scholarly journals Flood wave propagation in steep mountain rivers

2012 ◽  
Vol 15 (1) ◽  
pp. 120-137 ◽  
Author(s):  
Gabriella Petaccia ◽  
Luigi Natale ◽  
Fabrizio Savi ◽  
Mirjana Velickovic ◽  
Yves Zech ◽  
...  
Keyword(s):  
Wave Propagation ◽  
Shallow Water ◽  
Shallow Water Equations ◽  
Source Term ◽  
Term Treatment ◽  
Flood Wave ◽  
Numerical Schemes ◽  
Steady State Condition ◽  
One Dimensional ◽  
The One

Most of the recent developments concerning efficient numerical schemes to solve the shallow-water equations in view of real world flood modelling purposes concern the two-dimensional form of the equations or the one-dimensional form written for rectangular, unit-width channels. Extension of these efficient schemes to the one-dimensional cross-sectional averaged shallow-water equations is not straightforward, especially when complex natural topographies are considered. This paper presents different formulations of numerical schemes based on the HLL (Harten–Lax–van Leer) solver, and the adaptation of the topographical source term treatment when cross-sections of arbitrary shape are considered. Coupled and uncoupled formulations of the equations are considered, in combination with centred and lateralised source term treatment. These schemes are compared to a numerical solver of Lax Friedrichs type based on a staggered grid. The proposed schemes are first tested against two theoretical benchmark tests and then applied to the Brembo River, an Italian alpine river, firstly simulating a steady-state condition and secondly reproducing the 2002 flood wave propagation.

scholarly journals Deficiency of finite difference methods for capturing shock waves and wave propagation over uneven bottom seabed

2018 ◽  
Vol 7 (3.28) ◽  
pp. 97
Author(s):  
Mohammad Fadhli Ahmad ◽  
Mohd Sofiyan Suliman ◽  
. .
Keyword(s):  
Wave Propagation ◽  
Finite Difference ◽  
Shallow Water ◽  
Shallow Water Equations ◽  
Source Term ◽  
Finite Difference Methods ◽  
Dam Break ◽  
Numerical Schemes ◽  
Inaccurate Information ◽  
Uneven Bottom

The implementation of finite difference method is used to solve shallow water equations under the extreme conditions. The cases such as dam break and wave propagation over uneven bottom seabed are selected to test the ordinary schemes of Lax-Friederichs and Lax-Wendroff numerical schemes. The test cases include the source term for wave propagation and exclude the source term for dam break. The main aim of this paper is to revisit the application of Lax-Friederichs and Lax-Wendroff numerical schemes at simulating dam break and wave propagation over uneven bottom seabed. For the case of the dam break, the two steps of Lax-Friederichs scheme produce non-oscillation numerical results, however, suffering from some of dissipation. Moreover, the two steps of Lax-Wendroff scheme suffers a very bad oscillation. It seems that these numerical schemes cannot solve the problem at discontinuities which leads to oscillation and dissipation. For wave propagation case, those numerical schemes produce inaccurate information of free surface and velocity due to the uneven seabed profile. Therefore, finite difference is unable to model shallow water equations under uneven bottom seabed with high accuracy compared to the analytical solution.  

scholarly journals Convergence Improved Lax-Friedrichs Scheme Based Numerical Schemes and Their Applications in Solving the One-Layer and Two-Layer Shallow-Water Equations

2015 ◽  
Vol 2015 ◽  
pp. 1-10
Author(s):  
Xinhua Lu ◽  
Bingjiang Dong ◽  
Bing Mao ◽  
Xiaofeng Zhang
Keyword(s):  
High Resolution ◽  
Shallow Water ◽  
Shallow Water Equations ◽  
Small Time ◽  
Numerical Schemes ◽  
Time Step ◽  
Numerical Diffusion ◽  
First Order ◽  
Small Time Step ◽  
The One

The first-order Lax-Friedrichs (LF) scheme is commonly used in conjunction with other schemes to achieve monotone and stable properties with lower numerical diffusion. Nevertheless, the LF scheme and the schemes devised based on it, for example, the first-order centered (FORCE) and the slope-limited centered (SLIC) schemes, cannot achieve a time-step-independence solution due to the excessive numerical diffusion at a small time step. In this work, two time-step-convergence improved schemes, the C-FORCE and C-SLIC schemes, are proposed to resolve this problem. The performance of the proposed schemes is validated in solving the one-layer and two-layer shallow-water equations, verifying their capabilities in attaining time-step-independence solutions and showing robustness of them in resolving discontinuities with high-resolution.

scholarly journals A Proposed Implicit Friction Source Term Treatment for Simulating Overland Flow

10.29007/fsp8 ◽  
2018 ◽  
Author(s):  
Tian Wang ◽  
Jingming Hou ◽  
Peng Li ◽  
Jiaheng Zhao ◽  
Ilhan Özgen ◽  
...  
Keyword(s):  
Shallow Water ◽  
Shallow Water Equations ◽  
Overland Flow ◽  
Source Term ◽  
Specific Treatment ◽  
Term Treatment ◽  
Finite Volume Scheme ◽  
Flow Problems ◽  
Friction Term ◽  
Water Depths

The fully 2D dynamic shallow water equations have been widely applied for numerical simulation of overland flow in the recent years. However, most of the existing friction term discretisation schemes do not recover the correct asymptotic flow behaviour as water depths becomes small. In this model, the shallow water equations were discretized by the framework of the Godunov-type finite volume scheme. The hydrostatic reconstruction is applied to reconstruct non-negative water depths at wet- dry interfaces. Numerical fluxes are computed with a HLLC solver. The novel aspects of the model include the slope source term treatment. Specific treatment of friction source terms has been proposed to discretize the friction terms to recover the correct asymptotic behaviour of SWEs when the water depth becomes small. The accuracy and robustness of the proposed model are verified by comparing with analytical solutions. The results demonstrate that the proposed method treating friction source term is a relatively more accurate, efficient, straightforward and universal one for evaluating overland flow problems.

scholarly journals Implicit and time reversible CABARET schemes for quasilinear shallow water equations

2016 ◽  
pp. 402-414
Author(s):  
В.М. Головизнин ◽  
Д.Ю. Горбачев ◽  
А.М. Колокольников ◽  
П.А. Майоров ◽  
П.А. Майоров ◽  
...  
Keyword(s):  
Difference Scheme ◽  
Numerical Solution ◽  
Shallow Water ◽  
Riemann Problem ◽  
Shallow Water Equations ◽  
Unconditionally Stable ◽  
One Dimensional ◽  
Dispersion Properties ◽  
The One ◽  
Stable Scheme

Предложена новая неявная безусловно устойчивая схема для одномерных уравнений мелкой воды, сохраняющая все особенности явной схемы Кабаре. Проведен анализ диссипативных и дисперсионных свойств новой схемы и предложен алгоритм ее численного решения. Приведены примеры решения задачи о распаде разрыва. A new implicit unconditionally stable scheme for the one-dimensional shallow water equations is proposed. This implicit scheme retains all the features of the explicit CABARET (Compact Accurately Boundary Adjusting-REsolution Technique) difference scheme. Dissipative and dispersion properties of this new scheme are analyzed; an algorithm of its numerical solution is discussed. Some examples of solving the Riemann problem are considered.

CONSERVATION LAWS FOR ONE-DIMENSIONAL SHALLOW WATER MODELS FOR ONE AND TWO-LAYER FLOWS

2006 ◽  
Vol 16 (01) ◽  
pp. 119-137 ◽  
Author(s):  
RICARDO BARROS
Keyword(s):  
Conservation Laws ◽  
Shallow Water ◽  
Shallow Water Equations ◽  
Water Model ◽  
Shallow Water Model ◽  
Frobenius Problem ◽  
One Dimensional ◽  
Shallow Water Models ◽  
The One ◽  
Open Question

A full set of conservation laws for the two-layer shallow water equations is presented for the one-dimensional case. We prove that all the conservation laws are linear combination of the equations for the conservation of mass and velocity (in each layer), total momentum and total energy.This result generalizes that of Montgomery and Moodie that found the same conserved quantities by restricting their search to the multinomials expressions in the layer variables. Though the question of whether or not there are only a finite number of these quantities is left as an open question by the authors. Our work puts an end to this: in fact, no more conservation laws are admitted for the two-layer shallow water equations. The key mathematical ingredient of the method proposed leading to the result is the Frobenius problem. Moreover, we present a full set of conservation laws for the classical one-dimensional shallow water model with topography, by using the same techniques.

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