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In this paper, we propose a semi-Lagrangian discontinuous Galerkin methodcoupled with Runge-Kutta exponential integrators (SLDG-RKEI) for nonlinearVlasov dynamics. The commutator-free Runge-Kutta (RK) exponential integrators(EI) were proposed by Celledoni, et al. (FGCS, 2003). In the nonlineartransport setting, the RKEI can be used to decompose the evolution of thenonlinear transport into a composition of a sequence of linearized dynamics.The resulting linearized transport equations can be solved by thesemi-Lagrangian (SL) discontinuous Galerkin (DG) method proposed in Cai, et al.(JSC, 2017). The proposed method can achieve high order spatial accuracy viathe SLDG framework, and high order temporal accuracy via the RK EI. Due to theSL nature, the proposed SLDG-RKEI method is not subject to the CFL condition,thus they have the potential in using larger time-stepping sizes than those inthe Eulerian approach. Inheriting advantages from the SLDG method, the proposedSLDG-RKEI schemes are mass conservative, positivity-preserving, have nodimensional splitting error, perform well in resolving complex solutionstructures, and can be evolved with adaptive time-stepping sizes. We show theperformance of the SLDG-RKEI algorithm by classical test problems for thenonlinear Vlasov-Poisson system, as well as the Guiding center Vlasov model.Though that it is not our focus of this paper to explore the SLDG-RKEI schemefor nonlinear hyperbolic conservation laws that develop shocks, we show somepreliminary results on schemes' performance on the Burgers' equation.

High Order Semi-Lagrangian Discontinuous Galerkin Method Coupled with Runge-Kutta Exponential Integrators for Nonlinear Vlasov Dynamics