论文标题
浅水方程的均衡活性通量法，并干燥
A well-balanced Active Flux method for the shallow water equations with wetting and drying
论文作者
论文摘要
Active Flux是一种三阶准确的数值方法，它独立地在单元格接口处逐渐发展出细胞平均值和点值。它自然使用连续的重建，但是当应用于双曲线问题时是稳定的。在这项工作中，主动通量方法首次扩展到非线性双曲线系统的平衡定律系统，即具有底部地形的浅水方程。我们演示了如何实现均衡，阳性保存并允许在一个空间尺寸的干燥状态的主动通量方法。由于连续重建，所有这些属性都是使用新方法实现的。为了保持三阶准确性，我们还提出了一个新型的高阶近似演化操作员，以更新点值。各种测试问题也证明了该方法的良好性能，即使存在冲击。
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.
