Sparse grid-based adaptive noise reduction strategy for particle-in-cell schemes

We propose a sparse grid-based adaptive noise reduction strategy for electrostatic particle-in-cell (PIC) simulations. By projecting the charge density onto sparse grids we reduce the high-frequency particle noise. Thus, we exploit the ability of sparse grids to act as a multidimensional low-pass filter in our approach. Thanks to the truncated combination technique [1], [2], [3], we can reduce the larger grid-based error of the standard sparse grid approach for non-aligned and non-smooth functions. The truncated approach also provides a natural framework for minimizing the sum of grid-based and particle-based errors in the charge density. We show that our approach is, in fact, a filtering perspective for the noise reduction obtained with the sparse PIC schemes first introduced in [4]. This enables us to propose a heuristic based on the formal error analysis in [4] for selecting the optimal truncation parameter that minimizes the total error in charge density at each time step. Hence, unlike the physical and Fourier domain filters typically used in PIC codes for noise reduction, our approach automatically adapts to the mesh size, number of particles per cell, smoothness of the density profile and the initial sampling technique. It can also be easily integrated into high performance large-scale PIC code bases, because we only use sparse grids for filtering the charge density. All other operations remain on the regular grid, as in typical PIC codes. We demonstrate the efficiency and performance of our approach with two test cases: the diocotron instability in two dimensions and the three-dimensional electron dynamics in a Penning trap. Our run-time performance studies indicate that our approach can provide significant speedup and memory reduction to PIC simulations for achieving comparable accuracy in the charge density.