Efficient particle Simulation of a virtual cathode using a grid-free treecode Poisson solver

An efficient grid-free numerical method is developed for particle simulations in plasma dynamics. The method relies on a treecode algorithm to solve the Poisson equation for interacting charged particles. Such algorithms have been extensively used in astrophysics, fluid dynamics, and molecular dynamics, and our aim is to assess their capability in plasma dynamics in comparison with traditional mesh-based methods such as particle-in-cell (PIC). It is well-known that PIC simulations have difficulty resolving local particle interactions within a grid cell, and we expect the grid-free treecode algorithm to have an advantage for problems involving small-scale features and highly nonuniform particle distributions. To illustrate this point we apply direct summation, treecode, and PIC methods to simulate a virtual cathode in one dimension. The treecode and PIC methods are found to be much faster than direct summation, but the treecode yields the correct solution while spurious features are present in the PIC results. The present work treats a one-dimensional problem, but the treecode algorithm generalizes directly to higher dimensions.