The degrees of freedom of Active Flux are cell averages and point values along the cell boundaries. These latter are shared between neighbouring cells, which gives rise to a globally continuous reconstruction. The semi-discrete Active Flux method uses its degrees of freedom to obtain Finite Difference approximations to the spatial derivatives which are used in the point value update. The averages are updated using a quadrature of the flux and making use of the point values as quadrature points. The integration in time employs standard Runge-Kutta methods. We show that this generalization of the Active Flux method in two and three spatial dimensions is stationarity preserving for linear acoustics on Cartesian grids, and present an analysis of numerical diffusion and stability.

翻译：主动通量法的自由度包括单元平均值和沿单元边界的点值。后者在相邻单元之间共享，从而产生全局连续的重构。半离散主动通量法利用其自由度获得空间导数的有限差分近似，这些近似用于点值的更新。平均值则通过通量的数值积分进行更新，并将点值用作积分点。时间积分采用标准的龙格-库塔方法。我们证明，对于笛卡尔网格上的线性声学问题，这种主动通量法在二维和三维空间中的推广具有保持稳态的特性，并给出了数值耗散和稳定性的分析。