Three-dimensional shallow water system: A relaxation approach (original) (raw)
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A finite volume shock-capturing solver of the fully coupled shallow water-sediment equations
International Journal for Numerical Methods in Fluids, 2017
This paper describes a numerical solver of well-balanced, 2D depth-averaged shallow water-sediment equations. The equations permit variable variable horizontal fluid density and are designed to model watersediment flow over a mobile bed. A Godunov-type, HLLC finite volume scheme is used to solve the fully coupled system of hyperbolic conservation laws which describe flow hydrodynamics, suspended sediment transport, bedload transport and bed morphological change. Dependent variables are specially selected to handle the presence of the variable density property in the mathematical formulation. The model is verified against analytical and semi-analytical solutions for bedload transport and suspended sediment transport, respectively. The well-balanced property of the equations is verified for a variable-density dam break flow over discontinuous bathymetry. Simulations of an idealised dam-break flow over an erodible bed are in excellent agreement with previously published results ([1]), validating the ability of the model to capture the complex interaction between rapidly varying flow and an erodible bed and validating the eigenstructure of the system of variable-density governing equations. Flow hydrodynamics and final bed topography of a laboratory-based 2D partial dam breach over a mobile bed are satisfactorily reproduced by the numerical model. Comparison of the final bed topographies, computed for two distinct sediment transport methods, highlights the sensitivity of shallow water-sediment models to the choice of closure relationships.
Applied Mathematics and Computation, 2009
A class of central unstaggered finite volume methods for approximating solutions of hyperbolic systems of conservation laws is developed in this paper. The proposed method is an extension of the central, non-oscillatory, finite volume method of Nessyahu and Tadmor (NT). In contrast with the original NT scheme, the method we develop evolves the numerical solution on a single grid; however ghost cells are implicitly used to avoid the resolution of the Riemann problems arising at the cell interfaces. We apply our method and solve classical one and two-dimensional unsteady shallow water problems. Our numerical results compare very well with those obtained using the original NT method, and are in good agreement with corresponding results appearing in the recent literature, thus confirming the efficiency and the potential of the proposed method.
Communications in Computational Physics, 2012
Abstract. We construct a new first order central-upwind numerical method for solving systems of hyperbolic equations in conservative form. It applies in multidimensional structured and unstructured meshes. The proposed method is an extension of the UFORCE method developed by Stecca, Siviglia and Toro,[25] in which the upwind bias for the modification of the staggered mesh is evaluated taking into account the smallest and largest wave of the entire Riemann fan. The proposed first-order method is shown to be identical ...
The shallow water equations: An example of hyperbolic system
2008
Many problems of river management and civil protection consist of the evaluation of the maximum water levels and discharges that may be attained at particular locations during the development of an exceptional meteorological event. There is also the prevision of the scenario subsequent to the almost instantaneous release of a great volume of liquid. The situation is that of the breaking of a man made dam. There is therefore a necessity to develop adequate numerical models able to reproduce situations originated by the irregularities of a non-prismatic bed. It is also necessary to trace their applicability considering the difficulty of developing a model capable of producing solutions of the complete equations despite the irregular character of the river bed. When trying to use mathematical models as a predictive tool in the simulation of free surface flows, the hypothesis of one-dimensional models are not always valid. Such is the case when dealing with compound, or highly irregular, cross-section configurations, abrupt contractions and expansions, or rivers of high curvature. When trying to reproduce these hydraulic situations, it becomes necessary to use a two-dimensional formalism which takes into consideration the influence of transverse components of the flow. Many efforts have been recently devoted to the development of multidimensional techniques for free surface flows.
Journal of Applied & Computational Mathematics
The paper deals with the unsteady two-dimensional (2D) non-linear shallow-water equations (SWE) in conservation-law form to capture the fluid flow in transition. Numerical simulations of dam-break flood wave in channel transitions have been performed for inviscid and incompressible flow by using two new implicit higher-order compact (HOC) schemes. The algorithm is second order accurate in time and fourth order accurate in space, on the nine-point stencil using third order non-centered difference at the wall boundaries. To solve the algebraic system, bi-conjugate gradient stabilized method (BiCGStab) with preconditioning has been employed. Although, both the schemes are able to capture both transient and steady state solution of shallow water equations, the scheme expressed in conservative law form is unconditionally stable. The model results have been validated for dam-break problem and compared with the experimental data for dry and wet bed conditions. The model results are found to be in good agreement with the experimental observations. The proposed scheme is useful to solve to capture flow transition with minimal number of nodal points, particularly for hyperbolic system.
Journal of Scientific Computing, 2006
An energy-stable high-order central finite difference scheme is derived for the two-dimensional shallow water equations. The scheme is mathematically formulated using the semi-discrete energy method for initial boundary value problems described in Olsson (1995, Math. Comput. 64, 1035-1065): after symmetrizing the equations via a change to entropy variables, the flux derivatives are entropy-split enabling the formulation of a semi-discrete energy estimate. We show experimentally that the entropy-splitting improves the stability properties of the fully discretized equations. Thus, the dependence on numerical dissipation to keep the scheme stable for long term time integrations is reduced relative to the original unsplit form, thereby decreasing non-physical damping of solutions. The numerical dissipation term used with the entropy-split equations is in a form which preserves the semi-discrete energy estimate. A random onedimensional dam break calculation is performed showing that the shock speed is computed correctly for this particular case, however it is an open question whether the correct shock speed will be computed in general.
A Fast and Stable Well-Balanced Scheme with Hydrostatic Reconstruction for Shallow Water Flows
SIAM Journal on Scientific Computing, 2004
We consider the Saint-Venant system for shallow water flows, with nonflat bottom. It is a hyperbolic system of conservation laws that approximately describes various geophysical flows, such as rivers, coastal areas, and oceans when completed with a Coriolis term, or granular flows when completed with friction. Numerical approximate solutions to this system may be generated using conservative finite volume methods, which are known to properly handle shocks and contact discontinuities. However, in general these schemes are known to be quite inaccurate for near steady states, as the structure of their numerical truncation errors is generally not compatible with exact physical steady state conditions. This difficulty can be overcome by using the so-called well-balanced schemes. We describe a general strategy, based on a local hydrostatic reconstruction, that allows us to derive a well-balanced scheme from any given numerical flux for the homogeneous problem. Whenever the initial solver satisfies some classical stability properties, it yields a simple and fast well-balanced scheme that preserves the nonnegativity of the water height and satisfies a semidiscrete entropy inequality.
A new composite scheme for two-layer shallow water flows with shocks
Journal of Applied Mathematics and Computing, 2014
This paper is devoted to solve the system of partial differential equations governing the flow of two superposed immiscible layers of shallow water flows. The system contains source terms due to bottom topography, wind stresses, and nonconservative products describing momentum exchange between the layers. The presence of these terms in the flow model forms a nonconservative system which is only conditionally hyperbolic. In addition, twolayer shallow water flows are often accompanied with moving discontinuities and shocks. Developing stable numerical methods for this class of problems presents a challenge in the field of computational hydraulics. To overcome these difficulties, a new composite scheme is proposed. The scheme consists of a time-splitting operator where in the first step the homogeneous system of the governing equations is solved using an approximate Riemann solver. In the second step a finite volume method is used to update the solution. To remove the non-physical oscillations in the vicinity of shocks a nonlinear filter is applied. The method is well-balanced, non-oscillatory and it is suitable for both low and high values of the density ratio between the two layers. Several standard test examples for two-layer shallow water flows are used to verify high accuracy and good resolution properties for smooth and discontinuous solutions.
Finite-volume schemes for shallow-water equations
Acta Numerica, 2018
Shallow-water equations are widely used to model water flow in rivers, lakes, reservoirs, coastal areas, and other situations in which the water depth is much smaller than the horizontal length scale of motion. The classical shallow-water equations, the Saint-Venant system, were originally proposed about 150 years ago and still are used in a variety of applications. For many practical purposes, it is extremely important to have an accurate, efficient and robust numerical solver for the Saint-Venant system and related models. As their solutions are typically non-smooth and even discontinuous, finite-volume schemes are among the most popular tools. In this paper, we review such schemes and focus on one of the simplest (yet highly accurate and robust) methods: central-upwind schemes. These schemes belong to the family of Godunov-type Riemann-problem-solver-free central schemes, but incorporate some upwinding information about the local speeds of propagation, which helps to reduce an ex...