Finite-grid instability in non-uniform grids

The use of a discretization grid in particle-in-cell methods is responsible for the nite-grid instability. The instability is a spurious numerical e ect due to the aliasing of di erent Fourier modes, that are undistinguished by the computational grid but have di erent e ects on the particles. The nite-grid instability has been studied thoroughly in uniform grids [1, 2]; a linear theory has been developed and non-linear e ects have been observed in paradigmatic cases. In recent years, the nite-grid instability has been reconsidered for non-uniform grids [3, 4], as a number of codes use such kind of mesh in order to reduce the computational e ort. The e ect of the nite-grid instability has been shown to limit the range of variation of the grid spacing on non-uniform grids, unless methods to overcome the instability are used [3]. For example, in the simulation of 1D electromagnetic shocks using the computational code Celest1D [3], the nite-grid instability was found in the region with larger meshes, where the stability criterion was not ful lled, while the smaller cells in the shock region ful lled the criterion. Notwithstanding the work performed on the subject, a systematic study of the nite-grid instability in non-uniform grids had never been done. In the present communication, a theoretical analysis of the nite-grid instability in non-uniform grids is described and the ndings of an extensive investigation conducted with a simple explicit code are presented.