A high order semi-Lagrangian discontinuous Galerkin method for Vlasov–Poisson simulations without operator splitting

In this paper, we develop a high order semi-Lagrangian (SL) discontinuous Galerkin (DG) method for nonlinear Vlasov–Poisson (VP) simulations without operator splitting. In particular, we combine two recently developed novel techniques: one is the high order non-splitting SLDG transport method (Cai et al. (2017) [4]), and the other is the high order characteristics tracing technique proposed in Qiu and Russo (2017) [29]. The proposed method with up to third order accuracy in both space and time is locally mass conservative, free of splitting error, positivity-preserving, stable and robust for large time stepping size. The SLDG VP solver is applied to classic benchmark test problems such as Landau damping and two-stream instabilities for VP simulations. Efficiency and effectiveness of the proposed scheme is extensively tested. Tremendous CPU savings are shown by comparisons between the proposed SL DG scheme and the classical Runge–Kutta DG method.