A quasi–second-order scheme is developed to obtain approximate solutions of the two-dimensional shallow water equations (SWEs) with bathymetry. The scheme is based on a staggered finite volume space discretization: the scalar unknowns are located in the discretization cells while the vector unknowns are located on the edges of the mesh. A monotonic upwind-central scheme for conservation laws (MUSCL)-like interpolation for the discrete convection operators in the water height and momentum balance equations is performed in order to improve the accuracy of the scheme. The time discretization is performed either by a first-order segregated forward Euler scheme or by the second-order Heun scheme. Both schemes are shown to preserve the water height positivity under a Courants Friedrichs Lewy (CFL) condition and an important state equilibrium known as the lake at rest. Using some recent Lax–Wendroff type results for staggered grids, these schemes are shown to be LW-consistent with the weak formulation of the continuous equations, in the sense that if a sequence of approximate solutions is bounded and strongly converges to a limit, then this limit is a weak solution of the SWEs; besides, the forward Euler scheme is shown to be LW-consistent with a weak entropy inequality. Numerical results confirm the efficiency and accuracy of the schemes.

中文翻译：

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二维浅水方程的一致准二阶交错格式

开发了一种准二阶格式以获得具有水深测量的二维浅水方程 (SWE) 的近似解。该方案基于交错有限体积空间离散化：标量未知数位于离散化单元中，而矢量未知数位于网格边缘。为了提高方案的准确性，对水高和动量平衡方程中的离散对流算子进行了单调迎风中心守恒定律（MUSCL）类插值。时间离散化通过一阶分离前向 Euler 方案或通过二阶 Heun 方案执行。两种方案均显示在 Courants Friedrichs Lewy (CFL) 条件和称为静止湖的重要状态平衡下保持水位正性。使用交错网格的一些最近的 Lax-Wendroff 类型结果，这些方案被证明与连续方程的弱公式是 LW 一致的，在某种意义上，如果一系列近似解是有界的并且强烈收敛到一个极限，那么此限制是 SWE 的弱解；此外，前向欧拉方案被证明与弱熵不等式是 LW 一致的。数值结果证实了方案的效率和准确性。从某种意义上说，如果一系列近似解是有界的并且强烈收敛到一个极限，那么这个极限就是 SWE 的弱解；此外，前向欧拉方案被证明与弱熵不等式是 LW 一致的。数值结果证实了方案的效率和准确性。从某种意义上说，如果一系列近似解是有界的并且强烈收敛到一个极限，那么这个极限就是 SWE 的弱解；此外，前向欧拉方案被证明与弱熵不等式是 LW 一致的。数值结果证实了方案的效率和准确性。