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Well-Balanced Central Schemes for the Saint-Venant Systems

机译：Saint-Venant Systems的均衡中心方案

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In this paper we present a new well-balanced unstaggered central finite volume scheme for the numerical solution of the one-dimensional Saint-Venant systems on flat or variable bottom topographies. The Saint-Venant equations also known as the shallow water equations (SWE) are useful when modeling the hydrodynamics of coastal oceans and lakes and the simulation of tsunami waves and inundation floods. The Saint-Venant system is a hyperbolic system with a nonzero geometrical source term when flows over variable bottom topographies are considered. In this work we develop a new one-dimensional well-balanced central scheme for the numerical solution of SWE systems: The numerical base scheme is an unstaggered extension of the 1D Nessyahu and Tadmor central scheme that evolves the numerical solution on a unique grid and avoids the resolution of the Riemann problems arising at the cell interfaces thanks to a layer of ghost staggered cells. As is the case with general balance laws, the SWE system admits steady-state solutions in which the nonzero divergence of the flux is exactly balanced by the source term. This balance is very difficult to maintain at the discrete level, and in general most numerical schemes for hyperbolic systems fail to preserve it, and thus generate nonphysical waves. In this work we propose a special discretization of the source term according to the discretization of the divergence of the flux by the numerical base scheme, and then adapt the surface gradient method to the case of one-dimensional unstaggered central schemes. The resulting scheme ensures the well-balanced constraint at the discrete level and is consistent with the SWE system.
机译：在本文中，我们提出了一种新的平衡良好的未置换的中央有限体积方案，用于平板或可变底部地形上的一维圣宫内系统的数值解。诸如浅水方程（SWE）的圣文鸣方程在建模沿海海洋和湖泊的流体动力学以及海啸波浪和淹没洪水的模拟时是有用的。 Saint-Venant系统是一个具有非零几何源期间的双曲线系统，当考虑变量底部拓扑时，术语。在这项工作中，我们为SWE系统的数值解决方案开发了一种新的一维均衡的中央方案：数值基础方案是1D Nessyahu和Tadmor中央方案的未播放延伸，从而在唯一的网格上发展并避免由于一层重影交错细胞，细胞界面产生的riemann问题的分辨率。正如一般余额法则的情况一样，SWE系统承认稳态解决方案，其中源期完全平衡了通量的非零分歧。这种平衡非常难以在离散水平维持，并且一般来说，双曲线系统的大多数数值方案都没有保留它，从而产生非物理波。在这项工作中，我们提出了根据数值基准方案的离散化的离散化来提出了源期限的特殊离散化，然后将表面梯度法适应一维未播的中央方案的情况。所产生的方案确保了离散水平的良好平衡约束，与SWE系统一致。

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