No full-text available
To read the full-text of this research,
you can request a copy directly from the authors.
Unstructured finite volume discretisation of bed friction and convective flux in solute transport models linked to the shallow water equations
... Numerical modeling of the scalar transport process is a very useful and important tool for surface water pollution risk assessment which is of critical importance in a wide variety of applications for environmental protection and management [1,2]. The coupled system of shallow water equations (SWEs) and solute transport model is usually used to predict the dynamics of both water flow and contaminant transport in rivers and coastal regions [3][4][5][6][7][8]. In shallow flows, vertical scales are much smaller than horizontal ones, the velocity profile exhibits small changes throughout the depth, and the pressure is close to hydrostatic [9]. ...
... The nonzero components of the source term due to variable topography (2) and (3) are discretized using a special quadrature technique to guarantee the well-balanced property of the proposed scheme. The semi-discrete form of the scheme (7) should exactly preserve ''lake at rest'' steady state solutions where we used the initial condition (20) under wall-boundary conditions. ...
... The same procedure can be applied for (3) to deduce the following discretization: ...
We develop an unstructured numerical scheme for a coupled system modeling shallow water flows and solute transport over variable topography. A novel algorithm is introduced for the reconstructions of the variables of the system with variable density. These reconstructions are used in combination with the expression of the relative density of the mixture to guarantee the positivity required for some physical parameters of the coupled model. New discretization techniques are developed to guarantee the well-balanced property of the scheme and the consistency between the transport equation and the continuity equation at the discrete level. We prove the well-balanced property of the proposed method as well as the positivity preserving property of the scheme for both the water depth and concentrations. The performance of the proposed scheme is tested on a number of numerical examples, among which we consider nontrivial analytical solutions for the equilibrium with mixture constituents, parabolic wave with pollutant transport, and dam-break problem for modeling solute transport in rapidly varying flow. The numerical results confirm stability and well-balanced property of the scheme, consistency between the discretizations of the continuity and transport equations, positivity preserving property required for some physical parameters, and accuracy of the proposed method in modeling the dynamics of water flow and scalar transport.
... The shallow water equations(SWEs), also called the Saint-Venant system [17], is a system of partial-differential equations, commonly used to predict and describe flows where the water depth is much smaller than the horizontal length scale of motion and the variations of the flow in the vertical direction is negligible compared to its horizontal variations [43,41,6]. The SWEs are widely used in many applications involving free-surface flows in lakes, rivers, and oceans, and become an efficient tool for studying a wide range of hydraulic engineering problems, as well as tsunami and weather predictions [9,21,18,14,44]. The coupled system of SWEs and transport equation is used in many studies to predict the dynamics and the distribution of the concentration of pollutant in flows [14,3,2,32], which have various applications in environmental risk assessment for the development of solutions for sustainable water resources management. ...
... The SWEs are widely used in many applications involving free-surface flows in lakes, rivers, and oceans, and become an efficient tool for studying a wide range of hydraulic engineering problems, as well as tsunami and weather predictions [9,21,18,14,44]. The coupled system of SWEs and transport equation is used in many studies to predict the dynamics and the distribution of the concentration of pollutant in flows [14,3,2,32], which have various applications in environmental risk assessment for the development of solutions for sustainable water resources management. The evolution of pollutants is complex, and is subject to various phenomena such as advection, dispersion and diffusion. ...
... In the present study, we propose a new finite volume method for the simulation and prediction of pollution concentration in water bodies. We consider the coupled system of SWEs and the scalar transport equation [14,3,44,2,32], with different source terms, such as variable bottom topography, bottom friction effects and diffusion. Novel techniques are proposed to numerically solve the resulting non-linear system where we extended the methodology proposed in [5] to our system. ...
We develop a new finite volume method using unstructured mesh-vertex grids for coupledsystems modeling shallow water flows and solute transport over complex bottom topography.Novel well-balanced positivity preserving discretization techniques are proposed for the watersurface elevation and the concentration of the pollutant. For the hydrodynamic system, theproposed scheme preserves the steady state of a lake at rest and the positivity of the waterdepth. For the scalar transport equation, the proposed method guarantees the positivityand a perfect balance of the scalar concentration. The constant-concentration states arepreserved in space and time for any hydrodynamic field and complex topography in theabsence of source terms of the passive pollutant. Importantly and this is one of the mainfeatures of our approach is that the novel reconstruction techniques proposed for the watersurface elevation and concentration satisfy the discrete maximum-minimum principle for thesolute concentration. We demonstrate, in a series of numerical tests, the well-balanced andpositivity properties of the proposed method and the accuracy of our techniques and theirpotential advantages in predicting the solutions of the shallow water-transport model.
... Meanwhile, the DHD scheme provides more stable and rapid results in rainfall-runoff applications at the catchment and hillslope scales García-Alén et al., 2022;Sanz-Ramos et al., 2021). In both cases the bed friction is discretised with a semi-implicit scheme in order to enhance the numerical stability of the solver (Cea and Vázquez-Cendón, 2012). ...
... The suspended sediment transport equation is solved using the explicit finite volume scheme for scalar transport equations described in (Cea and Vázquez-Cendón, 2012), which ensures the conservation of the mass of sediment. The main particularity of equation (6) with respect to a typical scalar transport equation are the source terms, namely , , , , , , , , , . ...
We present the implementation of a new fully distributed multiclass soil erosion module. The model is based on a 2D finite volume solver (Iber+) for the 2D shallow water equations that computes the overland flow water depths and velocities. From these, the model evaluates the transport of sediment particles due to bed load and suspended load, including rainfall-driven and runoff-driven erosion processes, and using well-established physically-based formulations. The evolution of the mass of sediment particles in the soil layer is computed from a mass conservation equation for each sediment class. The solver is implemented using High Performance Computing techniques that take advantage of the computational capabilities of standard Graphical Processing Units, achieving speed-ups of two orders of magnitude relative to a sequential implementation on the CPU. We show the application and validation of the model at different spatial scales, ranging from laboratory experiments to meso-scale catchments.
... Iber + implements an unstructured finite volume solver to solve the hydrodynamic and sediment transport equations. Extensive details about the numerical schemes used to solve these equations in rainfallrunoff applications can be found in Cea and Bladé (2015) and Cea and Vázquez-Cendón (2012). In particular, the DHD scheme was used in this work, since it has proven to be very efficient for solving the 2D-SWE in rainfall-runoff applications , García-Alén et al., 2022, in which the presence of highly unsteady wet-dry fronts, the small water depths and the high bed slopes can promote numerical instabilities. ...
... In particular, the DHD scheme was used in this work, since it has proven to be very efficient for solving the 2D-SWE in rainfall-runoff applications , García-Alén et al., 2022, in which the presence of highly unsteady wet-dry fronts, the small water depths and the high bed slopes can promote numerical instabilities. The finite volume discretisation of the suspended sediment transport equation ensures the positiveness of the sediment concentration and the global conservation of the sediment mass (Cea and Vázquez-Cendón, 2012). ...
... In our study, the physically based model used for surface water flows is the shallow water equations (SWEs), also called the Saint Venant system (De St Venant 1871). The SWEs are widely used to study systems of shallow water flows where the water depth is much smaller than the horizontal length scale of these systems and the variations of the flow in the vertical direction is negligible compared to the variations in the horizontal one (Mugler, et al. 2011, Cea and Vazquez-Cendon 2012, Yoon and Kang 2004. For subsurface water flow, the Green-Ampt model (Green and Ampt 1911) is considered to describe the infiltration process through soils. ...
... The Manning formula is based on the assumption of uniform flow (Marcus, et al. 1992) and it is initially developed for flow in open channels and pipes. The Manning formula is widely used in many applications to compute the friction stresses in runoff models (Cea andVazquez-Cendon 2012, Fernández-Pato, Gracia andGarcía-Navarro 2018). In this study, we assume a constant Manning coefficient. ...
In this paper, we propose to use the HLL finite volume scheme combined with implicit techniques for modelling the coupled surface and subsurface water flows. In our approach, we used the shallow water equations modelling surface water flow with different source terms such as variable bottom topography, friction effects, precipitation and infiltration. For subsurface water flow, the Green-Ampt equation is used to simulate the infiltration process through soils. For solving the resulting nonlinear-coupled system of shallow water flow and the Green-Ampt infiltration equations, the HLL finite volume scheme with linear reconstructions of the solutions at the discrete level are implemented in order to achieve the second-order accuracy of the scheme. Appropriate discretization techniques are used for the source terms to guarantee the well-balanced property of our numerical method. Numerical experiments are performed to test the capability of the developed numerical scheme to simulate the coupled surface and subsurface water flows.
... In our study, the physically based model used for surface water flows is the shallow water equations (SWEs), also called the Saint Venant system (De St Venant 1871). The SWEs are widely used to study systems of shallow water flows where the water depth is much smaller than the horizontal length scale of these systems and the variations of the flow in the vertical direction is negligible compared to the variations in the horizontal one (Mugler, et al. 2011, Cea and Vazquez-Cendon 2012, Yoon and Kang 2004. For subsurface water flow, the Green-Ampt model (Green and Ampt 1911) is considered to describe the infiltration process through soils. ...
... The Manning formula is based on the assumption of uniform flow (Marcus, et al. 1992) and it is initially developed for flow in open channels and pipes. The Manning formula is widely used in many applications to compute the friction stresses in runoff models (Cea andVazquez-Cendon 2012, Fernández-Pato, Gracia andGarcía-Navarro 2018). In this study, we assume a constant Manning coefficient. ...
In this paper, we propose to use the HLL finite volume scheme combined with implicit techniques for modelling the coupled surface and subsurface water flows. In our approach, we used the shallow water equations modelling surface water flow with different source terms such as variable bottom topography, friction effect, precipitation and infiltration. For subsurface water flow, the Green-Amp equation is used to simulate the infiltration process through soils. For solving the resulting nonlinear-coupled system of shallow water flow and the Green-Ampt infiltration equations, the HLL finite volume schemes with linear reconstructions of the solutions at the discrete level are implemented in order to achieve the second-order accuracy of the scheme. Appropriate discretization techniques are used for the source terms to guarantee the well-balanced property of our numerical scheme. Numerical experiments are performed to test the capability of the developed numerical scheme to simulate the coupled surface and subsurface water flows.
... Both Cea et al. (2016) in 2D and Gordillo et al. (2019) in 1D are based on a coupled discretization of the hydraulic and water quality parts that ensures positivity by considering the effects of bed slope and friction source terms. On this subject, different techniques have been developed to control the two main basic properties that any numerical scheme with solute transport must comply with: C-property and positivity (Cea and Vázquez-Cendón, 2012;Liu, 2019;. ...
... Following the same philosophy, a first-order upwind scheme can be formulated for the transport of conservative and non-conservative solutes. Previous studies (Cea and Vázquez-Cendón, 2012;Murillo et al., 2009) have shown that the scheme is able to guarantee conservation. However, it may give rise to unrealistic values when dealing with problems that involve bed variations and transient flows over wet/dry fronts, leading to unbounded values in the final solute concentration (Morales-Hernández et al., 2012). ...
In this study, a 2D shallow water flow solver integrated with a water quality model is presented. The interaction between the main water quality constituents included is based on the Water Quality Analysis Simulation Program. Efficiency is achieved by computing with a combination of a CPU and a Graphics Processing Unit device. This technique is intended to provide robust and accurate simulations with high computation speedups with respect to a single-core CPU in real events. The proposed numerical model is evaluated in cases that include the transport and reaction of water quality components over irregular bed topography and dry–wet fronts, verifying that the numerical solution in these situations conserves the required properties (C-property and positivity). The model can operate in any steady or unsteady form allowing an efficient assessment of the environmental impact of water flows. The field data from an unsteady river reach test case are used to show that the model is capable of predicting the measured temporal distribution of dissolved oxygen and water temperature, proving the robustness and computational efficiency of the model, even in the presence of noisy signals such as wind speed.
... The equations are solved with a finite volume solver which implements a second order MUSCL extension of the Roe scheme (Roe, 1986). A detailed description of the numerical schemes implemented in the surface drainage model has already been presented in previous publications by the authors (Cea & Vázquez-Cendón, 2012) and is not included here. The numerical model has been validated under overland flow conditions, including rainfall-runoff transformation, in Cea et al. (2010aCea et al. ( , 2010bCea et al. ( , 2014, where it has proved to efficiently deal with some of the main numerical difficulties that appear in the modelling of overland flow, including the presence of highly unsteady wet -dry fronts, shallow water depths, and high bed friction. ...
... The bed slope source term is explicitly computed. Bed friction is the only term in Equation (2) that is computed with a semi-implicit scheme, which is a quite common technique to avoid numerical instabilities when the water depth approaches zero (Begnudelli et al., 2008;Cea & Vázquez-Cendón, 2012;Liang & Marche, 2009;Sanders et al., 2008;Wu 2004). ...
... (5.28). Another difficulty is related to the fact that this system admits not only 'lake at rest' steady states (5.25), but other steady-state solutions, which may become more relevant in many practical situations, for example, when drainage of rain water in urban areas is simulated; see, for example, Cea, Garrido and Puertas (2010) and Cea and Vázquez-Cendón (2012) and references therein. Let us first consider the simplest 1-D case, in which the system (5.27), ...
... are still physically relevant. A well-balanced Roe-type numerical scheme which is capable of exactly preserving the steady states (5.30), (5.31) and (5.32) was proposed in Cea and Vázquez-Cendón (2012). However, to maintain the positivity of the water depth h, the scheme in Cea and Vázquez-Cendón (2012) may require one to use very small time steps and thus may not be robust in certain settings. ...
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 excessive amount of numerical diffusion typically present in classical (staggered) non-oscillatory central schemes. Besides the classical one- and two-dimensional Saint-Venant systems, we will consider the shallow-water equations with friction terms, models with moving bottom topography, the two-layer shallow-water system as well as general non-conservative hyperbolic systems.
... Remark 2.1 (C-Property). As described in [13,50,51], instead of speaking of well-balanced schemes, one could alternatively say that a scheme enjoys the C-property if it preserves exactly the steady state (3). However one still speaks of C-property when referring to other steady states (4). ...
In this paper, we develop and present an arbitrary high order well-balanced finite volume WENO method combined with the modified Patankar Deferred Correction (mPDeC) time integration method for the shallow water equations. Due to the positivity-preserving property of mPDeC, the resulting scheme is unconditionally positivity preserving for the water height. To apply the mPDeC approach, we have to interpret the spatial semi-discretization in terms of production-destruction systems. Only small modifications inside the classical WENO implementation are necessary and we explain how it can be done. In numerical simulations, focusing on a fifth order method, we demonstrate the good performance of the new method and verify the theoretical properties.
... Mathematical details about the discretization schemes implemented in the model have been published in several previous papers and are therefore not reproduced here. The reader is referred to Bladé et al. (2014), Cea and Bladé (2015), and Cea and Vázquez-Cendón (2012) for detailed descriptions of the numerical schemes implemented in the solver. ...
The backwater effect generated by bridges can significantly increase the risk of flooding. In this work we compare two different methods to include the effect of bridges in two-dimensional (2D) shallow water models. The first method is based on empirical discharge equations that are implemented as internal conditions. The second method is the recently proposed 2D extension of the two-component pressure approach, which accounts for the vertical confinement of the flow. Both approaches are tested and compared using a new set of experimental data obtained in 32 laboratory tests, including four different bridge geometries under different flow conditions. The results show that both methods can reproduce the observed bridge afflux for a wide range of flow conditions, but the two-component pressure approach is less dependent on model calibration. On the other hand, both methods fail to correctly reproduce the 2D water depth patterns observed around the bridge.
... In this study, we consider the coupled system of SWEs and the scalar transport equation [2,3,14,34,48] with source terms due to variable topography, bottom friction effects, and diffusion. We introduce an artificial dissipation parameter to improve the original cell-vertex central-upwind scheme [5] in terms of numerical dissipation, especially for the scalar concentration. ...
We consider a system of coupled equations modeling a shallow water flow with solute transport and introduce an artificial dissipation in order to improve the dissipation properties of the original cell-vertex central-upwind numerical scheme applied to these equations. Namely, a formulation is proposed which involves an artificial dissipation parameter and guarantees a consistency property
between the continuity equation and the transport equation at the discrete level and, in addition, ensures the nonlinear stability and positivity of the scheme. A well-balanced positivity preserving reconstruction is stated in terms of the conservative variable describing the concentration. We establish that constant-concentration states are preserved in space and time for any hydrodynamic flow field in the absence of source terms in the transport equation. Furthermore, we prove the maximum and minimum principles for the concentration. A suitable discretization of the diffusion
term is used in combination with the proposed reconstruction procedure and artificial dissipation formulation and this allows us to prove the positivity of the concentration in the presence of diffusion effects. Finally, our numerical experiments confirm the well-balanced and positivity preserving properties when the artificial dissipation is introduced in the central-upwind scheme, and the accuracy of the scheme for modeling surface water flows with transport processes.
... Following this, the transport equation models the contaminant distribution over the lake under different wind conditions. The SWE can be written as (16,17) . ...
In this paper, we analyze the distribution of a non-reactive contaminant in Ypacarai Lake. We propose a shallow-water model that considers wind-induced currents, inflow and outflow conditions in the tributaries, and bottom effects due to the lakebed. The hydrodynamic is based on the depth-averaged Navier-Stokes equations considering wind stresses as force terms which are functions of the wind velocity. Bed (bottom) stress is based on Manning's equation, the lakebed characteristics, and wind velocities. The contaminant transportation is modeled by a 2D convection-diffusion equation taking into consideration water level. Comparisons between the simulation of the model, analytical solutions, and laboratory results confirm that the model captures the complex dynamic phenomenology of the lake. In the simulations, one can see the regions with the highest risk of accumulation of contaminants. It is observed the effect of each term and how it can be used them to mitigate the impact of the pollutants.
... Remark 2.1 (C-Property). As described in [14,52,53], instead of speaking of well-balanced schemes, one could alternatively say that a scheme enjoys the C-property if it preserves exactly the steady state (3). However one still speaks of C-property when referring to other steady states (4). ...
In this paper, we develop and present an arbitrary high order well-balanced finite volume WENO method combined with the modified Patankar Deferred Correction (mPDeC) time integration method for the shallow water equations. Due to the positivity-preserving property of mPDeC, the resulting scheme is unconditionally positivity preserving for the water height. To apply the mPDeC approach, we have to interpret the spatial semi-discretization in terms of production-destruction systems. Only small modifications inside the classical WENO implementation are necessary and we explain how it can be done. In numerical simulations, focusing on a fifth order method, we demonstrate the good performance of the new method and verify the theoretical properties.
... Numerical modelling of the scalar transport process is of a great importance for surface water pollution risk assessment which is used in a large variety of applications for environmental protection and management (Rekolainen et al. 2003, Horn et al. 2004). The coupled system of shallow water equations and solute transport model are used to predict the dynamics of both water flow and contaminant transport in rivers and coastal regions (Cea andVazquez-Cendon 2012, Begnudelli andSanders 2006). In shallow flows, horizontal scales are predominant over vertical ones and the pressure is nearly hydrostatic (Vreugdenhil 1994). ...
In this study, we focus on the modelling of coupled systems of shallow water flows and solute transport with source terms due to variable topography and friction effect. Our aim is to propose efficient and accurate numerical techniques for modelling these systems using unstructured triangular grids. We used a Riemann-solver free method for the hyperbolic shallow water system and a suitable discretization technique for the bottom topography. The friction source term is discretized using the techniques proposed by (Xia and Liang 2018). Our approach performs very well for stationary flow in the presence of variable topography, and it is well-balanced for the concentration in the presence of wet and dry zones. In our techniques, we used linear piecewise reconstructions for the variables of the coupled system. The proposed method is well-balanced, and we prove that it exactly preserves the nontrivial steady-state solutions of the coupled system. Numerical experiments are carried out to validate the performance and robustness of the proposed numerical method. Our numerical results show that the method is stable, well-balanced and accurate to model the coupled systems of shallow water flows and solute transport.
... The solver is explicit in time and thus, the computational time step is limited by the Courant-Friedrichs-Lewy (CFL) stability condition [22]. A detailed mathematical description of the discretization schemes used in the model can be found in [18,23,24]. The model has been validated extensively and applied to a large number of open channel flow and river inundation studies [25][26][27][28]. ...
2D models based on the shallow water equations are widely used in river hydraulics. However, these models can present deficiencies in those cases in which their intrinsic hypotheses are not fulfilled. One of these cases is in the presence of weirs. In this work we present an experimental dataset including 194 experiments in nine different weirs. The experimental data are compared to the numerical results obtained with a 2D shallow water model in order to quantify the discrepancies that exist due to the non-fulfillment of the hydrostatic pressure hypotheses. The experimental dataset presented can be used for the validation of other modelling approaches.
... Equation (2) is solved with an unstructured finite-volume solver using the same spatial discretization as for the hydrodynamic equations. For a detailed description of the numerical schemes used to solve Eq. (2) coupled to the shallow-water equations, the reader is referred to Cea and Vázquez-Cendón (2012). The solution of Eq. ...
Soil erosion and suspended sediment transport understanding is an important issue in terms of soil and water resources management in the critical zone. In mesoscale watersheds (>10 km2) the spatial distribution of potential sediment sources within the catchment associated with rainfall dynamics is considered to be the main factor in the observed suspended sediment flux variability within and between runoff events. Given the high spatial heterogeneity that can exist for such scales of interest, distributed physically based models of soil erosion and sediment transport are powerful tools to distinguish the specific effect of structural and functional connectivity on suspended sediment flux dynamics. As the spatial discretization of a model and its parameterization can crucially influence how the structural connectivity of the catchment is represented in the model, this study analyzed the impact of modeling choices in terms of the contributing drainage area (CDA) threshold to define the river network and of Manning's roughness parameter (n) on the sediment flux variability at the outlet of two geomorphologically distinct watersheds. While the modeled liquid and solid discharges were found to be sensitive to these choices, the patterns of the modeled source contributions remained relatively similar when the CDA threshold was restricted to the range of 15 to 50 ha, with n restricted to the range 0.4–0.8 on the hillslopes and to 0.025–0.075 in the river. The comparison of the two catchments showed that the actual location of sediment sources was more important than the choices made during discretization and parameterization of the model. Among the various structural connectivity indicators used to describe the geological sources, the mean distance to the stream was the most relevant proxy for the temporal characteristics of the modeled sedigraphs.
... Both, the upwind discretization of the bed slope and the semiimplicit discretization of the friction term are common practice in finite volume solvers for the shallow water equations. [45][46][47][48][49][50] The proposed numerical scheme is first order in time. ...
Numerical models based on the two‐dimensional shallow water equations (2D‐SWE) are routinely used in flood risk management and inundation studies. However, most of these models do not adequately account for vertically confined flow conditions that can appear during inundations, due to the presence of hydraulic structures such as bridges, culverts or underground river reaches. In this paper we propose a new mathematical modification of the standard 2D‐SWE, inspired by the Two‐Component Pressure Approach for 1D flows, to address the issue of transient vertically confined flows including transitions between free surface and pressurised conditions. A finite volume discretisation to solve the proposed system of equations is proposed and analysed. Various test cases are used to show the numerical stability and accuracy of the discretisation, and to validate the proposed formulation. Results show that the proposed method is numerically stable, accurate, mass conservative and preserves the C‐property. It can also handle subcritical, supercritical and transcritical flows under free surface or vertically confined conditions.
... The main difficulty is related to the fact that well-balanced approximations of the geometric sources now need to include terms that are small for smooth solutions, but may become artificially large at discontinuities. Among the papers, in which moving-water steady-state-preserving methods were developed, the schemes in [5,6,9,10,13,24,25,32,34,35] require non-trivial rootfinding based on the energy balance, and some methods require substantial effort on local reconstruction [4,5,6,19] or complex reconstruction of the geometric source terms [7,32,40]. ...
We develop a new moving-water equilibria preserving numerical scheme for the Saint-Venant system. The new scheme is designed in two major steps. First, the geometric source term is incorporated into the discharge flux, which results in a hyperbolic system with a global flux. Second, the discharge equation is relaxed so that the nonlinearity is moved into the stiff right-hand side of the added auxiliary equation. The main advantages of the new scheme are that (i) no special treatment of the geometric source term is required, and (ii) no nonlinear (cubic) equations should be solved to obtain the point values of the water depth out of the reconstructed equilibrium variables, as it must be done in the existing alternative methods. We also develop a hybrid numerical flux, which helps to handle various flow regimes in a stable manner. Several numerical experiments are performed to verify that the proposed scheme is capable of exactly preserving general moving-water steady states and accurately capturing their small perturbations.
... En la literatura existe mucha investigación numérica en el área del transporte de contaminantes en aguas poco profundas, aplicando métodos de diferencias finitas [6], métodos de volúmenes finitos [7,8,11,12,13], métodos de elementos finitos [4,9,10], métodos tipo-Godunov [18] y métodos de Boltzmann [14,15,16]. ...
En este artículo presentamos la aproximación del modelo acoplado de las ecuaciones del movimiento de un fluido en aguas poco profundas con la ecuación convección-difusión-reacción (CDR) del transporte de contaminantes. Dicha aproximación se realiza mediante elementos finitos de alto orden y usando métodos variacionales estabilizados de subescalas. El sistema acoplado de ecuaciones, previamente discretizado en el tiempo y linealizado, lo escribimos como una ecuación vectorial transitoria de CDR. Los métodos estabilizados de elementos finitos utilizados son los conocidos métodos de subescalas ASGS y OSS, los mismos que nos permiten usar igual interpolación para todas las incógnitas, así como tratar con flujos de convección y reacción dominantes. Consideramos la posibilidad de no linealidad tanto en el término convectivo como en el de reacción. No consideraremos el posible desarrollo de choques en la solución. Con el fin de examinar la precisión y robustez de los métodos ASGS y OSS, presentamos cuatro casos de prueba: convergencia en malla, transporte de un contaminante en una cavidad cuadrada, transporte de un contaminante en el golfo de Roses y en la desembocadura del río Guadalquivir, y el modelo depredador-presa, que puede escribirse como una ecuación vectorial de CDR transitoria con no linealidad en el término de reacción. 1 Introducción Un contaminante es aquel componente que está presente en el agua a niveles perjudiciales para la vida de los seres humanos, plantas y animales ([1], véase también los informes de la U.S. Environmental Protecction Agency, https://www.epa.gov/). La simulación del transporte de contaminantes para la predicción de la concentración de contaminantes en ríos, lagos, lagunas y regiones costeras tiene importancia estratégica en el análisis y diseño de soluciones de los problemas de contaminación ambiental del planeta. En el fenómeno físico del movimiento de contaminantes se deben distinguir tres procesos: la difusión, la convección y la reacción. La difusión es el proceso físico debido al cual el contaminante se mueve como resultado del movimiento intermolecular de las partículas de ambas sustancias, el fluido que la transporta (agua) y el contaminante. La convección es el movimiento del soluto (contaminante) debido al movimiento del agua, por lo cual si el agua permanece en reposo no hay convección. Finalmente, la reacción tiene en cuenta el posible efecto de crecimiento o decrecimiento de un contaminante por factores externos (calor, especies con las que puede combinarse); en el caso de haber distintas especies de contaminantes, la reacción puede modelar la interacción entre ellas. La formulación del problema del transporte de contaminantes se fundamenta, como la de todos los fenó-menos físicos, en las ecuaciones de equilibrio y las ecuaciones constitutivas. Las ecuaciones del transporte de sustancias disueltas o en suspensión en el flujo se basan en el principio de conservación de la masa de dichas sustancias en caso de que no haya reacción o en el modelo de crecimiento o decrecimiento en caso de que la haya. En el problema que queremos abordar, la dinámica del transporte de contaminantes involucra los modelos de flujos del campo de velocidades en aguas someras y del transporte, difusión y reacción de contaminantes, los cuales tienen características de la ecuación transitoria de convección, difusión y reacción (CDR).
... This module solved the 2D Saint Venant equations using an upwind Godunov scheme with the approximate Riemann solver of Roe. In order to transfer the water velocity and depth from the hydraulic module to the water quality advection-diffusion equations, the mass conservative scheme detailed in [26] was used. ...
Numerical models are useful tools to analyze water quality by computing the concentration of physical, chemical and biological parameters. The present work introduces a two-dimensional depth-averaged model that computes the most relevant and frequent parameters used to evaluate water quality. High performance computing (HPC) techniques based on graphic processing unit (GPU) parallelization have been applied to improve the efficiency of the package, providing speed-ups of two orders of magnitude in a standard PC. Several test cases were analyzed to show the capabilities and efficiency of the model to evaluate the environmental status of rivers and non-stratified estuaries. IberWQ will be freely available through the package Iber.
... The first-order scheme above is the basis of many discontinuity capturing approximations for shallow water flows (see, e.g. [11,12,15]). The strategy proposed here is to introduce a cell-wise mass-matrix limiter that allows to switch from this first -order non-oscillatory method across shocks, to the full finite element approximation, which has been shown to be third-order accurate and is appropriate for propagation [36]. ...
The aim of the present work has been to develop a model able to represent the propagation and transformation of waves in nearshore areas. The focus is on the phenomena of wave breaking, shoaling, and run-up. These different phenomena are represented through a hybrid approach obtained by the coupling of non-linear Shallow Water equations with the extended Boussinesq equations of Madsen and Sørensen. The novelty is the switch tool between the two modelling equations: a critical free surface Froude criterion. This is based on a physically meaningful new approach to detect wave breaking, which corresponds to the steepening of the wave’s crest which turns into a roller. To allow for an appropriate discretization of both types of equations, we consider a finite element Upwind Petrov Galerkin method with a novel limiting strategy that guarantees the preservation of smooth waves as well as the monotonicity of the results in presence of discontinuities. We provide a detailed discussion of the implementation of the newly proposed detection method, as well as of two other well-known criteria which are used for comparison. An extensive benchmarking on several problems involving different wave phenomena and breaking conditions allows to show the robustness of the numerical method proposed, as well as to assess the advantages and limitations of the different detection methods.
... The first-order scheme above is the basis of many discontinuity capturing approximations for shallow water flows (see e.g. [11,15,12]). The strategy proposed here is to introduce a cell-wise mass-matrix limiter, that allows to switch from this first order non-oscillatory method across shocks, to the full finite element approximation, which has been shown to be third-order accurate and is appropriate for propagation [36]. ...
... Furthermore, previous authors have indicated the necessity to implement a coupled discretization of the shallow water equations and the transport equations in order to avoid numerical difficulties. This idea was included in Burguete and Garcia-Navarro (2001); Murillo et al. (2009) for the 1D case and in Cea and Vázquez-Cendón (2012) for the 2D case using the pure advection of a passive solute. In the present work this has been extended to include a water quality model with the ambition to be able to deal with unsteady flow in complex and realistic river cross sections. ...
In this work, an one-dimensional (1D) finite volume numerical model for the unsteady simulation of the flow hydrodynamics and water quality is developed. The water dynamics is formulated with the 1D shallow water equations, and the water quality evolution is described by the Water Quality Analysis Simulation Program (WASP) model, allowing us to interpret and predict the transport and fate of various biochemical substances along any river reach. This combined system is solved with an explicit finite volume scheme based on Roe's linearization for the advection component of both the flow and the solute transport equations. The proposed model is able to consider temporal variations in tributaries and abstractions occurring in the river basin. This feature is transcendent in order to predict the chemical composition of natural water bodies during winter and summer periods, leading to an improvement in the agreement between computed and observed water quality evolutions. The combined model has been evaluated using literature tests in a steady state and a real-field case of the Ebro river (Spain), characterized by a marked unsteady regime. In the real case, we found that the water temperature was very sensitive to both the solar radiation and the average air temperature, requiring a careful calibration of these parameters. The numerical results also demonstrate to be reasonably accurate, conservative and robust in real-scale field cases, showing that the model is able to predict the evolution of quality parameters as well as hydrodynamic variables in complex scenarios.
... There is a vast variety of commercial software that has been developed for simulating river dynamics. In this work we have resorted to use the software IBER, since it is a freeware very useful to deal with 2D 4 water simulation [7], [10], although other numerical codes could have been chosen, perfectly well. Although this software is widely used, to the best of our knowledge it has not been previously applied to the type of regions of interest in this work and, therefore, there are no references in literature on this particular application. ...
The river floods happening in populated areas are serious natural risks that give rise to human and economic losses. In order to predict the consequences of river floods and to implement preventive and corrective measures, the mathematical modelling and numerical simulation play, nowadays, a very important role. Among the wide variety of software available for the numerical simulation in fluvial dynamics we have used, in this work, the hydrodynamic model IBER, which is free access simulation software for solving 2D shallow water models. In this paper we focus our attention in floods happening in the vicinity of the confluence of two rivers where there are also crops, with economic importance for the inhabitants of the area that may be affected by the inundation. As an example of this type of geographical region we have used data obtained from the region Las Omañas in NW Spain where, although there is the confluence of rivers Luna and Omaña, the confluence region does not belong to the study area, since we pay our attention to the inundations happening in the village Las Omañas, which is due to the action of one of both rivers, namely the river Omaña.
... Recently, full 2D hydrodynamic models have also been developed for simulating rainfall-runoff induced overland flows (e.g. ( Cea and Vázquez-Cendón, 2012;Costabile et al., 2012;Lacaster et al., 2015;Simons et al., 2014;Xia et al., 2017;Duan, 2017, 2014 )). However, due to inhibitive computational cost, no attempt has been reported to use a full 2D hydrodynamic model to simulate, at sufficiently high resolution ( < 10 m), the full-scale fluvial flooding process from the upstream rainfall-runoff to the downstream inundation across a large catchment > 1000 km 2 . ...
Full-scale fluvial flood modelling over large catchments has traditionally been carried out using coupled hydrological and hydraulic/hydrodynamic models. Such a traditional modelling approach is not well suited for the simulation of extreme floods induced by intense rainfall, which is usually featured with highly transient and dynamic rainfall-runoff and flooding process. This work aims to develop and demonstrate a modelling framework to predict the full-scale process of fluvial flooding from the source (rainfall) to impact (inundation) over a large catchment using a single high-performance hydrodynamic model driven by rainfall inputs. The modelling framework is applied to reproduce the flood event caused by the 2015 Storm Desmond in the 2500 km2 Eden Catchment at 5 m resolution. Without intensive model calibration, the predicted results compare well with field observations in terms of inundation extent and gauged water levels across the catchment. Sensitivity tests reveal that high-resolution grid is essential for accurate simulation of fluvial flood events using a 2D hydrodynamic model. Accelerated by multiple modern GPUs, the simulation is more than 2.5 times faster than real time although it involves 100 million computational cells inside the computational domain. This work provides a novel and promising approach to assess and forecast at real time the risk of extreme fluvial floods from intense rainfall.
Full text is available from
https://authors.elsevier.com/c/1Zavk_71vnhve3
... Kuang et al. (2011) presented semi-implicit Eulerian-Lagrangian method, unstructured SWAN model, CurWaC2D-Sed andFVM to develop 2D morphological model. Cea andVázquez-Cendón (2012) presented a novel unstructured upwind FVM for discretization of the bed friction term. Gharehbaghi, Kaya, and Saadatnejadgharahassanlou (2017) established a new depth-averaged 2D non-equilibrium coupled model. ...
A coupled model is developed to simulate two dimensional water surface profile, suspended sediment load and bed deformation in unsteady open channels. The hydrodynamical component employs the two dimensional shallow water equations to obtain the hydraulic variables. These, in turn, are used in the morphdynamical component to determine the bed deformation. For the turbulence variables; two turbulence models are supervened to the governing equations. Triangular meshes were developed to discretize the domain of open channel. In order to discretize the governing equations, the explicit finite volume method is used by the total variation diminishing (TVD) schemes. The performance of the developed model is compared to that of the Flow3D software. The comparison results are in good agreement.
... The hydrodynamic characteristics of water bodies such as coastal lagoons are governed by a slight balance between tidal forces, currents flow, wind stresses, and density, which induce pressure and friction forces at the bottom [74,75], in addition to other factors such as the geometry and flow, which is predominantly turbulent with a horizontal length scale much greater than the vertical one [76]. ...
This paper describes the development of a two-dimensional water quality model that solves hydrodynamic equations tied to transport equations with reactions mechanisms inherent in the processes. This enables us to perform an accurate assessment of the pollution in a coastal ecosystem. The model was developed with data drawn from the ecosystem found in Mexico's southeast state of Tabasco. The coastal ecosystem consists of the interaction of El Yucateco lagoon with Chicozapote and Tonalá rivers, that connect the lagoon with the Gulf of Mexico. The results of pollutants transport simulation in the coastal ecosystem are presented, focusing on toxic parameters for two hydrodynamic scenarios: wet and dry seasons. As it of interest in the zone, the transport of four metals is studied: Cadmium, Chromium, Nickel and Lead. In order to address this objectives, a self-posed mathematical problem is solved numerically, which is based on the measured data. The performed simulations show to characterise metals transport with an acceptable accuracy, agreeing well with measured data in total concentrations in four control points along the water body. Although for the accurate implementation of the hydrodynamic-based water quality model herein presented , boundary (geometry, tides, wind, etc.) and initial (concentrations measurements) conditions are required, it poses as an excellent option when the distribution of solutes with high accuracy is required, easing environmental, economic and social management of coastal ecosystems. It ought to be remarked that this constitutes a robust differential equation-based water quality model for the transport of heavy metals. Models with this characteristics are not common to be found elsewhere.
... The difficulty is related to the fact that well-balanced approximations of the geometric source now need to include terms that are small for smooth solutions, but may become artificially large at discontinuities. In the nonzero friction case, only certain particular moving-water equilibria could be exactly preserved by existing numerical methods; see, e.g., [9,13,22]. ...
We construct a new second-order moving-water equilibria preserving central-upwind scheme for the one-dimensional Saint-Venant system of shallow water equations. The idea is based on a reformulation of the source terms as integral in the flux function. Reconstruction of the flux variable yields then a third order equation that can be solved exactly. This procedure does not require any further modification of existing schemes. Several numerical tests are performed to verify the ability of the proposed scheme to accurately capture small perturbations of steady states. © 2019, Springer Science+Business Media, LLC, part of Springer Nature.
... The first-order scheme above is the basis of many discontinuity capturing approximations for shallow water flows (see e.g. [10,14,11]). The strategy proposed here is to introduce a cell-wise mass-matrix limiter, that allows to switch from this first order non-oscillatory method across shocks, to the full finite element approximation, which has been shown to be third-order accurate and is appropriate for propagation [36]. ...
The aim of the present work is to develop a model able to represent the propagation and transformation of waves in nearshore areas. The focus is on the phenomena of wave breaking, shoaling and run-up. These different phenomena are represented through a hybrid approach obtained by the coupling of non-linear Shallow Water equations with the extended Boussinesq equations of Madsen and Sørensen. The novelty is the switch tool between the two modelling equations: a critical free surface Froude criterion. This is based on a physically meaningful new approach to detect wave breaking, which corresponds to the steepening of the wave's crest which turns into a roller. To allow for an appropriate discretization of both types of equations, we consider a finite element Upwind Petrov Galerkin method with a novel limiting strategy, that guarantees the preservation of smooth waves as well as the monotonicity of the results in presence of discontinuities. We provide a detailed discussion of the implementation of the newly proposed detection method, as well as of two other well known criteria which are used for comparison. An extensive benchmarking on several problems involving different wave phenomena and breaking conditions allows to show the robustness of the numerical method proposed, as well as to assess the advantages and limitations of the different detection methods.
[Link to the preprint: https://arxiv.org/abs/1902.03021 ]
... The first-order scheme above is the basis of many discontinuity capturing approximations for shallow water flows (see e.g. [10,14,11]). The strategy proposed here is to introduce a cell-wise mass-matrix limiter, that allows to switch from this first order non-oscillatory method across shocks, to the full finite element approximation, which has been shown to be third-order accurate and is appropriate for propagation [36]. ...
The aim of the present work is to develop a model able to represent the propagation and transformation of waves in nearshore areas. The focus is on the phenomena of wave breaking, shoaling and run-up. These different phenomena are represented through a hybrid approach obtained by the coupling of non-linear Shallow Water equations with the extended Boussinesq equations of Madsen and Sorensen. The novelty is the switch tool between the two modelling equations: a critical free surface Froude criterion. This is based on a physically meaningful new approach to detect wave breaking, which corresponds to the steepening of the wave's crest which turns into a roller. To allow for an appropriate discretization of both types of equations, we consider a finite element Upwind Petrov Galerkin method with a novel limiting strategy, that guarantees the preservation of smooth waves as well as the monotonicity of the results in presence of discontinuities. We provide a detailed discussion of the implementation of the newly proposed detection method, as well as of two other well known criteria which are used for comparison. An extensive benchmarking on several problems involving different wave phenomena and breaking conditions allows to show the robustness of the numerical method proposed, as well as to assess the advantages and limitations of the different detection methods.
... The hydrodynamic characteristics of water bodies such as coastal lagoons are governed by a slight balance between tidal forces, currents flow, wind stresses, and density, which induce pressure and friction forces at the bottom [74,75], in addition to other factors such as the geometry and flow, which is predominantly turbulent with a horizontal length scale much greater than the vertical one [76]. ...
This paper describes the development of a two-dimensional water quality model that solves hydrodynamic equations tied to transport equations with reactions mechanisms inherent in the processes. This enable us to perform an accurate assessment of the pollution in a coastal ecosystem. The model was developed with data drawn from the ecosystem found in Mexico's southeast state of Tabasco. The coastal ecosystem consists of the interaction of El Yucateco lagoon with the Chicozapote and Tonalá rivers, that connect the lagoon with the Gulf of Mexico. We present the results of pollutants transport simulation in the coastal ecosystem, focusing on toxic parameters for two hydrodynamic scenarios: wet and dry seasons. As it of interest in the zone, we study the transport of four metals: Cadmium, Chromium, Nickel and Lead. In order to address our objectives we solved numerically a self-posed mathematical problem,which is based on the measured data. The performed simulations show to characterise metal transport within the acceptable range of accuracy and in accordance with the measured data. The performed simulations show to characterise metals transport with an acceptable accuracy, agreeing well with measured data in total concentrations in four control points along the water body. Although for the accurate implementation of the hydrodynamic-based water quality model herein presented, boundary (geometry, tides, wind, etc.) and initial (concentrations measurements) conditions are required, it poses as an excellent option when the distribution of solutes with high accuracy is required, easing environmental, economic and social management of coastal ecosystems.
... Similar schemes have been introduced for the shallow water equations in [19] and [11] . An important feature is that it preserves the positivity of partial densities ρ k if the CFL condition is satisfied, as the computations below show. ...
A first-order well-balanced finite volume scheme for the solution of a multi-component gas flow model in a pipe on non-flat topography is introduced. The mathematical model consists of Euler equations with source terms which arise from heat exchange, and gravity and viscosity forces, coupled with the mass conservation equations of species. We propose a segregated scheme in which the Euler and species equations are solved separately. This methodology leads to a flux vector in the Euler equations which depends not only on the conservative variables but also on time and space variables through the gas composition. This fact makes necessary to add some artificial viscosity to the usual numerical flux which is done by introducing an additional source term. Besides, in order to preserve the positivity of the species concentrations, we discretize the flux in the mass conservation equations for species, in accordance with the upwind discretization of the total mass conservation equation in the Euler system. Moreover, as proposed in a previous reference by the authors, [5], the discretizations of the flux and source terms are made so as to ensure that the full scheme is well-balanced. Numerical tests including both academic and real gas network problems are solved, showing the performance of the proposed methodology.
... The transport of active or passive scalar quantities within a given flow domain is governed by advective and diffusive processes where turbulence plays a key role in their mixing and distribution. In fact, processes such as the transport of pollutants or the injection of a tracer [7,8] are often treated as passive scalar quantities. Understanding how a turbulent separated flow impacts the transport of a passive scalar can be of particular interest in environmental applications such as sediment deposition, channel scour [27], sidewall erosion, tracer dye transport [3,4] and pollutant dispersion [16]. ...
The results of large-eddy simulations of flow and transient solute transport over a backward facing step and through a 180° bend are presented. The simulations are validated successfully in terms of hydrodynamics and tracer transport with experimental velocity data and measured residence time distribution curves confirming the accuracy of the method. The hydrodynamics are characterised by flow separation and subsequent recirculation in vertical and horizontal directions and the solute dispersion process is a direct response to the significant unsteadiness and turbulence in the flow. The turbulence in the system is analysed and quantified in terms of power density spectra and covariance of velocity fluctuations. The injection of an instantaneous passive tracer and its dispersion through the system is simulated. Large-eddy simulations enable the resolution of the instantaneous flow field and it is demonstrated that the instabilities of intermittent large-scale structures play a distinguished role in the solute transport. The advection and diffusion of the scalar is governed by the severe unsteadiness of the flow and this is visualised and quantified. The analysis of the scalar mass transport budget quantifies the mechanisms controlling the turbulent mixing and reveals that the mass flux is dominated by advection.
Los ejemplos visuales y gráficos interactivos de experimentos controlados permiten al alumno resolver problemas, hacer demostraciones, mediciones y otras actividades prácticas y teóricas. Iber es un modelo numérico bidimensional para la simulación del flujo superficial que combina módulos de hidrodinámica, turbulencia, transporte de sedimentos y calidad de aguas, utilizando el método de volúmenes finitos para solucionar las ecuaciones correspondientes. Su interfaz amigable para el pre y pos procesamiento permite crear videos con resultados y visualizar variables en 3D, convirtiéndose en una herramienta docente didáctica, robusta, gratuita, relativamente fácil de usar, práctica y eficaz para modelaciones hidráulicas e hidrodinámicas. En este trabajo presentamos algunos ejemplos numéricos de simulación de flujo en lámina libre como complementos a las prácticas de docencia experimental que se dificultan en muchas instituciones por falta de recursos económicos y humanos, o por el excesivo tiempo de montaje y operación. Los resultados muestran el ahorro de un tiempo considerable frente a los laboriosos e inevitables cálculos iterativos de las numerosas ecuaciones; la versatilidad para introducir, corregir y visualizar datos es un gran avance al momento de tabular y presentar los resultados como si se tratara de una práctica real. Los tiempos de simulación también son muy razonables y están asociados al tipo de problema, método de cálculo, esquema numérico, resultados a obtener, aspectos que pueden modificarse fácilmente durante las sesiones de clase.
Article Cite this article: Sanz-Ramos M, Bladé E, Oller P, Furdada G (2023). Numerical modelling of dense snow avalanches with a well-balanced scheme based on the 2D shallow water equations. Journal of Glaciology 1-17. https:// Abstract A common technique for simulating non-Newtonian fluid dynamics, such as snow avalanches, is to solve the Shallow Water Equations (SWE), together with a rheological model describing the momentum dissipation by shear stresses. Friction and cohesion terms are commonly modelled using the Voellmy friction model and, recently, the Bartelt cohesion model. Here, an adaptation of the Roe scheme that ensures the balance between the flux and pressure gradients and the friction source term is presented. An upwind scheme was used for the discretisation of the SWE numerical fluxes and the non-velocity-dependent terms of the friction-cohesion model, whereas a centred scheme was used for the velocity-dependent source terms. The model was tested in analytically solvable settings, laboratory experiments and real cases. In all cases, the model performed well, avoiding numerical instabilities and achieving stable and consistent solution even for an avalanche stopping on a sloping terrain.
Physically-based soil erosion models are valuable tools for the understanding and efficient management of soil erosion related problems at the basin and river reach scales, as soil loss, muddy floods, freshwater pollution or reservoir siltation, among others. We present the implementation of a new fully distributed multiclass soil erosion module. The model is based on a 2D finite volume solver (Iber+) for the 2D shallow water equations that computes the overland flow water depths and velocities. From these, the model evaluates the transport of sediment particles due to bed load and suspended load, including rainfall-driven and runoff-driven erosion processes, and using well-established physically-based formulations. The evolution of the mass of sediment particles in the soil layer is computed from a mass conservation equation for each sediment class. The solver is implemented using High Performance Computing techniques that take advantage of the computational capabilities of standard Graphical Processing Units, achieving speed-ups of two orders of magnitude relative to a sequential implementation on the CPU. We show the application and validation of the model at different spatial scales, ranging from laboratory experiments to meso-scale catchments.
In this study, a new numerical treatment of the friction source term is proposed for modeling shallow water
flows over complex bed topography. We design a novel semi-implicit temporal discretization technique based
on Taylor series expansion of friction source terms. A well-balanced positivity-preserving finite volume scheme
on unstructured triangular grids is used to solve the shallow water model. We present a series of numerical
tests to demonstrate the robustness and effectiveness of the proposed numerical model where we used rapidly
varying flows over complex topography as well as flows involving small values of water depth and wet/dry
zones. The simulation results of the numerical model based on the proposed approach for friction source
terms are compared to experimental data and to those obtained by using other existing techniques. The
proposed numerical model is stable and second-order accurate and our predictions are in good agreement
with experimental observations
A large amount of contaminants enter marine systems with river runoff, so the purpose of the study is to model the transport of suspended particles in the estuary area. To describe hydrodynamic and hydrophysical processes, the mathematical model of the suspended particles transport was used, supplemented by a three-dimensional mathematical model of hydrodynamics, used to calculate the fields of the aquatic environment movement velocity vector, and equation for calculating the variable density. The approximation of the equations for calculating the velocity field by spatial variables is based on the splitting schemes for physical processes with fluid volume of the control areas, which allows for us to consider the complex geometry of the coastline and the bottom. The suspended particles transport model is approximated using splitting schemes for two-dimensional and one-dimensional problems. Numerical experiments were carried out to simulate the aquatic environment movement in the estuary area, the multicomponent suspension deposition, as well as mixing of waters in the mouth, taking into account the different density of the aquatic environment. The used models and methods allow to significantly improve the accuracy of modeling suspended particle transport and consider the factors influencing the studied processes.
In this study, we focus on the modelling of coupled systems of shallow water flows and solute transport with source terms due to variable topography and friction effect. Our aim is to propose efficient and accurate numerical techniques for modelling these systems using unstructured triangular grids. We used a Riemann-solver free method for the hyperbolic shallow water system and a suitable discretization technique for the bottom topography. The friction source term is discretized using the techniques proposed by (Xia and Liang in Adv Water Resour 117:87–97, 2018) [19]. Our approach performs very well for stationary flow in the presence of variable topography, and it is well-balanced for the concentration in the presence of wet and dry zones. In our techniques, we used linear piecewise reconstructions for the variables of the coupled system. The proposed method is well-balanced, and we prove that it exactly preserves the nontrivial steady-state solutions of the coupled system. Numerical experiments are carried out to validate the performance and robustness of the proposed numerical method. Our numerical results show that the method is stable, well-balanced and accurate to model the coupled systems of shallow water flows and solute transport.
The main goal of this work is the resolution of the two-dimensional shallow water equations of water-sediment mixture coupled to the transport diffusion equation for the total sediment load, and bed change rate equation by Roe scheme with a new discretization of the source term. The proposed discretization is well-balanced with the flux gradient and uses data right and left on the interfaces between two control volumes and satisfies the C-property. The numerical method uses unstructured meshes and incorporates minmod limiter and runge-kutta method to reach second order spatial and temporal accuracy. We also use an adaptive mesh based on gradient concentration of sediments to refine the study domain with a lower computational cost. We present some numerical results in order to verify and validate the performance of the numerical scheme, particular attention is given to the treatment of the dam-break problem over mobile beds. The numerical scheme demonstrates the intended accuracy and robustness to modelize dam-break flows over erodible bed.
To improve the modeling quality of pollutant transport in shallow waters, different reconstruction schemes have been proposed to better link the edge values to the centroid values of a pollutant concentration in finite-volume shallow water models: a scheme of higher (lower) order generally has a better (poorer) quantitative accuracy but lower (higher) computational efficiency. Here, a numerical comparative study of several classical schemes is first conducted under a variety of pollutant distribution conditions. The results reveal that, for the condition of relatively uniform pollutant distribution, the numerical accuracy of a lower-order scheme (such as the first-order scheme or the MUSCL scheme) may be similar to that of a higher-order scheme (such as the WENO scheme). The second-order derivative of the concentration, here termed the nonlinear indicator (NI), correlates well with the discrepancies between the numerical solutions and analytical solutions. A threshold value of approximately 10−7∼10−6m−2 for the NI is identified, above which a higher-order scheme may be required. Based on this understanding, a hybrid first-order and WENO scheme is proposed. Numerical case studies show that the hybrid scheme can successfully combine the efficiency of the first-order scheme with the high accuracy of the WENO scheme for pollutant modeling.
The study of soil erosion by water and the transfer of suspended solids from watersheds to rivers is crucial given the environmental and socio-economic issues with regards to growing human influence and the expected intensification of these processes under climate change. The objective of this thesis is to understand how rainfall variability controls the activation of different sediment source zones and the dynamics of hydro-sedimentary flows in two mesoscale Mediterranean catchments, i.e. the Claduègne (42 km², subcatchment of the Ardèche) and the Galabre (20 km2 , subcatchment of the Durance) which are members of the OZCAR critical zone research infrastructure.In the first part, the contributions of the erosion zones to sediment fluxes at the outlet of the Claduègne catchment were quantified at high temporal resolution with a low-cost sediment fingerprinting approach. Two sets of tracers (Color and X-ray fluorescence tracers) and three mixing models were compared to assess the sensitivity of estimated source contributions to these methodological choices. Marly-calcareous badlands were identified as the main sediment source. A similar approach carried out on the Galabre catchment area showed that badlands on molasses were the main source. The comparison of tracer sets and mixing models, showed that the methodological choices generated important differences. Thus, we suggest a multi-tracer-multi-model ensemble approach to obtain more robust results. The application of this approach to a large number of sediment samples highlighted the important within and between event variability in the contributions of different sediment sources, raising questions about the hydro-sedimentary processes that cause this variability.We hypothesized that this variability resulted from variable suspended sediment transit time distributions governed by the interplay of (i) catchment characteristics such as the location of different sources and how they are linked to the outlet (referred to as structural sediment connectivity) and (ii) the spatio-temporal characteristics of rain events that activate and impact transfer velocities (i.e. functional connectivity).Thus, in the second part, a distributed numerical model based on the resolution of Saint Venant equations coupled to a multi-source erosion module was used to evaluate the respective roles of structural and functional connectivity. Sensitivity analysis of the discretization and parameterization choices (i.e. threshold of contributing drainage area to identify the river network, values of roughness coefficients on hillslopes and the river) showed that the location of the sediment sources in the watershed was more important than the modeling choices when the parameters were limited to realistic range. A general temporal pattern of source contributions was observed. This was consistent with the results of the fingerprinting approach and the distribution of distances from the sources to the river and the outlet. The same pattern persists for different rainfall durations or intensities but became much more variable when bimodal hyetographs or spatially variable precipitation was applied. In addition, the location of the rainfall with respect to the sources determined the average contributions of the sources and thus differences between rainfall events.The two approaches, sediment fingerprinting and numerical modeling, were found to complement each other and their combined application has a high potential for understanding how interactions between structural and functional connectivity control the dynamics of sediment fluxes in mesoscale catchments.
At the early 1980s, the research group in Mathematical Engineering, mat+i, started working on finite volume methods for the simulation of environmental issues concerning Galician rias (Spain). The focus was on the study of hyperbolic balance laws due to the presence of source terms related to the bathymetry. A correct treatment of these terms, an upwind discretization, was presented in [2, 7, 22]. The transfer of this knowledge has motivated the registration of the software Iber (http://www.iberaula.es). Latterly, under different research problems, we have been working on the development of a numerical algorithm for the resolution of Euler and Navier–Stokes equations. A hybrid projection finite volume/finite element method is employed making use of unstructured staggered grids (see [3, 9]). To attain second order of accuracy ADER methodology is employed [10]. On the other hand, with numerical simulation of gas transportation networks in view, a first-order well balanced finite volume scheme for the solution of a model, for the flow of a multicomponent gas in a pipe on non-flat topography, is introduced. The mathematical model consists of Euler equations, with source terms, coupled with the mass conservation equations of species. We propose a segregated scheme in which Euler and species equations are solved separately [6].
Physically-based urban wash-off models are a promising means of studying the transport of finer suspended solids and their associated pollutants during rain events, considering spatial and temporal heterogeneities. This study contributes to the understanding of these models through an in-depth sensitivity analysis to provide the necessary information to simplify the model and deal with parameter identifiability. First, based on twelve tailored high-resolution experiments, the accurate measurement of input variables was used to study the parameters of the Hairsine-Rose sediment transport model through a global sensitivity analysis. Using Standardized Regression Coefficients (SRC) and Extended Fourier Amplitude Sensitivity Test (EFAST) methods, the analysis showed that both the total washed-off mass and the TSS peaks concentration are highly sensitive to the critical mass, which considers the reduction in the detachment of particles when the sediment available decreases and is scattered over the surface. In addition, the rain- and flow-driven detachment parameters were presented as key for smaller and larger sediment particles, respectively. Then, those uncertainties that are associated in field studies with the determination of the model input variables were also considered by conducting a local sensitivity analysis. The initial load of sediment and the mean grain size were seen to be the most important variables, thus underlining the need for very accurate measurements here. Moreover, a precise definition of Harsine-Rose parameters is also necessary to achieve reliable results in order to work on treatment and management techniques to minimize the impact of urban surface contaminants on urban environments.
We focus on the development and analysis of staggered schemes for the two-dimensional non-linear Shallow Water equations with varying bathymetry. Semi-implicit and fully explicit time-discretizations are proposed. Particular attention is paid on non-linear stability results, principally considered here through discrete energy dissipation arguments. To address such an issue, specific convective fluxes are employed, implying diffusive terms relying on the pressure gradient. In addition of providing an explicit control of the discrete energy budget, the method is shown to preserve motionless steady states as well as the positivity of the water height. These properties are still satisfied in a fully explicit context, provided an appropriate discretization of the pressure gradient. These stability results make the approach particularly robust and efficient, for both coastal flows and low-Froude number regimes. As a result, in addition of a great ease of implementation, the presented schemes meet the operational requirements attached to the simulation of large and small scale oceanic flows.
In this paper, we develop third-order conservative sign-preserving and steady-state-preserving time integrations and seek their applications in multispecies and multireaction chemical reactive flows. In this problem, the density and pressure are nonnegative, and the mass fraction for the ith species, denoted as zi, 1 ≤ i ≤ M should be between 0 and 1, where M is the total number of species. There are four main difficulties in constructing high-order bound-preserving techniques for multispecies and multireaction detonations. First of all, most of the bound-preserving techniques available are based on Euler forward time integration. Therefore, for problems with stiff source, the time step will be significantly limited. Secondly, the mass fraction does not satisfy a maximum-principle and hence it is not easy to preserve the upper bound 1. Thirdly, in most of the previous works for gaseous denotation, the algorithm relies on second-order Strang splitting methods where the flux and stiff source terms can be solved separately, and the extension to high-order time discretization seems to be complicated. Finally, most of the previous ODE solvers for stiff problems cannot preserve the total mass and the positivity of the numerical approximations at the same time. In this paper, we will construct third-order conservative sign-preserving Rugne-Kutta and multistep methods to overcome all these difficulties. The time integrations do not depend on the Strang splitting, i.e. we do not split the flux and the stiff source terms. Moreover, the time discretization can handle the stiff source with large time step and preserves the steady-state. Numerical experiments will be given to demonstrate the good performance of the bound-preserving technique and the stability of the scheme for problems with stiff source terms.
In this paper, a hybrid artificial viscosity–central-upwind (AV-CU) scheme is proposed for simulating recirculating turbulent shallow water flows by combining the artificial viscosity (AV) technique with the central-upwind (CU) scheme. The two-dimensional (2D) depth-averaged Reynolds-averaged Navier Stokes (DA-RANS) equations are solved using the AV technique, whereas the CU scheme is employed to compute the $\kappa-\epsilon$ model. The model is of spatially and temporally second-order accurate. The scalable wall functions (ScWF) are employed, thus becoming flexible in generating meshes without having to estimate the wall friction velocity at initial time step as if the standard wall functions (StWF) are used. The results benefit strongly from this hybrid approach being more accurate than the CU and Harten-Lax-van Leer-Contact (HLLC) schemes—and 1.52x cheaper than the HLLC scheme for such recirculating turbulent flows. As such, the proposed approach could become a promising method for practical engineering purposes to simulate turbulent shallow water flows more efficiently and accurately.
We consider nonlinear hyperbolic conservation laws posed on a curved geometry, referred to as “geometric Burgers equations” after Ben-Artzi and LeFloch (2007), when the underlying geometry is the sphere and the flux vector field is determined from a potential function. Despite its apparent simplicity, this model exhibits complex wave phenomena which are not observed in absence of geometrical effects. To study the late-time asymptotic behavior of the solutions of this model, we consider a finite volume method based on a generalized Riemann solver. We provide a numerical validation of the accuracy and efficiency of the method in presence of nonlinear waves and a curved geometry and, especially, demonstrate the contraction, time-variation monotonicity, and entropy monotonicity properties. The late-time asymptotic behavior of the solutions is studied and discussed in terms of the properties of the flux. A new classification of the flux vector field is introduced where we distinguish between foliated flux and generic flux, and the character of linearity of the flux which are expected to be sufficient to predict the late-time asymptotic behavior of the solutions. When the flux is foliated and linear, the solutions are transported in time within the level sets of the potential. If the flux is foliated and is genuinely nonlinear, the solutions converge to their (constant) average within each level set. For generic flux, the solutions evolve with large variations which depend on the geometry and converge to constant values within certain “independent” domains on the sphere. The number of constant values depends on curves that “split” the sphere into independent domains. For fluxes which are linear, foliated or generic only on parts of the sphere, combinations of the late-time asymptotic behavior of the solutions can be obtained which depends also on the interaction between the fluxes at boundaries of these parts of the sphere.
Exact solutions of the nonlinear shallow water wave equations for forced flow involving linear bottom friction in a region with quadratic bathymetry have been found. These solutions also involve moving shorelines. The motion decays over time. In the solution of the three simultaneous nonlinear partial differential shallow water wave equa-tions it is assumed that the velocity is a function of time only and along one axis. This assumption reduces the three simultaneous non-linear partial differential equations to two simultaneous linear ordinary differential equations. The analytical model has been tested against a numerical solution with good agreement between the numerical and analytical solutions. The analytical model is useful for testing the accuracy of a moving boundary shallow water numerical model.
This work extends and improves a one-dimensional shallow flow model to two-dimensional (2D) for real-world flood simulations. The model solves a prebalanced formulation of the fully 2D shallow water equations, including friction source terms using a finite volume Godunov-type numerical scheme. A reconstruction method ensuring nonnegative depth is used along with a Harten, Lax, and van Leer approximate Riemann solver with the contact wave restored for calculation of interface fluxes. A local bed modification method is proposed to maintain the well-balanced property of the algorithm for simulations involving wetting and drying. Second-order accurate scheme is achieved by using the slope limited linear reconstruction together with a Runge-Kutta time integration method. The model is applicable to calculate different types of flood wave ranging from slow-varying inundations to extreme and violent floods, propagating over complex domains including natural terrains and dense urban areas. After validating against an analytical case of flow sloshing in a domain with a parabolic bed profile, the model is applied to simulate an inundation event in a 36 km 2 floodplain in Thamesmead near London. The numerical predictions are compared with analytical solutions and alternative numerical results.
A model based on the finite-volume method is developed for unsteady, two-dimensional, shallow-water flow over arbitrary topography with moving lateral boundaries caused by flooding or recession. The model uses Roe's approximate Riemann solver to compute fluxes, while the monotone upstream scheme for conservation laws and predictor-corrector time stepping are used to provide a second-order accurate solution that is free from spurious oscillations. A robust, novel procedure is presented to efficiently and accurately simulate the movement of a wet/dry boundary without diffusing it. In addition, a new technique is introduced to prevent numerical truncation errors due to the pressure and bed slope terms from artificially accelerating quiescent water over an arbitrary bed. Model predictions compare favorably with analytical solutions, experimental data, and other numerical solutions for one- and two-dimensional problems.
Cited By (since 1996): 37, Export Date: 28 February 2013, Source: Scopus, doi: 10.1061/(ASCE)0733-9429(2006)132:7(652), Language of Original Document: English, Correspondence Address: Valiani, A.; Dept. of Engineering, Univ. of Ferrara, via G. Saragat 1, 44100 Ferrara, Italy; email: alessandro.valiani@unife.it, References: Alcrudo, F., (1999), Proc., 4th CADAM Meeting Zaragoza, SpainAlcrudo, F., Garcia-Navarro, P., A high-resolution Godunov-type scheme in finite volumes for the 2D shallow water equations (1993) Int. J. Numer. Methods Fluids, 16, pp. 489-505. , IJNFDW0271-209110.1002/fld.1650160604;
Godunov-based shallow-water models utilize a discontinuous reconstruction of data at cell faces even for smooth flow, which can cause energy dissipation and degrade accuracy. Analysis of discrete equations shows that jumps and therefore error can be minimized by adaptively selecting either primitive or conservative variables for slope limiting and reconstruction according to the local Froude number. Therefore, a Godunov-based model with an adaptive scheme of slope limiting and variable reconstruction is presented. Two practical flood modeling applications are used to compare the performance of the adaptive scheme against two nonadaptive schemes. In addition, performance of second-order accurate schemes is compared to first-order schemes that utilize a second-order accurate description of terrain. Results show that the first-order adaptive scheme possesses the best combination of robustness, efficiency, and accuracy of the models tested.
A numerical technique for the modelling of shallow water flow in one and two dimensions is presented in this work along with the results obtained in different applications involving unsteady flows in complex geometries. A cell-centred finite volume method based on Roe's approximate Riemann solver across the edges of both structured and unstructured cells is presented. The discretization of the bed slope source terms is done following an upwind approach. In some applications a problem arises when the flow propagates over adverse dry bed slopes, so a special procedure has been introduced to model the advancing front. It is shown that this modification reproduces exactly steady state of still water in configurations with strong variations in bed slope and contour. The applications presented are mainly related with unsteady flow problems. The scheme is capable of handling complex flow domains as will be shown in the simulations corresponding to the test cases that are going to be presented. Comparisons of experimental and numerical results are shown for some of the tests. Copyright © 2002 John Wiley & Sons, Ltd.
A wetting–drying condition (WDC) for unsteady shallow water flow in two dimensions leading to zero numerical error in mass conservation is presented in this work. Some applications are shown which demonstrate the effectiveness of the WDC in flood propagation and dam break flows over real geometries. The WDC has been incorporated into a cell centred finite volume method based on Roe's approximate Riemann solver across the edges of both structured and unstructured meshes. Previous wetting–drying condition based on steady-state conditions lead to numerical errors in unsteady cases over configurations with strong variations on bed slope. A modification of the wetting–drying condition including the normal velocity to the cell edge enables to achieve zero numerical errors. The complete numerical technique is described in this work including source terms discretization as a complete and efficient 2D river flow simulation tool. Comparisons of experimental and numerical results are shown for some of the applications. Copyright © 2004 John Wiley & Sons, Ltd.
We present a comparison of two discretization methods for the shallow water equations, namely the finite volume method and the finite element scheme. A reliable model for practical interests includes terms modelling the bottom topography as well as the friction effects. The resulting equations belong to the class of systems of hyperbolic partial differential equations of first order with zero order source terms, the so-called balance laws. In order to approximate correctly steady equilibrium states we need to derive a well-balanced approximation of the source term in the finite volume framework. As a result our finite volume method, a genuinely multidimensional finite volume evolution Galerkin (FVEG) scheme, approximates correctly steady states as well as their small perturbations (quasi-steady states). The second discretization scheme, which has been used for practical river flow simulations, is the finite element method (FEM). In contrary to the FVEG scheme, which is a time explicit scheme, the FEM uses an implicit time discretization and the Newton-Raphson iterative scheme for inner iterations. We show that both discretization techniques approximate correctly steady and quasi-steady states with bottom topography and friction and compare their accuracy and performance.
This paper presents a well-balanced numerical scheme for simulating frictional shallow flows over complex domains involving wetting and drying. The proposed scheme solves, in a finite volume Godunov-type framework, a set of pre-balanced shallow water equations derived by considering pressure balancing. Non-negative reconstruction of Riemann states and compatible discretization of slope source term produce stable and well-balanced solutions to shallow flow hydrodynamics over complex topography. The friction source term is discretized using a splitting implicit scheme. Limiting value of the friction force is derived to ensure stability. This new numerical scheme is validated against four theoretical benchmark tests and then applied to reproduce a laboratory dam break over a domain with irregular bed profile.
A novel scheme has been developed for data reconstruction within a Godunov-type method for solving the shallow-water equations with source terms. In contrast to conventional data reconstruction methods based on conservative variables, the water surface level is chosen as the basis for data reconstruction. This provides accurate values of the conservative variables at cell interfaces so that the fluxes can be accurately calculated with a Riemann solver. The main advantages are: (1) a simple centered discretization is used for the source terms; (2) the scheme is no more complicated than the conventional method for the homogeneous terms; (3) small perturbations in the water surface elevation can be accurately predicted; and (4) the method is generally suitable for both steady and unsteady shallow-water problems. The accuracy of the scheme has been verified by recourse to both steady and unsteady flow problems. Excellent agreement has been obtained between the numerical predictions and analytical solutions. The results indicate that the new scheme is accurate, simple, efficient, and robust.
We consider the numerical solution of the shallow water equations on unstructured grids. We focus on flows over wet areas. The extension to the case of dry bed will be reported elsewhere. The shallow water equations fall into the category of systems of conservation laws which can be symmetrized thanks to the existence of a mathematical entropy coinciding, in this case, with the total energy. Our aim is to show the application of a particular class of conservative residual distribution (RD) schemes to the discretization of the shallow water equations and to analyze their discrete accuracy and stability properties. We give a review of conservative RD schemes showing relations between different approaches previously published, and recall L∞ stability and accuracy criteria characterizing the schemes. In particular, the accuracy of the RD method in presence of source terms is analyzed, and conditions to construct rth order discretizations on irregular triangular grids are proved. It is shown that the RD approach gives a natural way of obtaining high order discretizations which, moreover, preserves exactly the steady lake at rest solution independently on mesh topology, nature of the variation of the bottom and polynomial order of interpolation used for the unknowns. We also consider more general analytical solutions which are less investigated from the numerical view point. On irregular grids, linearity preserving RD schemes yield a truly second order approximation. We also sketch a strategy to achieve discretizations which preserve exactly some of these solutions. Numerical results on steady and time-dependent problems involving smooth and non-smooth variations of the bottom topology show very promising features of the approach.
Pollutant transport by shallow water flows on non-flat topography is presented and numerically solved using a finite volume scheme. The method uses unstructured meshes, incorporates upwinded numerical fluxes and slope limiters to provide sharp resolution of steep bathymetric gradients that may form in the approximate solution. The scheme is non-oscillatory and possesses conservation property that conserves the pollutant mass during the transport process. Numerical results are presented for three test examples which demonstrate the accuracy and robustness of the scheme and its applicability in predicting pollutant transport by shallow water flows. In this paper, we also apply the developed scheme for a pollutant transport event in the Strait of Gibraltar. The scheme is efficient, robust and may be used for practical pollutant transport phenomena.
A depth-averaged two-dimensional (2D) numerical model for unsteady flow and nonuniform sediment transport in open channels is established using the finite volume method on a nonstaggered, curvilinear grid. The 2D shallow water equations are solved by the SIMPLE(C) algorithms with the Rhie and Chow's momentum interpolation technique. The proposed sediment transport model adopts a nonequilibrium approach for nonuniform total-load sediment transport. The bed load and suspended load are calculated separately or jointly according to sediment transport mode. The sediment transport capacity is determined by four formulas which are capable of accounting for the hiding and exposure effects among different size classes. An empirical formula is proposed to consider the effects of the gravity on the sediment transport capacity and the bed-load movement direction in channels with steep slopes. Flow and sediment transport are simulated in a decoupled manner, but the sediment module adopts a coupling procedure for the computations of sediment transport, bed change, and bed material sorting. The model has been tested against several experimental and field cases, showing good agreement between the simulated results and measured data.
The paper describes a modelling study of near-shore coastal waters, undertaken to assess the impact of various bacterial input loads on the receiving waters in a coastal basin in the UK. Total and faecal coliforms, used as the indicators for bathing water quality under the European Union (EU) Bathing Water Directive, were numerically modelled using a 2D depth integrated hydro-environmental model. Details are given of the governing equations and solution methods used in the numerical model, together with a discussion of the recent development in faecal bacterial indicator modelling. Details are also given of a field data collection exercise, which involved initially collecting existing information on effluent input loads and followed by an intensive field survey. Using the water quality model, the mortality rate of the pathogen bacteria was investigated. Three methods were used to represent the relationship between the decay rate and the level of solar radiation including: a constant decay rate, day- and night-time decay rates and a solar radiation related time varying decay rate. Relatively close agreement between model predicted and measured total and faecal coliform concentration distributions were obtained for different day- and night-time decay rates and time varying decay rates. No significant differences were found in the optimum decay rates for total and faecal coliform levels. Finally, the impact of the individual inputs on the bathing water quality of the basin was also statistically and numerically investigated. Results showed that the River Irvine was the most significant input during high river flows, and that under these conditions the bathing waters were likely to fail to comply with the European Union Bathing Water Directive. For base river flow conditions the Meadowhead effluent input was found to be critical for both total and faecal coliform level predictions.
In recent years upwind differencing has gained acceptance as a robust and accurate technique for the numerical approximation of the one-dimensional shallow water equations. In two dimensions the benefits have been less marked due to the reliance of the methods on standard operator splitting techniques. Two conservative genuinely multidimensional upwind schemes are presented which have been adapted from flux balance distribution methods recently proposed for the approximation of steady state solutions of the Euler equations on unstructured triangular grids. A method for dealing with source terms, such as those introduced by modelling bed slope and friction, is also suggested and results are presented for two-dimensional steady state channel flows to illustrate the accuracy and robustness of the new algorithms.
Dam-break flows usually occur in domains with complex geometric and topographic features and involve abrupt flow patterns. A dam-break model must therefore be able to effectively handle different flow types including transcritical flows or hydraulic jumps, deal with complex domain topography, capture repeating wet–dry interface and represent high roughness values in the floodplain. Herein, all of these objectives are achieved by extending a recent one-dimensional finite volume Godunov-type model into two dimensions for solving the shallow-water equations. While doing so, a much simplified condition to maintain well-balanced solutions around a wet–dry front is proposed and a two-dimensional friction source term discretization is derived under a suitable stability condition in relation to practical simulations. The two-dimensional model is successfully validated against three analytical benchmark tests and then assessed for predicting realistic dam-break flood events.
When resistance is neglected, the solution of the simple dam-break problem is readily obtained on the basis of shallow-water theory, and the results are well known. However, near the head of the wave, where the water surface meets the ground, resistance effects cannot be neglected; there is, in fact, a type of boundary layer near the wave-front. In this paper, the Pohlhausen method (which is used in conventional boundary-layer problems) is applied to a study of the effect of this 'boundary layer'. In particular, the retardation of the wave-front behind the position predicted by the simple theory is found.
An important test of the quality of a computational model is its ability to reproduce standard test cases or benchmarks. For steady open-channel flow based on the Saint Venant equations some benchmarks exist for simple geometries from the work of Bresse, Bakhmeteff and Chow but these are tabulated in the form of standard integrals. This paper provides benchmark solutions for a wider range of cases, which may have a nonprismatic cross section, nonuniform bed slope, and transitions between subcritical and supercritical flow. This makes it possible to assess the underlying quality of computational algorithms in more difficult cases, including those with hydraulic jumps. Several new test cases are given in detail and the performance of a commercial steady flow package is evaluated against two of them. The test cases may also be used as benchmarks for both steady flow models and unsteady flow models in the steady limit.
A numerical model based upon a second-order upwind finite volume method on unstructured triangular grids is developed for solving shallow water equations. The HLL approximate Riemann solver is used for the computation of inviscid flux functions, which makes it possible to handle discontinuous solutions. A multidimensional slope-limiting technique is employed to achieve second-order spatial accuracy and to prevent spurious oscillations. To alleviate the problems associated with numerical instabilities due to small water depths near a wet/dry boundary, the friction source terms are treated in a fully implicit way. A third-order total variation diminishing Runge-Kutta method is used for the time integration of semidiscrete equations. The developed numerical model has been applied to several test cases as well as to real flows. Numerical tests prove the robustness and accuracy of the model.
A mathematical model of rainfall runoff formation in small catchments is developed; the model is physically valid. The model gives an adequate description of such processes as interception of rainfall by vegetation; storage in relief micro-depressions; infiltration; overland runoff; wash off, transport and re-deposition of soil particles. Numerical simulation of formation of the surface runoff involves the solution of two-dimensional nonstationary shallow water equations, the infiltration equation, and the sediment transport equation. The shallow water equations and the sediment transport equation are integrated numerically using conservative implicit first-order finite-difference schemes. The finite-difference scheme for the shallow water equations allows simulating an open flow with a free boundary. Verification of the model is based on observed data for rain-induced high water in catchment areas of the Butenya river.
Complex isochoric flows in a domain of space that is long compared to its width are studied for a viscoplastic and perfectly rigid Herschel-Bulkley model. It is argued here that no continuous yield surface can exist along the flow direction in these either confined or open channel flows. A similarity analysis is performed that shows that normal stresses cannot be neglected. For open channel flows the influence of normal stresses can be estimated through comparison of the yield stress value to the hydrostatic pressure value at the channel bed. Generalized Barré de Saint Venant one-dimensional equations are obtained. The influence of the yield stress value on wave velocity and on gradually varried flows and critical depth has been deduced.
Although many numerical techniques such as approximate Riemann solvers can be used to analyze subcritical and supercritical flows modeled by hyperbolic-type shallow-water equations, there are some difficulties in practical applications clue to the numerical unbalance between source and flux terms. In this study, a revised surface gradient method is proposed that balances source and flux terms. The new numerical model employs the MUSCL-Hancock scheme and the HLLC approximate Riemann solver. Several verifications are conducted, including analyses of transcritical steady-state flows, unsteady dam break flows on a wet and dry bed, and flows over an irregular bathymetry. The model consistently returns accurate and reasonable results comparable to those obtained through analytical methods and laboratory experiments. The revised surface gradient method may be a simple but robust numerical scheme appropriate for solving hyperbolic-type shallow-water equations over an irregular bathymetry.
The present work concerns the derivation of a suitable discretization to approximate the friction source terms in the shallow-water model. Such additional source terms are known to be very stiff as soon as the water height is vanishing. The proposed numerical procedure comes from a relevant correction of a Godunov-type scheme that approximates the solutions of hyperbolic systems of conservation laws. The adopted correction gives a discretization of the source term which preserves the robustness and does not change the CFL condition. The scheme is shown to be particularly efficient for wet/dry transition simulations. In addition, this numerical procedure can be used together with any robust and well-balanced discretization of the topography source term. Second order extension is also investigated. Extensive numerical validations illustrate the interest of this new approach.
This paper reports the development of an enhanced two-dimensional (2D) numerical model for the simulation of flow hydrodynamics and mass transport in meandering channels. The hydrodynamic model is based on the solution of the depth-averaged flow continuity and momentum equations where the density of flow varies with the concentration of transported mass. The governing equation for mass transport model is the depth-averaged convection and diffusion equation. The dispersion terms arisen from the integration of the product of the discrepancy between the mean and the actual vertical velocity distribution were included in the momentum equations to take into account the effect of secondary current. Two laboratory experimental cases, flow in mildly and sharply curved channels, were selected to test the hydrodynamic model. The comparison of the simulated velocity and water surface elevation with the measurements indicated that the inclusion of the dispersion terms has improved the simulation results. A laboratory experiment study of dye spreading in a sine-generated channel, in which dye was released at the inner bank, centerline, and outer bank, respectively, was chosen to verify the mass transport model. The simulated concentration field indicated that the Schmidt number can be used as a calibration parameter when dispersion is computed using a 2D approach with a simplified turbulence model.
This paper presents the analysis of two different finite volume schemes for hyperbolic conservation laws: the Kolgan high-resolution scheme, and a new Kolgan-type scheme in which the high-order extrapolation of the conservative variables is used just in the upwind contribution of the numerical flux and source terms. Both schemes are compared in terms of the local truncation error, the stability conditions and the C-property. The schemes are applied to different hyperbolic conservation equations, including the one-dimensional scalar transport equation, the Burgers equation and the 2D shallow water equations, in order to compute the observed order of accuracy and to verify the C-property. When applied to the 2D shallow water equations, the new approach avoids spurious oscillations in the solution without the need of using high-order corrections in the definition of the bed slope source term.
This paper presents an experimental validation of two widely used numerical models in urban flood inundation studies, the two-dimensional dynamic and diffusive wave models. Instead of using the common approach in flood inundation modelling, which consists of computing the water depth and velocity fields for a given water discharge, in this study the rainfall intensity is imposed directly in the model, the surface runoff being generated automatically. Both the dynamic and diffusive wave models are implemented in the same unstructured finite volume code, removing in such a way any differences in the numerical discretisation other than the wave approximation used to compute the water velocity. Two different methods for representing buildings are used and compared, the so-called building-block and building-hole approaches. Experimental validation of the models is presented in several simplified laboratory configurations of urban catchments, in which the surface runoff has been measured for different hyetographs. For this purpose, 72 experiments were undertaken in a rainfall simulator, including eight catchment configurations and nine hyetographs. Numerical results show that the dynamic wave model is able to predict the peak discharge and its arrival time, as well as the shape of the outlet hydrograph, while the diffusive wave model gives less accurate results. The experimental validation confirms that, when the geometry of the problem is well defined, depth-averaged dynamic wave models may be used to predict rainfall–runoff from direct precipitation data in urban environments.
The aim of this paper is to present a finite volume kinetic method to compute the transport of a passive pollutant by a flow modeled by the shallow water equations using a new time discretization that allows large time steps for the pollutant computation. For the hydrodynamic part the kinetic solver ensures – even in the case of a non flat bottom – the preservation of the steady state of a lake at rest, the non-negativity of the water height and the existence of an entropy inequality. On an other hand the transport computation ensures the conservation of pollutant mass, a non-negativity property and a maximum principle for the concentration of pollutant and the preservation of discrete steady states associated with the lake at rest equilibrium. The interest of the developed method is to preserve these theoretical properties with a scheme that allows to disconnect the hydrodynamic time step – related to a classical CFL condition – and the transport one – related to a new CFL condition – and further the hydrodynamic calculation and the transport one. The CPU time is very reduced and we can easily solve different transport problems with the same hydrodynamic solution without large storage. Moreover the numerical results exhibit a better accuracy than with a classical method especially when using 1D or 2D regular grids.
The accuracy, stability, and reliability of a numerical model based on a Godunov-type scheme are verified in this paper, through a comparison between calculated results and observed data for the Malpasset dam-break event, which occurred in southern France in 1959. This event is an unique opportunity for code validation because of the availability of extensive field data on the flooding wave due to the dam failure. In the code the shallow water equations are discretized using the finite volume method, and the numerical model allows second order accuracy, both in space and time. The classical Godunov approach is used. More specifically, the Harten, Lax, and van Leer Riemann solver is applied. The resulting scheme is of high resolution and satisfies the total variation diminishing condition. For the numerical treatment of source terms relative to the friction slope, a semi-implicit technique is used, while for the source terms relative to the bottom slope a new explicit method is developed and tested.
This paper deals with the numerical discretization of two-dimensional depth-averaged models with porosity. The equations solved by these models are similar to the classic shallow water equations, but include additional terms to account for the effect of small-scale impervious obstructions which are not resolved by the numerical mesh because their size is smaller or similar to the average mesh size. These small-scale obstructions diminish the available storage volume on a given region, reduce the effective cross section for the water to flow, and increase the head losses due to additional drag forces and turbulence. In shallow water models with porosity these effects are modelled introducing an effective porosity parameter in the mass and momentum conservation equations, and including an additional drag source term in the momentum equations. This paper presents and compares two different numerical discretizations for the two-dimensional shallow water equations with porosity, both of them are high-order schemes. The numerical schemes proposed are well-balanced, in the sense that they preserve naturally the exact hydrostatic solution without the need of high-order corrections in the source terms. At the same time they are able to deal accurately with regions of zero porosity, where the water cannot flow. Several numerical test cases are used in order to verify the properties of the discretization schemes proposed. Copyright © 2009 John Wiley & Sons, Ltd.
Two distributed parameter models, a one-dimensional (1D) model and a two-dimensional (2D) model, are developed to simulate overland flow in two small semiarid shrubland watersheds in the Jornada basin, southern New Mexico. The models are event-based and represent each watershed by an array of 1-m2 cells, in which the cell size is approximately equal to the average area of the shrubs.
Each model uses only six parameters, for which values are obtained from field surveys and rainfall simulation experiments. In the 1D model, flow volumes through a fixed network are computed by a simple finite-difference solution to the 1D kinematic wave equation. In the 2D model, flow directions and volumes are computed by a second-order predictor–corrector finite-difference solution to the 2D kinematic wave equation, in which flow routing is implicit and may vary in response to flow conditions.
The models are compared in terms of the runoff hydrograph and the spatial distribution of runoff. The simulation results suggest that both the 1D and the 2D models have much to offer as tools for the large-scale study of overland flow. Because it is based on a fixed flow network, the 1D model is better suited to the study of runoff due to individual rainfall events, whereas the 2D model may, with further development, be used to study both runoff and erosion during multiple rainfall events in which the dynamic nature of the terrain becomes an important consideration. Copyright
A two-dimensional model for the simulation of solute transport by convection and diffusion into shallow water flow over variable bottom is presented. It is based on a finite volume method over triangular unstructured grids. A first-order upwind technique, a second order in space and time and an extended first-order method are applied to solve the non-diffusive terms in both the flow and solute equations and a centred implicit discretization is applied to the diffusion terms. The stability constraints are studied and the form to avoid oscillatory results in the solute concentration in the presence of complex flow situations is detailed. Some comparisons are carried out in order to show the performance in terms of accuracy of the different options. Copyright © 2007 John Wiley & Sons, Ltd.
The paper describes a new approach to approximating the convection term found in typical steady-state transport equations. A polynomial-based discretization scheme is constructed around a technique called ‘curvature compensation’; the resultant curvature-compensated convective transport approximation is essentially third-order accurate in regions of the solution domain where the concept of order is meaningful. In addition, in linear scalar transport problems it preserves the boundedness of solutions. Sharp changes in gradient in the dependent variable are handled particularly well. But above all, the scheme, when used in conjunction with an ADI pentadiagonal solver, is easy to implement with relatively low computational cost, representing an effective algorithm for the simulation of multi-dimensional fluid flows. Two linear test problems, for the case of transport by pure convection, are employed in order to assess the merit of the method.
Numerical hydrodynamic and sediment transport models provide a means of extending inferences from direct observation and for advancing our understanding of estuarine processes. However, their parametric complexity invites questions concerning the extent to which model output can be assessed with respect to data. This paper examines the basis for evaluating the performance of complex hydrodynamic and sediment transport models, with reference to a case study of a muddy meso-tidal estuary. Sophisticated and computationally-intensive models should be evaluated using robust objective functions, but conventional measures of fit and model efficiency invoke restrictive assumptions about the nature of the errors. Furthermore, they offer little insight into causes of poor performance. Optimization of tidal hydrodynamic models can usefully combine conventional performance measures with harmonic analysis of modelled shallow water tidal constituents that are diagnostic of the interactions between tidal propagation, bathymetry and bottom friction. Models with similar efficiencies can thus be distinguished and likely sources of error pinpointed. Hydrodynamic models have a predictive power that is rooted in a more-or-less complete representation of the physical processes and boundary conditions that are well-constrained with respect to data. In contrast, fine sediment models rely on a less complete conceptualization of a broader set of processes and, crucially, have a parametric complexity that is unmatched by the quantity and quality of observational data. Their performance as measured by conventional objective functions is weaker and it is important to match the structural complexity of model errors with analyses that can localize the scales and times of poor performance. Wavelet analysis is potentially useful here as a means of identifying aspects of the model that need improvement. The context in which such models are deployed is also important. Used heuristically, what might otherwise be dismissed as weak models can still provide mechanistic support for empirically-derived inferences concerning specific aspects of system behaviour. Copyright © 2009 John Wiley & Sons, Ltd.
Two-dimensional shallow water models with porosity appear as an interesting path for the large-scale modelling of floodplains with urbanized areas. The porosity accounts for the reduction in storage and in the exchange sections due to the presence of buildings and other structures in the floodplain. The introduction of a porosity into the two-dimensional shallow water equations leads to modified expressions for the fluxes and source terms. An extra source term appears in the momentum equation. This paper presents a discretization of the modified fluxes using a modified HLL Riemann solver on unstructured grids. The source term arising from the gradients in the topography and in the porosity is treated in an upwind fashion so as to enhance the stability of the solution. The Riemann solver is tested against new analytical solutions with variable porosity. A new formulation is proposed for the macroscopic head loss in urban areas. An application example is presented, where the large scale model with porosity is compared to a refined flow model containing obstacles that represent a schematic urban area. The quality of the results illustrates the potential usefulness of porosity-based shallow water models for large scale floodplain simulations. Copyright © 2005 John Wiley & Sons, Ltd.
An integral form of the shallow-water equations suitable for urban flood mod-eling is derived by applying Reynolds transport theorem to a finite control volume encom-passing buildings on a flood plain. The effect of buildings on storage and conveyance is modeled with a binary density function iðx; yÞ that equals unity when ðx; yÞ corresponds to a void, and nil otherwise, and can be measured using remote sensing data such as clas-sified aerial imagery; the effect of buildings on flow resistance is modeled with a drag for-mulation. Discrete equations are obtained by applying the integral equations to a computational cell and adopting a Godunov-type, piecewise linear distribution of flow variables. The discrete equations include a volumetric porosity / that represents the inte-gral of i over the cell, normalized by the cell area, and an areal porosity w that represents the integral of i over an edge of the mesh, normalized by the edge length. The latter is directionally dependent which introduces anisotropy to the shallow-water equations and captures sub-grid preferential flow directions which occur in urban settings due to asymmetric building shapes and spacings and the alignment of buildings along streets. A important implication is that model predictions are necessarily grid dependent; therefore, a mesh design strategy is proposed. First-and second-order accurate numerical methods are presented to solve the discrete equations, and applications are shown for verification and validation purposes including the ability of the model to resolve preferential flow directions.
In this paper an unstructured finite volume model for quasi-2D tidal flow with wet–dry fronts and turbulence modelling is presented, and applied to the Crouch–Roach estuarine system (Essex, U.K.). Two depth averaged turbulence models, a mixing length model and a k–ε model, are used in the numerical computations. An additional limiter to the production of turbulence due to bed friction is introduced in order to improve the performance and numerical stability of the model near wet–dry fronts. In addition to a first-order and a second-order schemes, an hybrid second-order/first-order upwind scheme which improves the accuracy of the first-order scheme while maintaining a good numerical stability is used to discretize the convective flux. Numerical results are compared with observed current speed and water level data, with particular reference to the ability of the model to reproduce shallow water tidal harmonics. Copyright © 2006 John Wiley & Sons, Ltd.
Depth averaged models are widely used in engineering practice in order to model environmental flows in river and coastal regions, Depth averaged models are widely used in engineering practice in order to model environmental flows in river and coastal regions,
as well as shallow flows in hydraulic structures. This paper deals with depth averaged turbulence modelling. The most important as well as shallow flows in hydraulic structures. This paper deals with depth averaged turbulence modelling. The most important
and widely used depth averaged turbulence models are reviewed and discussed, and a depth averaged algebraic stress model is and widely used depth averaged turbulence models are reviewed and discussed, and a depth averaged algebraic stress model is
presented. A finite volume model for solving the depth averaged shallow water equations coupled with several turbulence models presented. A finite volume model for solving the depth averaged shallow water equations coupled with several turbulence models
is described with special attention to the modelling of wet-dry fronts. In order to asses the performance of the model, several is described with special attention to the modelling of wet-dry fronts. In order to asses the performance of the model, several
flows are modelled and the numerical results are compared with experimental data. flows are modelled and the numerical results are compared with experimental data.
Problems involving the classical linear partial differential equations of mathematical physics can be reduced to algebraic ones of a very much simpler structure by replacing the differentials by difference quotients on some (say rectilinear) mesh. This paper will undertake an elementary discussion of these algebraic problems, in particular of the behavior of the solution as the mesh width tends to zero. For present purposes we limit ourselves mainly to simple but typical cases, and treat them in such a way that the applicability of the method to more general difference equations and to those with arbitrarily many independent variables is made clear.
In this paper we review recent progress in the use of reduced complexity models for predicting floodplain inundation. We review the theoretical basis for modelling floodplain flow with simplified hydraulic treatments based on a dimensional analysis of the one-dimensional shallow water equations. We then review how such schemes can be applied in practice and consider issues of space discretization, time discretization and model parameterisation, before going on to consider model assessment procedures. We show that a key advantage of reduced complexity codes is that they force modellers to think about the minimum process representation necessary to predict particular quantities and act as a check on any tendency to reductionism. At the same time, however, the use (compared to standard hydraulic codes) of strong simplifying assumptions requires us to also address the question “how simple can a model be and still be physically realistic?” We show that by making explicit this debate about acceptable levels of abstraction, reduced complexity codes allow progress to be made in addressing a number of long-standing debates in hydraulics.
Upwind schemes are very well adapted to advection dominated flows and have become popular for applications involving the Euler system of equations. Recently, Riemann solver-based techniques such as Roe’s scheme have become a successful tool for numerical simulation of other conservation laws like the shallow water equations. One of the disadvantages of this technique is related to the treatment of the source terms of the equations. The conservativity of the scheme can be seriously damaged if a careless treatment is applied. Previous papers studied the way to treat the terms arising from bed level changes. This paper deals with the analysis of the main reasons leading to a correct treatment of the geometrical source terms, that is, those representing the changes in cross-section which may be linked to the specific dependence of the flux function on the geometry.
This paper concerns the development of high-order accurate centred schemes for the numerical solution of one-dimensional hyperbolic systems containing non-conservative products and source terms. Combining the PRICE-T method developed in [Toro E, Siviglia A. PRICE: primitive centred schemes for hyperbolic system of equations. Int J Numer Methods Fluids 2003;42:1263–91] with the theoretical insights gained by the recently developed path-conservative schemes [Castro M, Gallardo J, Parés C. High-order finite volume schemes based on reconstruction of states for solving hyperbolic systems with nonconservative products applications to shallow-water systems. Math Comput 2006;75:1103–34; Parés C. Numerical methods for nonconservative hyperbolic systems: a theoretical framework. SIAM J Numer Anal 2006;44:300–21], we propose the new PRICE-C scheme that automatically reduces to a modified conservative FORCE scheme if the underlying PDE system is a conservation law. The resulting first-order accurate centred method is then extended to high order of accuracy in space and time via the ADER approach together with a WENO reconstruction technique. The well-balanced properties of the PRICE-C method are investigated for the shallow water equations. Finally, we apply the new scheme to the shallow water equations with fix bottom topography and with variable bottom solving an additional sediment transport equation.
This paper explores a possible technique for extending to multidimensional flows some of the upwind-differencing methods that have proved highly successful in the one-dimensional case. Attention here is concentrated on the two-dimensional case, and the flow domain is supposed to be divided into polygonal computational elements. Inside each element the flow is represented by a local superposition of elementary solutions consisting of plane waves not necessarily aligned with the element boundaries.
This paper is about the extension of the gas-kinetic BGK scheme to the shallow-water equations with source terms. In the current study, the particle velocity change due to the gravitational force and variable river bottom is implemented explicitly in the flux evaluation. The current scheme is a well-balanced method, which presents accurate and robust results in both steady and unsteady flow simulations.
Flux difference splitting methods are widely used for the numerical approximation of homogeneous conservation laws where the flux depends only on the conservative variables. However, in many practical situations this is not the case. Not only are source terms commonly part of the mathematical model, but also the flux can vary spatially even when the conservative variables do not. It is the discretisation of the additional terms arising from these two situations which is addressed in this work, given that a specific flux difference splitting method has been used to approximate the underlying conservation law. The discretisation is constructed in a manner which retains an exact balance between the flux gradients and the source terms when this is appropriate. The effectiveness of these new techniques, in both one and two dimensions, is illustrated using the shallow water equations, in which the additional terms arise from the modelling of bed slope and, in one dimension, breadth variation. Roe's scheme is chosen for the approximation of the conservation laws and appropriate discrete forms are constructed for the additional terms, not only in the first-order case but also in the presence of flux- and slope-limited high-resolution corrections. The method is then extended to two-dimensional flow where it can be applied on both quadrilateral and triangular grids.
A two-dimensional (2D) numerical model of wave run-up and overtopping is presented. The model (called OTT-2D) is based on the 2D nonlinear shallow water (NLSW) equations on a sloping bed, including bed shear stress. These equations are solved using an upwind finite volume technique and a hierarchical Cartesian Adaptive Mesh Refinement (AMR) algorithm. The 2D nature of the model means that it can be used to simulate wave transformation, run-up, overtopping and regeneration by obliquely incident and multi-directional waves over alongshore-inhomogeneous sea walls and complex, submerged or surface-piercing features. The numerical technique used includes accurate shock modeling, and uses no special shoreline-tracking algorithm or shoreline coordinate transformation, which means that noncontiguous flows and multiple shorelines can easily be simulated. The adaptivity of the model ensures that only those parts of the flow that require higher resolution (such as the region of the moving shoreline) receive it, resulting in a model with a high level of efficiency. The model is shown to accurately reproduce analytical and benchmark numerical solutions. Existing wave flume and wave basin datasets are used to test the ability of the model to approximate 1D and 2D wave transformation, run-up and overtopping. Finally, we study a 2D dataset of overtopping of random waves at off-normal incidence to investigate overtopping of a sea wall by long-crested waves. The data set is interesting as it has not been studied in detail before and suggests that, in some instances, overtopping at an angle can lead to more flooding than at normal incidence.
In this paper, certain well-known upwind schemes for hyperbolic equations are extended to solve the two-dimensional Saint-Venant (or shallow water) equations. We consider unstructured meshes and a new type of finite volume to obtain a suitable treatment of the boundary conditions. The source term involving the gradient of the depth is upwinded in a similar way as the flux terms. The resulting schemes are compared in terms of a conservation property. For the time discretization we consider both explicit and implicit schemes. Finally, we present the numerical results for tidal flows in the Pontevedra ria, Galicia, Spain.