Particle-In-Cell (PIC) methods are widely used in plasma physics simulations, originating from early interest in discretizing kinetic plasma models. These methods combine Eulerian-Lagrangian characteristics, with the plasma properties represented by numerical particles sampling the distribution function, while Maxwell’s equations governing electromagnetic field changes are discretized by Finite Differences or Finite Elements on a mesh. PIC methods offer computational efficiency compared to Eulerian discretizations, especially for problems with high dimensionality in phase space. The linear complexity of PIC methods contrasts with the exponential complexity of Eulerian methods. 

However, PIC methods introduce statistical noise due to the limited number of particles sampling the distribution function. This noise affects the precision of computed quantities, analyzed through error decomposition into grid-based error and variance of the statistical estimator. The latter decreases slowly with the number of particles, necessitating a significant increase to reduce numerical noise effectively.

PIC methods with sparse grid reconstructions address this issue by modifying distribution moment projection and field interpolation to control numerical noise better. Utilizing a hierarchy of sparse grids called component grids, Sparse-PIC methods reconstruct an interpolant from data deposited onto each component grid. These grids, with coarse resolutions in at least one direction, increase the number of particles per cell, damping numerical noise effectively.

The figure above illustrates the electron density for a strong Landau damping 2D-3V test case [1]. The left panel depicts results obtained using standard PIC methods with over 16,000,000 particles, while the right panel showcases outcomes from sparse-PIC methods employing only 2,000,000 particles. Despite the reduced particle count, sparse-PIC methods effectively mitigate statistical noise, as evidenced by the noticeably clearer results.

The figure above illustrates the electron density for a diocotron instability 3D-3V test case [2] . The upper panel depicts results obtained using standard PIC method, while the bottom panel showcases outcomes from sparse-PIC methods with the offset combination method, an improvement proposed in [1] based on the elimation of the most anisotropic component grids. 

Publications :

• [1] F. Deluzet, G. Fubiani, L. Garrigues, C. Guillet, and J. Narski. Sparse grid reconstructions for Particle-In-Cell methods. ESAIM: M2AN, 56(5):1809–1841, (2022).  Open access : https://www.esaim-m2an.org/articles/m2an/pdf/2022/05/m2an220004.pdf; prepublication : https://hal.science/hal-03355720.

• [5] F. Deluzet, C. Guillet, P. Pace and J. Narski. High-order Particle-In-Cell schemes with sparse grid reconstructions. Submitted to  SIAM:SINUM (2024). Prepublication: https://hal.science/hal-04437088.