In this paper, we propose a new well-balanced fifth-order finite volume WENO method for solving one- and two-dimensional shallow water equations with bottom topography. The well-balanced property is crucial to the ability of a scheme to simulate perturbation waves over the ``lake-at-rest'' steady state such as waves on a lake or tsunami waves in the deep ocean. We adopt the constant subtraction technique such that both the flux gradient and source term in the new pre-balanced form vanish at the lake-at-rest steady state, while the well-balanced WENO method by Xing and Shu [Commun. Comput. Phys., 2006] uses high-order accurate numerical discretization of the source term and makes the exact balance between the source term and the flux gradient, to achieve the well-balanced property. The scaling positivity-preserving limiter is used for the water height near the dry areas. The fifth-order WENO-AO reconstruction is used to construct the solution since it has better resolution than the WENO-ZQ and WENO-MR reconstructions for the perturbation of steady state flows. Extensive one- and two-dimensional numerical examples are presented to demonstrate the well-balanced, fifth-order accuracy, non-oscillatory, and positivity-preserving properties of the proposed method.

翻译：本文针对带底部地形的二维浅水方程，提出了一种新的保平衡五阶有限体积WENO方法。保平衡特性对于格式模拟“静止湖泊”稳态上的扰动波（如湖泊波浪或深海海啸波）至关重要。我们采用常数相减技术，使得新预处理平衡形式中的通量梯度和源项在静止湖泊稳态下同时为零；而Xing与Shu的保平衡WENO方法（Commun. Comput. Phys., 2006）通过对源项进行高阶精确数值离散，并实现源项与通量梯度间的精确平衡来获得保平衡特性。针对近干涸区域的水位计算，采用了缩放型保正限制器。在解重构方面，选用五阶WENO-AO重构方法，因其对稳态流动扰动的分辨率优于WENO-ZQ和WENO-MR重构。通过大量二维数值算例，验证了所提方法具备保平衡性、五阶精度、无振荡特性及保正性。