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

Abstract

Abstract. 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., J Sci Comput, 2017], and the other is the high order characteristics tracing technique proposed in [Qiu and Russo, J Sci Comput, 2017]. 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.

A High Order Semi-Lagrangian Discontinuous Galerkin Method for the Two-Dimensional Incompressible Euler Equations and the Guiding Center Vlasov Model Without Operator Splitting

“…In this paper, we propose a class of high order semi-Lagrangian discontinuous Galerkin (SLDG) methods for the two-dimensional (2D) time dependent incompressible Euler equation in the vorticity stream-function formulation and the guiding center Vlasov model. This is a continuation of our previous research effort on the development of high order non-splitting SLDG methods for 2D linear transport equations [4] and the Vlasov-Poisson (VP) system [5].…”

“…In [4,5], we formulated a class of high order conservative SLDG schemes for solving 2D transport problems with application to the VP system. To the authors' best knowledge, such a method is the first SLDG scheme in the literature that is high order accurate (up to third order accurate), unconditionally stable, mass conservative and free of splitting error for 2D transport simulations.…”

A High Order Conservative Semi-Lagrangian Discontinuous Galerkin Method for Two-Dimensional Transport Simulations

“…Recently, a class of high order non-splitting SLDG methods is under great development [6,7,8] and in [31] for unstructured meshes. They are unconditionally stable and mass conservative.…”

“…We adopt the two-stage multi-derivative prediction-correction algorithms as proposed in [38] for the VP system and in [50] for the Euler equation and the guiding center model. We refer the reader to [7,8] for details.…”

“…Recently, a line of research has been carried out for the development of non-splitting SLDG methods [31,6,7,8]. The proposed methods are unconditionally stable, leading to immense computational efficiency.…”