A novel linear analysis of grid instabilities in PIC

A meshfree (MF) particle approach to plasma kinetic theory is used as an approach for analyzing grid instabilities in PIC methods, both of the momentum-conserving PIC (MCP) and the energy conserving PIC (ECP) variety. A method of linearizing is introduced, about an equilibrium prescribed by a uniform particle lattice. Particle positions are perturbed about this lattice. The linear equations are then formulated on a grid, in both MCP and ECP forms. For a cold plasma, the ECP linearization leads to equations of motion with a symmetric, positive definite (SPD) matrix, showing stability. In the MCP form, similar linear equations are derived but the matrix is not SPD and varies with the displacement of the particle lattice relative to the grid. Complex eigenvalues, indicating instability, arise with lattice to grid displacement. These analytical results are compared with PIC simulations and show agreement. Studies show that this MCP grid instability is related to the presence of degenerate modes in the MF limit. This variation with lattice displacement for a cold plasma is closely related to the variation in time for the matrix elements for both MCP and ECP discretizations of a cold drifting beam. Issues related to grid-induced instabilities for more general plasmas are discussed.