Particle-In-Cell (PIC) methods, traditionally implemented with explicit time discretization of the Vlasov equation, face challenges due to temporal and spatial stability constraints. Implicit formulations, like the implicit-moment method and direct implicit method, offer stability with larger time-steps and grid discretizations. 

PIC methods also suffer from statistical errors due to finite particle sampling, requiring a large number of particles and substantial computational resources. Sparse grid reconstructions, involving a hierarchy of component grids with a coarse resolution, have shown significant memory and time savings compared to standard grids by mitigating these errors, while preserving momentum conservation.

Sparse grid reconstructions have been merged into a  semi-implicit PIC scheme. The method is based on an electrostatic Vlasov-(div)Ampere formulation and a linearization of the equations so that the implicit particle response to the electric field can be obtained by solving a linear system. The method features the following properties:

• The scheme is unconditionally stable with respect to the plasma period: the time step can be chosen irrespective to this value.

• The aliasing or finite grid instability is eliminated, allowing grid discretization without any constraints related to the Debye length.

• The statistical error is significantly reduced compared to the ECSIM scheme and explicit scheme carried out with a Cartesian grid of comparable resolution and the same number of particles. The reduction of the statistical noise is achieved thanks to both the sparse grid reconstructions and the Vlasov-(div)Ampere formulation. This is a valuable contribution since, as a consequence of their usual Vlasov-Ampere formulation, semi-implicit methods tends to create more statistical noise than explicit methods.

Publications:

[4] C. Guillet. Semi-implicit Particle-In-Cell methods embedding sparse grid reconstructions. Submitted to SIAM:MMS, (2023).  Prepublication: https://hal.science/hal-04137654v1.