Conservative and non-conservative methods based on Hermite weighted essentially non-oscillatory reconstruction for Vlasov equations

We develop weighted essentially non-oscillatory reconstruction techniques based on Hermite interpolation both for semi-Lagrangian and finite difference methods. We apply these methods to transport equations in the context of plasma physics and the numerical simulation of turbulence phenomena. On the one hand the non-conservative semi-Lagrangian methods with high order reconstructions are particularly efficient and accurate in linear phase of simulations before the appearance of small structures. However in the nonlinear phase, the lack of conservations may generate inaccurate numerical simulations. At contrast, the conservative finite difference methods are more stable in nonlinear phase and the Hermite WENO reconstruction avoids spurious oscillations.