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

Abstract

In this paper, we consider a finite difference grid-based semi-Lagrangian approach for 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 Cheng and Knorr (J Comput Phys 22:330---351, 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 density via 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 propose to use weighted essentially non-oscillatory interpolation to recover function values between grid points, therefore to update the solution at the next time level. The proposed method does not have time step restriction as the Eulerian approach and enjoys high order spatial and temporal accuracy. The performance of the proposed schemes are numerically demonstrated via classical test problems such as Landau damping and two stream instabilities.