley

We present an analytic framework for understanding the errors associated with deposition in particle-in-cell methods applied to kinetics problems in one configuration space dimension. We begin by considering a one-dimensional advection model problem. Results of the analysis includes a O(1) error, also referred to as a barrier error, that is dependent on the deformation gradient of the particle locations. The presence of this barrier is confirmed by numerical experiments. The techniques developed in the analysis of the one-dimensional advection model problem are used to analyze a deposition model problem in the context of kinetics. This analysis shows an error that is similar to the error that arose in the one-dimensional advection model problem but is dependent only on the configuration space deformation gradient of the particle locations. A set of ODEs are derived for the Vlasov-Poisson system that can be used to compute the configuration space deformation gradient without explicit knowledge of particle neighbors. This allows for computation of the O(1) error at each time-step. Remapping is then introduced to control this error. Numerical experiments confirm that remapping when the O(1) error is comparable to the other errors inherent in the simulation reduces the "particle noise” phenomena that plagues many PIC methods and maintains the high-order convergence. With a fourth-order kernel this leads to an overall reduction in the number of remaps compared to a forward semi-Lagrangian method that remaps every few time steps. We also implement adaptive-in-time-and-space (local) remapping where only a portion of the phase-space is remapped and the O(1) error along with the value of the distribution serve as an implicit boundary between the remapped and non-remapped regions. The adaptive-in-time and locally remapped PIC methods are then applied to the Dory-Guest-Harris instability which, unlike the other test problems considered, has two velocity space dimensions.