Vlasov--Poisson System Tackled by Particle Simulation Utilizing Boundary Element Methods

Abstract

This paper presents a grid-free simulation algorithm for the fully three-dimensional Vlasov-Poisson system for collisionless electron plasmas. We employ a standard particle method for the numerical approximation of the distribution function. Whereas the advection of the particles is grid-free by its very nature, the computation of the acceleration involves the solution of the non-local Poisson equation. To circumvent a volume mesh, we utilise the Fast Boundary Element Method, which reduces the three-dimensional Poisson equation to a system of linear equations on its twodimensional boundary. This gives rise to fully populated matrices which are approximated by the H 2 -technique, reducing the computational time from quadratic to linear complexity. The approximation scheme based on interpolation has shown to be robust and flexible, allowing a straightforward generalisation to vector-valued functions. In particular, the Coulomb forces acting on the particles are computed in linear complexity. In first numerical tests, we validate our approach with the help of classical non-linear plasma phenomena. Furthermore, we show that our method is able to simulate electron plasmas in complex three-dimensional domains with mixed boundary conditions in linear complexity.