A positivity-preserving Active Flux method for the Vlasov-Poisson system

The Active Flux method is a finite volume method which uses point values as well as cell average values as degrees of freedom. The point values are evolved in time using the characteristic form of the equation while the conservative form of the equations is used to evolve cell average values. Here we present an Active Flux method for the 1 + 1 dimensional Vlasov-Poisson system. The resulting scheme is third order accurate and uses a compact stencil in space and time. This leads to accurate approximations on relatively coarse grids, a desirable property for high dimensional kinetic equations. To avoid negative values in the approximation of the Vlasov equation we introduce a new limiting approach for Active Flux methods that is motivated by positivity-preserving flux limiters.