Well-balanced positivity preserving cell-vertex central-upwind scheme for shallow water flows

Abstract

a b s t r a c tWe 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. Centralupwind 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 nonoscillatory reconstruction in which the gradient of each variable is computed using a modified minmodtype 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.

Hybrid Numerical Scheme of Preissmann Slot Model for Transient Mixed Flows

“…Because of their superior performance, the upwind flux solver has been preferred over general hyperbolic conservation systems such as compressible flow, hydrodynamics systems, and shallow water systems, etc. [3,13,16,[19][20][21]. However, for the Preissmann slot model, the monotonicity is not guaranteed by the widely used Riemann flux solvers because of the highly irregular shapes of pipes connected to the hypothetical slot, as presented in numerous previous works [5,10,22], whereas it is preserved for a regular-shaped flow domain.…”

“…However, this remedy can reduce the accuracy of the numerical solution because of the wider slot width and lower wave speed compared with the physical values. [13] presents an exact Riemann flux solver of the Preissmann slot model for highly transient mixed flows. However, the exact Riemann flux solver is not practical because it requires a large amount of computation for the iteration procedure.…”

Comparison of Shallow Water Solvers: Applications for Dam-Break and Tsunami Cases with Reordering Strategy for Efficient Vectorization on Modern Hardware

“…Instead, the local one-sided speeds of propagation at every edge, which can be computed in a straight-forward manner, are used. This scheme has been proven to be sufficiently robust and at the same time can satisfy both the well-balanced and positivity preserving properties, see [13][14][15].…”

“…The first loop corresponds to the edges 1-16 and the second one to the edges 17-31. In the first loop (lines 1-7), each flux computation accesses the array with the farthest alignment of seg_y, whereas the arrays are designed in the second loop (lines [8][9][10][11][12][13][14][15][16][17] to have contiguous patterns. Every edge has a certain pattern for its two corresponding cells, where no data-dependency exists, thus enabling an efficient vectorization.…”

A central-upwind geometry-preserving method for hyperbolic conservation laws on the sphere

“…Kurganov and Petrova [18] extended the central-upwind schemes to triangular grids for solving two-dimensional Cartesian systems of conservation laws. Next, Beljadid et al [4] proposed a two-dimensional well-balanced and positivity preserving cell-vertex central-upwind scheme for the computation of shallow water equations with source terms due to bottom topography.…”