Optimal generalized finite difference solution to the particle-in-cell problem

A new adaptive Particle-in-Cloud (AP-Cloud) method for obtaining optimal numerical solutions to the Vlasov-Poisson equation has been proposed. The traditional particle-in-cell (PIC) method, commonly used for solving this problem, is not optimal in terms of the balance of errors of the differential operator discretization and source integration; it is also inaccurate when the particle distribution is highly non-uniform. Our method replaces the Cartesian grid in the traditional PIC with adaptive computational nodes or particles, to which the charges from the physical macroparticles are assigned by a weighted least-square approximations. The partial differential equation is then discretized using a generalized finite difference (GFD) method and solved with fast linear solvers. The density of computational particles is chosen adaptively, so that the error from GFD and that from the source integration are balanced and the total error is approximately minimized. The method is independent of geometrical shape of computational domains and free of artificial parameters. Results of verification tests using electrostatic problems of particle beams with halo and comparison of accuracy and solution time of the AP-Cloud method with the traditional PIC are presented.