Well-balanced positivity preserving cell-vertex central-upwind scheme for shallow water flows. (10th September 2016)

• Record Type: Journal Article
• Authors: Beljadid, Abdelaziz; Mohammadian, Abdolmajid; Kurganov, Alexander
• Is Part Of: Computers & fluids. Volume 136(2016)
• Page Start: 193
• Page End: 206
• Publication Date: 2016-09-10
• Subjects: Shallow water equations -- Central-upwind schemes -- Finite-volume methods -- Cell-vertex grids -- Well-balanced schemes -- Positivity preserving property; Fluid dynamics -- Data processing -- Periodicals; 532.050285
• DOI: 10.1016/j.compfluid.2016.06.005
• Languages: English
• ISSNs: 0045-7930

Abstract

Highlights: A cell-vertex central-upwind scheme is proposed for shallow water system. The proposed method is free of any Riemann solver. New techniques are proposed for the water surface elevation and the topography. The proposed scheme preserves the positivity of the water depth. The numerical scheme is stable, well-balanced and second-order accurate. Abstract: We develop a new second-order two-dimensional central-upwind scheme on cell-vertex grids for approximating solutions of the Saint-Venant system with source terms due to bottom topography. Central-upwind schemes are developed based on the information about the local speeds of wave propagation. Compared to the triangular central-upwind schemes, the proposed cell-vertex one has an advantage of using more cell interfaces which provide more information on the waves propagating in different directions. We propose a new piecewise linear approximation of the bottom topography and a novel non-oscillatory reconstruction in which the gradient of each variable is computed using a modified minmod-type method to ensure the stability of the scheme. A new technique is proposed for the correction of the water surface elevation which guarantees the positivity of the water depth. The well-balanced property of the proposed central-upwind scheme is ensured using a special discretization for the cell averages of the topography source terms. The proposed scheme is tested on a number of numerical examples, among which we consider steady-state solutions with almost dry areas and their perturbations and solutions with rapidly varying flows over discontinuous bottom topography. Our numerical experiments confirm stability, well-balanced, positivity preserving properties and second-order accuracy of the proposed method. This scheme can be applied to shallow water models when the bed topography is discontinuous and/or highly oscillatory, and on complicated domains where the use of unstructured grids is advantageous.