Abstract
A second-order accurate, Godunov-type upwind finite volume method on dynamic refinement grids is developed in this paper for solving shallow-water equations. The advantage of this grid system is that no data structure is needed to store the neighbor information, since neighbors are directly specified by simple algebraic relationships. The key ingredient of the scheme is the use of the prebalanced shallow-water equations together with a simple but effective method to track the wet/dry fronts. In addition, a second-order spatial accuracy in space and time is achieved using a two-step unsplit MUSCL-Hancock method and a weighted surface-depth gradient method (WSDM) which considers the local Froude number is proposed for water depths reconstruction. The friction terms are solved by a semi-implicit scheme that can effectively prevent computational instability from small depths and does not invert the direction of velocity components. Several benchmark tests and a dam-break flooding simulation over real topography cases are used for model testing and validation. Results show that the proposed model is accurate and robust and has advantages when it is applied to simulate flow with local complex topographic features or flow conditions and thus has bright prospects of field-scale application.
Highlights
Solution to the two-dimensional (2D) hyperbolic conservation laws of the shallow-water equations (SWE) is relevant to many real-life hydrodynamic problems, such as free surface flows in shallow lakes, dam breaks, dyke breaches, wide rivers, flash floods, tidal bores, and coastal inundation, among others
The results show that the stationary solutions are well maintained, and the current numerical scheme is able to preserve the lake at rest solution involving wet/dry interface and is well balanced
The results prove that weighted surface-depth gradient method (WSDGM) performs better than the schemes based on pure depth gradient method (DGM) or surface gradient method (SGM) reconstructions
Summary
Solution to the two-dimensional (2D) hyperbolic conservation laws of the shallow-water equations (SWE) is relevant to many real-life hydrodynamic problems, such as free surface flows in shallow lakes, dam breaks, dyke breaches, wide rivers, flash floods, tidal bores, and coastal inundation, among others. When solving the shallow-water equations in the context of a Godunov-type framework, it is generally necessary and efficient to use the well-balanced methods, which employ a special treatment for the bed slope source terms that balanced the source terms and flux gradients. This well-balanced concept is known by exact conservation property (Cproperty) [4]. Liang and Borthwick [6] presented a Godunov-type shallow-flow model to solve the well-balanced SWEs with wet/dry fronts by a local bed slope modification method.
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