A conservative semi-Lagrangian HWENO method for the Vlasov equation

There are several new ingredients in this paper. Firstly we propose a mass-conservative SL HWENO scheme for a 1-D equation by working with a flux-difference form, following the work of Qiu and Christlieb (2010) [25]. Secondly, we propose a proper splitting for equations of partial derivatives in HWENO framework to ensure local mass conservation. The proposed fifth order SL HWENO scheme with dimensional splitting has been tested to work well in capturing filamentation structures without oscillations when the time step size is within the Eulerian CFL constraint. However, when the time stepping size becomes larger, numerical oscillations are observed for the 'mass conservative' dimensional splitting HWENO scheme, as there are extra source terms in equations of partial derivatives. In this case, we introduce WENO limiters to control oscillations. Classical numerical examples on linear passive transport problems, as well as the nonlinear Vlasov-Poisson system, have been tested to demonstrate the performance of the proposed scheme.