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 almost no oscillations with high frequencies for discontinuous solutions. In numerical experiments we compare the different methods and show an application to network flows.

翻译：本文发展了适用于标量平流方程的隐式主动流通量格式。在每个单元界面处，我们通过时间多项式逼近解，从而可利用特征线演化节点值，并通过对该多项式积分获得的通量更新单元平均值。所得格式具有高达五阶的收敛精度，且对间断解的高频振荡几乎完全抑制。通过数值实验比较不同方法，并展示了在网络流中的应用。