Implicit Active Flux Methods for Linear Advection

In this work we develop implicit Active Flux schemes for the scalar advection equation. At every cell interface we approximate the solution by a polynomial in time. This allows to evolve the point values using characteristics and to update the cell averages using fluxes obtained by integrating this polynomial. The resulting schemes have order of convergence up to five, but show only moderate oscillations with high frequencies for discontinuous solutions. In numerical experiments we compare the different methods and show an application to network flows.