A particle-in-Fourier method with semi-discrete energy conservation for non-periodic boundary conditions

We introduce a novel particle-in-Fourier (PIF) scheme based on [1], [2] that extends its applicability to non-periodic boundary conditions. Our method handles free space boundary conditions by replacing the Fourier Laplacian operator in PIF with a mollified Green's function as first introduced by Vico-Greengard-Ferrando [3]. This modification yields highly accurate free space solutions to the Vlasov-Poisson system, while still maintaining energy conservation up to an error bounded by the time step size. We also explain how to extend our scheme to arbitrary Dirichlet boundary conditions via standard potential theory, which we illustrate in detail for Dirichlet boundary conditions on a circular boundary. We support our approach with proof-of-concept numerical results from two-dimensional plasma test cases to demonstrate the accuracy, efficiency, and conservation properties of the scheme. By avoiding grid heating and finite grid instability we are able to show an order of magnitude speedup compared to the standard PIC scheme for a long time integration cyclotron simulation.