Active Flux is a third order accurate numerical method which evolves cell averages and point values at cell interfaces independently. It naturally uses a continuous reconstruction, but is stable when applied to hyperbolic problems. In this work, the Active Flux method is extended for the first time to a nonlinear hyperbolic system of balance laws, namely to the shallow water equations with bottom topography. We demonstrate how to achieve an Active Flux method that is well-balanced, positivity preserving, and allows for dry states in one spatial dimension. Because of the continuous reconstruction all these properties are achieved using new approaches. To maintain third order accuracy, we also propose a novel high-order approximate evolution operator for the update of the point values. A variety of test problems demonstrates the good performance of the method even in presence of shocks.

翻译：活性通量是第三级精确数字方法,它独立地在单元格界面中演变单元格平均数和点值。它自然地使用连续重建,但在应用双曲问题时是稳定的。在这项工作中,主动通量法首次扩展至非线性平衡法双曲法系统,即浅水方程和海底地形。我们展示了如何在一个空间层面实现平衡、相对性保存和允许干状态的主动通量法。由于持续重建,所有这些特性都是用新的方法实现的。为了保持第三级准确性,我们还建议为更新点值建立一个新的高端近似进化操作器。许多测试问题都表明了即使在发生冲击的情况下这种方法的好性能。