An adaptive sparse grids scheme for reducing noise in Particle-In-Cell simulations

The computational complexity of grid based numerical schemes grows exponentially with the number of dimensions. This is the curse of dimensionality, which is the source of the high computational cost of grid based solvers for kinetic equations. The Particle-In-Cell (PIC) scheme partially avoids this curse by combining a grid-based approach for the computation of the electromagnetic fields with a particle approach for the evolution of the distribution function. However, the slow decay of the intrinsic numerical noise in PIC with the number of simulated particles leads to codes which may be more computationally intensive than grid based codes when a high level of accuracy is required.

We recently demonstrated how the sparse grids combination technique could be combined with the PIC algorithm to increase the number of particles per cell without increasing computational cost, and thus significantly reduce the noise of PIC simulations [1]. In that work, we also highlighted the limitations of the method for some types of distribution functions and some dynamical evolutions of the plasma. We now propose an improvement on our sparse PIC scheme in order to obtain robust noise reduction regardless of the details of the plasma dynamics [2]. Our algorithm relies on the truncated sparse grids combination technique, and we dynamically choose the optimal sparse grids truncation level through a heuristic based on formal error analysis, in order to reduce the total numerical error. We demonstrate the strong potential of our approach with a simulation of the diocotron instability, and of the plasma dynamics in a Penning trap.