In this paper, high order semi-implicit well-balanced and asymptotic preserving finite difference WENO schemes are proposed for the shallow water equations with a non-flat bottom topography. We consider the Froude number ranging from O(1) to 0, which in the zero Froude limit becomes the "lake equations" for balanced flow without gravity waves. We apply a well-balanced finite difference WENO reconstruction, coupled with a stiffly accurate implicit-explicit (IMEX) Runge-Kutta time discretization. The resulting semi-implicit scheme can be shown to be well-balanced, asymptotic preserving (AP) and asymptotically accurate (AA) at the same time. Both one- and two-dimensional numerical results are provided to demonstrate the high order accuracy, AP property and good performance of the proposed methods in capturing small perturbations of steady state solutions.

翻译：在本文中,为浅水方程提出了高排序半隐含的平衡和无症状的保存有限差异WENO计划,且没有表层底地貌,我们认为从O(1)到0不等的Froude数量,零Froude限制为“湖方程”,用于没有重力波的平衡流动。我们应用了平衡的有限差异WENO重建,同时采用了精确准确的隐含(IMEX)Runge-Kutta时间分解法。由此形成的半隐含计划可以证明十分平衡,无症状保护(AP)和无症状准确(AAAA),同时提供了一维和二维的数字结果,以证明拟议的方法在捕捉稳定状态解决方案的小扰时的高度顺序准确性、AP属性和良好性能。