A High Order Multi-Dimensional Characteristic Tracing Strategy for the Vlasov-Poisson System

In this paper, we consider a finite difference grid-based semi-Lagrangian approach in solving the Vlasov-Poisson (VP) system. Many of existing methods are based on dimensional splitting, which decouples the problem into solving linear advection problems, see {\em Cheng and Knorr, Journal of Computational Physics, 22(1976)}. However, such splitting is subject to the splitting error. If we consider multi-dimensional problems without splitting, difficulty arises in tracing characteristics with high order accuracy. Specifically, the evolution of characteristics is subject to the electric field which is determined globally from the distribution of particle densities via the Poisson's equation. In this paper, we propose a novel strategy of tracing characteristics high order in time via a two-stage multi-derivative prediction-correction approach and by using moment equations of the VP system. With the foot of characteristics being accurately located, we proposed to use weighted essentially non-oscillatory (WENO) interpolation to recover function values between grid points, therefore to update solutions at the next time level. The proposed algorithm does not have time step restriction as Eulerian approach and enjoys high order spatial and temporal accuracy. However, such finite difference algorithm does not enjoy mass conservation; we discuss one possible way of resolving such issue and its potential challenge in numerical stability. The performance of the proposed schemes are numerically demonstrated via classical test problems such as Landau damping and two stream instabilities.