scholarly journals A New Well-Balanced Reconstruction Technique for the Numerical Simulation of Shallow Water Flows with Wet/Dry Fronts and Complex Topography

Water ◽  
2018 ◽  
Vol 10 (11) ◽  
pp. 1661 ◽  
Author(s):  
Zhengtao Zhu ◽  
Zhonghua Yang ◽  
Fengpeng Bai ◽  
Ruidong An
Keyword(s):  
Shallow Water ◽  
Water System ◽  
Reconstruction Technique ◽  
Test Cases ◽  
Bed Topography ◽  
Surface Gradient ◽  
Shallow Water Flows ◽  
Shallow Water System ◽  
The One ◽  
Flow Variables

This study develops a new well-balanced scheme for the one-dimensional shallow water system over irregular bed topographies with wet/dry fronts, in a Godunov-type finite volume framework. A new reconstruction technique that includes flooded cells and partially flooded cells and preserves the non-negative values of water depth is proposed. For the wet cell, a modified revised surface gradient method is presented assuming that the bed topography is irregular in the cell. For the case that the cell is partially flooded, this paper proposes a special reconstruction of flow variables that assumes that the bottom function is linear in the cell. The Harten–Lax–van Leer approximate Riemann solver is applied to evaluate the flux at cell faces. The numerical results show good agreement with analytical solutions to a set of test cases and experimental results.

WELL-BALANCED NUMERICAL SCHEMES BASED ON A GENERALIZED HYDROSTATIC RECONSTRUCTION TECHNIQUE

2007 ◽  
Vol 17 (12) ◽  
pp. 2055-2113 ◽  
Author(s):  
MANUEL J. CASTRO ◽  
ALBERTO PARDO MILANÉS ◽  
CARLOS PARÉS
Keyword(s):  
Shallow Water ◽  
Numerical Scheme ◽  
Source Term ◽  
Hyperbolic Systems ◽  
Water System ◽  
The Other ◽  
Reconstruction Technique ◽  
Numerical Schemes ◽  
Shallow Water System ◽  
The One

The goal of this paper is to generalize the hydrostatic reconstruction technique introduced in Ref. 2 for the shallow water system to more general hyperbolic systems with source term. The key idea is to interpret the numerical scheme obtained with this technique as a path-conservative method, as defined in Ref. 35. This generalization allows us, on the one hand, to construct well-balanced numerical schemes for new problems, as the two-layer shallow water system. On the other hand, we construct numerical schemes for the shallow water system with better well-balanced properties. In particular we obtain a Roe method which solves exactly every stationary solution, and not only those corresponding to water at rest.

A Numerical Study for the Solution of Time Fractional Nonlinear Shallow Water Equation in Oceans

2013 ◽  
Vol 68 (8-9) ◽  
pp. 547-553 ◽  
Author(s):  
Sunil Kumar
Keyword(s):  
Analytical Solution ◽  
Shallow Water ◽  
Homotopy Perturbation Method ◽  
Numerical Study ◽  
Shallow Water Equation ◽  
Water System ◽  
General Characteristic ◽  
Convergent Series ◽  
Shallow Water Flows ◽  
Shallow Water System

In this paper, an analytical solution for the coupled one-dimensional time fractional nonlinear shallow water system is obtained by using the homotopy perturbation method (HPM). The shallow water equations are a system of partial differential equations governing fluid flow in the oceans (sometimes), coastal regions (usually), estuaries (almost always), rivers and channels (almost always). The general characteristic of shallow water flows is that the vertical dimension is much smaller than the typical horizontal scale. This method gives an analytical solution in the form of a convergent series with easily computable components, requiring no linearization or small perturbation. A very satisfactory approximate solution of the system with accuracy of the order 10-4 is obtained by truncating the HPM solution series at level six.

scholarly journals Fast and slow resonant triads in the two-layer rotating shallow water equations

2018 ◽  
Vol 850 ◽  
pp. 18-45
Author(s):  
Alex Owen ◽  
Roger Grimshaw ◽  
Beth Wingate
Keyword(s):  
Shallow Water ◽  
Water System ◽  
Layer System ◽  
Interaction Coefficients ◽  
Shallow Water System ◽  
Rotating Shallow Water ◽  
Resonant Triads ◽  
The One ◽  
Baroclinic Modes ◽  
Fast Waves

In this paper, we examine triad resonances in a rotating shallow water system when there are two free interfaces. This allows for an examination in a relatively simple model of the interplay between baroclinic and barotropic dynamics in a context where there is also a geostrophic mode. In contrast to the much-studied one-layer rotating shallow water system, we find that as well as the usual slow geostrophic mode, there are now two fast waves, a barotropic mode and a baroclinic mode. This feature permits triad resonances to occur between three fast waves, with a mixture of barotropic and baroclinic modes, an aspect that cannot occur in the one-layer system. There are now also two branches of the slow geostrophic mode, with a repeated branch of the dispersion relation. The consequences are explored in a derivation of the full set of triad interaction equations, using a multiscale asymptotic expansion based on a small-amplitude parameter. The derived nonlinear interaction coefficients are confirmed using energy and enstrophy conservation. These triad interaction equations are explored, with an emphasis on the parameter regime with small Rossby and Froude numbers.

scholarly journals Exploring a Multiresolution Modeling Approach within the Shallow-Water Equations

2011 ◽  
Vol 139 (11) ◽  
pp. 3348-3368 ◽  
Author(s):  
Todd D. Ringler ◽  
Doug Jacobsen ◽  
Max Gunzburger ◽  
Lili Ju ◽  
Michael Duda ◽  
...  
Keyword(s):  
Shallow Water ◽  
Shallow Water Equations ◽  
Truncation Error ◽  
Water System ◽  
Test Cases ◽  
Grid Spacing ◽  
Variable Resolution ◽  
Mesh Resolution ◽  
Water Test ◽  
Shallow Water System

Abstract The ability to solve the global shallow-water equations with a conforming, variable-resolution mesh is evaluated using standard shallow-water test cases. While the long-term motivation for this study is the creation of a global climate modeling framework capable of resolving different spatial and temporal scales in different regions, the process begins with an analysis of the shallow-water system in order to better understand the strengths and weaknesses of the approach developed herein. The multiresolution meshes are spherical centroidal Voronoi tessellations where a single, user-supplied density function determines the region(s) of fine- and coarse-mesh resolution. The shallow-water system is explored with a suite of meshes ranging from quasi-uniform resolution meshes, where the grid spacing is globally uniform, to highly variable resolution meshes, where the grid spacing varies by a factor of 16 between the fine and coarse regions. The potential vorticity is found to be conserved to within machine precision and the total available energy is conserved to within a time-truncation error. This result holds for the full suite of meshes, ranging from quasi-uniform resolution and highly variable resolution meshes. Based on shallow-water test cases 2 and 5, the primary conclusion of this study is that solution error is controlled primarily by the grid resolution in the coarsest part of the model domain. This conclusion is consistent with results obtained by others. When these variable-resolution meshes are used for the simulation of an unstable zonal jet, the core features of the growing instability are found to be largely unchanged as the variation in the mesh resolution increases. The main differences between the simulations occur outside the region of mesh refinement and these differences are attributed to the additional truncation error that accompanies increases in grid spacing. Overall, the results demonstrate support for this approach as a path toward multiresolution climate system modeling.

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