使用傅里叶变换分析多维半离散 Active Flux 方法
Analysis of the multi-dimensional semi-discrete Active Flux method using the Fourier transform
AI总结：
本文利用傅里叶变换分析二维和三维半离散 Active Flux 方法，证明其在线性声学笛卡尔网格上保稳态，并研究其数值扩散与稳定性。
摘要：
Active Flux 的自由度是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建。半离散 Active Flux 方法利用其自由度获得空间导数的有限差分近似，并将其用于点值更新。平均值通过通量的数值积分进行更新，并利用点值作为积分节点。时间积分采用标准 Runge-Kutta 方法。我们证明，这种 Active Flux 方法在二维和三维空间中的推广对于笛卡尔网格上的线性声学问题是保稳态的，并给出了数值扩散和稳定性的分析。
英文摘要：
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 approxi\-mations 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.
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