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.

翻译：本文针对标量对流方程发展了隐式Active Flux格式。在每个单元界面处，我们采用时间多项式逼近解。这一方法使得能通过特征线推进点值，并利用对该多项式积分得到的通量更新单元平均值。所得格式具有高达五阶的收敛阶，但对于间断解仅呈现高频适度振荡。通过数值实验比较了不同方法，并展示了在网络流中的应用。