A binary filter inspired from the PIC sparse grid technique - Illustration on the XTOR-K code

It is known [1,2] that the sparse grid method for Particle-In-Cell (PIC) solvers acts as a filter to reduce the PIC noise. In this paper, a simple rule to discard or keep modes in Fourier space (a binary filter with values either 0 or 1) is derived using the sparse grid combination formula. Its relation to the standard sparse grid filter, which is characterized quantitatively, is explained. The relations between the sparse grid filters on grids of arbitrary levels are also outlined. Namely, in two (resp. three) dimensions and for bi-linear (resp. tri-linear) moment deposition, it is proven rigorously that the sparse grid filter, for a grid of size equal to an arbitrary power of two, can be expressed in terms of two (resp. three) unique real valued functions. The advantage of the binary filter over the standard sparse grid filter is the reduction of signal deformation introduced by the latter, for the same noise reduction capability. By applying the filter to moments of a marker distribution coming from the XTOR-K code, it appears the noise could be significantly reduced, with a moderate overhead in the moment deposition part of the algorithm.