High-Order Sparse-PIC Methods: Analysis and Numerical Investigations

SIAM Journal on Numerical Analysis, Volume 63, Issue 3, Page 1281-1314, June 2025.
Abstract. Particle-in-cell (PIC) methods embedding sparse grids have been recently introduced to decrease the statistical noise inherent to PIC approximations. In sparse-PIC methods, the numerical noise is filtered out from the approximation thanks to a reconstruction of the grid quantities on a hierarchy of coarse meshes. This procedure introduces a significant gain in the precision of the numerical approximation with respect to the mean number of particles in a grid cell, this parameter controlling the numerical noise, but also introduces a slight discrepancy of the method precision with respect to the mesh resolution. In previous studies, this issue is addressed by a careful tuning of the grids composing the sparse grid hierarchy, to define a trade-off between the gain in the numerical noise and the loss in the grid error. The present work is dedicated to improving the precision of sparse-PIC methods with respect to the mesh resolution and, contrary to the previous achievements, without deteriorating the gains with respect to the statistical noise. A refined error estimate is proposed. It permits one to control the number of numerical particles to obtain a comparable statistical noise in the approximation carried out by either a standard or a sparse-PIC method (and thus assess the true merits of the methods).