A new well-balanced non-oscillatory central scheme for the shallow water equations on rectangular meshes

Capilla, M.T.; Balaguer-Beser, Ángel

doi:10.1016/j.cam.2013.01.014

Cited by 20 publications

(15 citation statements)

References 25 publications

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“…Next, we investigate the positivity-preserving property of the current scheme using Equation (28). We begin with the following Lemma 4.1, which gives that the positivity-preserving property may be missed when the limited values of the water height is far less than the values computed on the nonstaggered cells.…”

Section: The Positivity-preserving Propertymentioning

confidence: 99%

Well‐balanced nonstaggered central schemes based on hydrostatic reconstruction for the shallow water equations with Coriolis forces and topography

Dong

2020

Math Methods in App Sciences

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The objective of this paper is to present a nonstaggered central scheme based on hydrostatic reconstruction (HR) for the shallow water equations with Coriolis forces and nonflat bottom topography. The discretization of the bed slope source term is based on the HR method to ensure the well-balanced property for the stationary solution. We use the cell average of the discharge on the staggered cell to discretize the source term due to the effect of Coriolis forces. The positivity-preserving property of the water depth is obtained by providing an appropriate CFL condition with the help of a "draining" time-step technique. Finally, several two-dimensional numerical results of classical problems of the system are presented to test these properties of the current scheme.

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Section: The Positivity-preserving Propertymentioning

confidence: 99%

Well‐balanced nonstaggered central schemes based on hydrostatic reconstruction for the shallow water equations with Coriolis forces and topography

Dong

2020

Math Methods in App Sciences

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“…Many upwind (see, e.g., [1,2,5,9,10,18,22,31,33,37] ) and central (see, e.g., [6,8,15,23,28,40,41,45] ) schemes for the shallow water system (1) , which is a hyperbolic system of conservation (if B x ≡ B y ≡ 0) or balance (if B is not a constant) laws, have been proposed in the past two decades. Roughly speaking, the main difference between upwind and central schemes is that upwind schemes use characteristic information and utilize (approximate) Riemann problem solvers to determine nonlinear wave propagation, while central schemes are based on averaging over the waves without using their detailed structures.…”

Section: Introductionmentioning

confidence: 99%

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

2016

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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.

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“…Shu and Osher [44] made it more efficient and used it for sub-pixel interpolation in curve evolution problems. An adaptive window around each pixel is used in ENO interpolation method to define a support window for the 2D piecewise reconstruction of point values that avoids high gradient regions whenever possible [45,46]. The reconstruction scheme selects an interpolating support window whose solution is the smoothest in the sense of divided differences.…”

Section: Introductionmentioning

confidence: 99%

A New Adaptive Image Interpolation Method to Define the Shoreline at Sub-Pixel Level

et al. 2019

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This paper presents a new methodological process for detecting the instantaneous land-water border at sub-pixel level from mid-resolution satellite images (30 m/pixel) that are freely available worldwide. The new method is based on using an iterative procedure to compute Laplacian roots of a polynomial surface that represents the radiometric response of a set of pixels. The method uses a first approximation of the shoreline at pixel level (initial pixels) and selects a set of neighbouring pixels to be part of the analysis window. This adaptive window collects those stencils in which the maximum radiometric variations are found by using the information given by divided differences. Therefore, the land-water surface is computed by a piecewise interpolating polynomial that models the strong radiometric changes between both interfaces. The assessment is tested on two coastal areas to analyse how their inherent differences may affect the method. A total of 17 Landsat 7 and 8 images (L7 and L8) were used to extract the shorelines and compare them against other highly accurate lines that act as references. Accurate quantitative coastal data from the satellite images is obtained with a mean horizontal error of 4.38 ± 5.66 m and 1.79 ± 2.78 m, respectively, for L7 and L8. Prior methodologies to reach the sub-pixel shoreline are analysed and the results verify the solvency of the one proposed.

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A new well-balanced non-oscillatory central scheme for the shallow water equations on rectangular meshes

Cited by 20 publications

References 25 publications

Well‐balanced nonstaggered central schemes based on hydrostatic reconstruction for the shallow water equations with Coriolis forces and topography

Well‐balanced nonstaggered central schemes based on hydrostatic reconstruction for the shallow water equations with Coriolis forces and topography

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

A New Adaptive Image Interpolation Method to Define the Shoreline at Sub-Pixel Level

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