The present work focuses on the numerical approximation of the weak solutions of the shallow water model over a non-flat topography. In particular, we pay close attention to steady solutions with nonzero velocity. The goal of this work is to derive a scheme that exactly preserves these stationary solutions, as well as the commonly preserved lake at rest steady solution. These moving steady states are solution to a nonlinear equation. We emphasize that the method proposed here never requires solving this nonlinear equation; instead, a suitable linearization is derived. To address this issue, we propose an extension of the well-known hydrostatic reconstruction. By appropriately defining the reconstructed states at the interfaces, any numerical flux function, combined with a relevant source term discretization, produces a well-balanced scheme that preserves both moving and non-moving steady solutions. This eliminates the need to construct specific numerical fluxes. Additionally, we prove that the resulting scheme is consistent with the homogeneous system on flat topographies, and that it reduces to the hydrostatic reconstruction when the velocity vanishes. To increase the accuracy of the simulations, we propose a well-balanced high-order procedure, which still does not require solving any nonlinear equation. Several numerical experiments demonstrate the effectiveness of the numerical scheme.

中文翻译：

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完全平衡的水动力重建

目前的工作重点是非平坦地形上浅水模型弱解的数值近似。特别是，我们密切关注非零速度的稳定解。这项工作的目标是推导出一个方案，准确地保留这些稳态解，以及通常保留的静止湖稳态解。这些移动稳态是非线性方程的解。我们强调，这里提出的方法不需要求解这个非线性方程；相反，会导出合适的线性化。为了解决这个问题，我们提出了著名的静水力重建的扩展。通过适当地定义界面处的重构状态，任何数值通量函数与相关源项离散化相结合，都会产生一个平衡良好的方案，该方案可以保留移动和非移动稳定解。这消除了构建特定数值通量的需要。此外，我们证明所得到的方案与平坦地形上的均质系统一致，并且当速度消失时它简化为流体静力重建。为了提高模拟的准确性，我们提出了一种平衡良好的高阶程序，它仍然不需要求解任何非线性方程。多个数值实验证明了数值方案的有效性。