Two-dimensional Shallow Water Equations Research Articles

A partitioned finite element method for power-preserving discretization of open systems of conservation laws

AbstractThis paper presents a structure-preserving spatial discretization method for distributed parameter port-Hamiltonian systems. The class of considered systems are hyperbolic systems of two conservation laws in arbitrary spatial dimension and geometries. For these systems, a partitioned finite element method (PFEM) is derived, based on the integration by parts of one of the two conservation laws written in weak form. The non-linear one-dimensional shallow-water equation (SWE) is first considered as a motivation example. Then, the method is investigated on the example of the non-linear two-dimensional SWE. Complete derivation of the reduced finite-dimensional port-Hamiltonian system (pHs) is provided and numerical experiments are performed. Extensions to curvilinear (polar) coordinate systems, space-varying coefficients and higher-order pHs (Euler–Bernoulli beam equation) are provided.

Upscaling the shallow water equations for fast flood modelling

This paper presents a new sub-grid flood inundation model aimed at high computational performance. The model solves the two-dimensional shallow water equations (SWE) by a Godunov-type finite volume (FV) method that uses two nested meshes. Runtime computations are performed at a coarse computational mesh, while a fine mesh is used to incorporate fine resolution information into the solution at pre-processing level. New upscaling methods are separately derived for each of the terms in the SWE based on the integration of the governing equations over subdomains defined by the coarse resolution grid cells. The accuracy and performance of the model are tested through artificial and real-world test problems. Results showed that (i) for the same computational (coarse mesh) resolution, the inclusion of sub-grid information delivers more accurate results than a single-mesh FV model and (ii) for the same accuracy and at low resolution, the proposed methods improve computational performance.

Consistent section-averaged shallow water equations with bottom friction

In this paper, we present a general framework to construct section-averaged models when the flow is constrained – e.g. by topography – to be almost one-dimensional. These models are consistent with the two-dimensional shallow water equations. After rewriting the two-dimensional shallow water equations in a suitable set of coordinates allowing to take care of a meandering configuration, we consider the quasi one-dimensional regime. Then, we expand the water elevation and velocity field in the spirit of the diffusive wave equations and establish a set of one-dimensional equations made of a mass, momentum and energy equations, which are close to the ones usually used in hydraulic engineering. Our model reduces to classical shallow water models with variable sections found in the literature. Out of these configurations, there is an O(1) deviation of our model from the classical ones. Finally, we present the main mathematical properties of our model and carry out numerical simulations to validate our approach by comparing the results to the full two-dimensional shallow water equations.

A quasi three dimensional hydrodynamic model for velocity distribution in open channel

ABSTRACT This work aims at developing a fully coupled numerical model by integrating the two-dimensional shallow water equations with entropy theory. TVD Mac-Cormack predictor-corrector finite difference scheme is employed to solve the modified form of shallow water equation in MATLAB. Computed two-dimensional hydraulic parameters are substituted in the entropy theory to calculate the vertical velocity profile in the domain. The coupled model applied at different locations in the Brahmaputra river and the computed outputs validated with the observed field data. The robustness and stability of the linked model are found satisfactory. The influence of the hydraulic structures on secondary current generation and the velocity dip phenomenon is investigated during the study.

Enriched Galerkin method for the shallow-water equations

This work presents an enriched Galerkin (EG) discretization for the two-dimensional shallow-water equations. The EG finite element spaces are obtained by extending the approximation spaces of the classical finite elements by discontinuous functions supported on elements. The simplest EG space is constructed by enriching the piecewise linear continuous Galerkin space with discontinuous, element-wise constant functions. Similar to discontinuous Galerkin (DG) discretizations, the EG scheme is locally conservative, while, in multiple space dimensions, the EG space is significantly smaller than that of the DG method. This implies a lower number of degrees of freedom compared to the DG method. The EG discretization presented for the shallow-water equations is well-balanced, in the sense that it preserves lake-at-rest configurations. We evaluate the method’s robustness and accuracy using various analytical and realistic problems and compare the results to those obtained using the DG method. Finally, we briefly discuss implementation aspects of the EG method within our MATLAB / GNU Octave framework FESTUNG.

Research on Flood Propagation for Different Dam Failure Modes: A Case Study in Shenzhen, China

Dam-break flood simulation can evaluate the impact of a dam-break, and is significant in conducting risk analyses, creating emergency plans, and mitigating calamities. This study investigates the dam-break flood propagation in the downstream urban areas of the Minzhi Reservoir in Shenzhen, China, based on a two-dimension shallow-water model (SWM-2D). The SWM-2D solves two-dimensional shallow water equations for free surface flow using the finite-volume method. A refined grid with an unstructured mesh of triangular cells, constrained by building walls, is constructed to represent urban structures in this model. The flood hydrographs of dam-break and the flood propagation under different failure modes are calculated by adopting two breach mechanisms, namely instantaneous dam-break and gradual dam-break. The results indicate that the peak flow of the instantaneous dam-break is relatively large at the beginning of the dam-break. The gradual dam-break peak flow is relatively small when seepage failure deformation develops towards the upper part of the dam. The inundation information during the evolution of the dam-break flood, including the regularity of the flood propagation, distribution of maximum water depth, and variation of depth and velocity with time are also analysed.

FESTUNG 1.0: Overview, usage, and example applications of the MATLAB/GNU Octave toolbox for discontinuous Galerkin methods

The present work documents the current state of development for our MATLAB/GNU Octave-based open source toolbox FESTUNG (Finite Element Simulation Toolbox for UNstructured Grids). The goal of this project is to design a user-friendly, research-oriented, yet computationally efficient software tool for solving partial differential equations (PDEs). Since the release of its first version, FESTUNG has been actively used for research and teaching purposes such as the design of novel algorithms and discretization schemes, benchmark studies, or just providing students with an easy-to-learn software package to study advanced numerical techniques and good programming practices. For spatial discretization, the package employs various discontinuous Galerkin (DG) methods, while different explicit, implicit, or semi-implicit Runge–Kutta schemes can be used for time stepping. The current publication discusses the most important aspects of our toolbox such as the code design concepts and various discretization procedures illustrated in some detail using a standard advection–diffusion–reaction equation. Moreover, we present selected applications already supported in FESTUNG including solvers for the two-dimensional shallow-water equations, the Cahn–Hilliard equation, and a coupled multi-physics model of free surface/subsurface flow.

2D Shallow Water Model for Dam Break and Column Interactions

Dam break causes disastrous effects on the surrounding area, especially at the downstream, therefore, there is a need for accurate and timely predictions of dam break propagation to prevent both property damage and loss of life. This study aimed to determine the movement of dam-break flow in the downstream area by solving the Shallow Water Equations (SWE) or Saint Venant Equations which are based on the conservation of mass and momentum derived from Navier Stokes equation. The model was generated using a finite difference scheme which is the most common and simplest method for dam-break modeling while Forward Time Central Space (FTCS) numerical scheme was applied to simulate two-dimensional SWE. Moreover, the accuracy of the numerical model was checked by comparing its results with the analytic results of one-dimensional cases and a relatively small value of error was found in comparison to the analytic models as indicated with the RMSE values close to 0. The numerical to the two-dimensional models were also compared to a simple dam break in a flume and dam break with column interactions and the wave propagation in both cases was observed to become very close at a certain time. The model, however, used numerical filter (Hansen) to reduce the oscillations or numerical instability. The simulation and analysis, therefore, showed the ability of the numerical scheme of FTCS to resolve both cases of the simple dam break and dam break with column interactions in the Two-dimensional Shallow Water.

Trajectory and weathering of oil spill in Daya bay, the South China sea

With the development of marine resources and marine transportation, oil spill accidents occur frequently which threaten the marine ecological environment and human life. In this paper, an oil spill model was established. The two-dimensional shallow water equation was discretized by the finite element weighted lumped mass method, and the time is discretized by the forward Euler scheme, then the planar two-dimensional hydrodynamic model was established. The model was verified by measure tidal level data. The oil particle drift model and oil spill weathering model were established in this paper, and it can be used to simulate the oil spill accidents on the sea area by inputting the terrain data, environmental conditions and oil spill information into the hydrodynamic model and oil spill model. The model is applied to Daya Bay, South China Sea, the oil spill behavior and destination under different residual currents were simulated and calculated, the pollution area of oil spill under clockwise residual flow is larger than that under anti-clockwise residual flow. The oil spill model is mainly used to simulate oil spill accidents on the sea surface such as ship oil spill accidents, and the simulation results can provide theoretical basis for taking effective emergency measures and risk assessment after oil spill.

Port-Hamiltonian model of two-dimensional shallow water equations in moving containers

Abstract The free surface motion in moving containers is an important physical phenomenon for many engineering applications. One way to model the free surface motion is by employing shallow water equations (SWEs). The port-Hamiltonian systems formulation is a powerful tool that can be used for modeling complex systems in a modular way. In this work, we extend previous work on SWEs using the port-Hamiltonian formulation, by considering the two-dimensional equations under rigid body motions. The resulting equations consist of a mixed-port-Hamiltonian system, with finite and infinite-dimensional energy variables and ports. 2000 Math Subject Classification: 34K30, 35K57, 35Q80, 92D25

Urban Flood Modeling Using 2D Shallow-Water Equations in Ouagadougou, Burkina Faso

Appropriate methods and tools accessibility for bi-dimensional flow simulation leads to their weak use for floods assessment and forecasting in West African countries, particularly in urban areas where huge losses of life and property are recorded. To mitigate flood risks or to elaborate flood adaptation strategies, there is a need for scientific information on flood events. This paper focuses on a numerical tool developed for urban inundation extent simulation due to extreme tropical rainfall in Ouagadougou city. Two-dimensional (2D) shallow-water equations are solved using a finite volume method with a Harten, Lax, Van Leer (HLL) numerical fluxes approach. The Digital Elevation Model provided by NASA’s Shuttle Radar Topography Mission (SRTM) was used as the main input of the model. The results have shown the capability of the numerical tool developed to simulate flow depths in natural watercourses. The sensitivity of the model to rainfall intensity and soil roughness coefficient was highlighted through flood spatial extent and water depth at the outlet of the watershed. The performance of the model was assessed through the simulation of two flood events, with satisfactory values of the Nash–Sutcliffe criterion of 0.61 and 0.69. The study is expected to be useful for flood managers and decision makers in assessing flood hazard and vulnerability.

A new fifth-order finite difference well-balanced multi-resolution WENO scheme for solving shallow water equations

In this paper, a new fifth-order finite difference well-balanced multi-resolution weighted essentially non-oscillatory (WENO) scheme is designed to solve for one-dimensional and two-dimensional shallow water equations with or without source terms on structured meshes. We only use the information defined on a hierarchy of nested central spatial stencils without introducing any equivalent multi-resolution representation once again. For the shallow flow problems with smooth or discontinuous bed, we combine with the well-balanced procedure developed by Xing and Shu (2005) for balancing the flux gradients and the source terms, and then the new fifth-order well-balanced multi-resolution WENO scheme (Zhu and Shu, 2018) could satisfy the exact C-property for still stationary solutions, maintain the fifth-order accuracy in smooth regions, and keep essentially non-oscillatory property in non-smooth regions. Compared with the classical well-balanced WENO schemes (Xing and Shu, 2005), the new features of this finite difference well-balanced multi-resolution WENO scheme is its simplicity and hierarchical structure in obtaining higher-order accuracy and its linear weights could be arbitrarily chosen with one requirement that their summation is one. It is the first time that a series of unequal-sized hierarchical central spatial stencils are used in designing finite difference well-balanced WENO scheme for solving shallow water equations. This new well-balanced WENO scheme is simple to construct and can be easily implemented to arbitrary high-order accuracy and in higher dimensions. Some benchmark numerical examples are performed to illustrate the good performances of this new well-balanced WENO scheme which could get high-order accuracy, keep exact C-property, sustain good convergence property, and obtain sharp shock transitions in the whole computational field.

Two-dimensional numerical modelling of shallow water flows over multilayer movable beds

The two-dimensional modelling of shallow water flows over multi-sediment erodible beds is presented. A novel approach is developed for the treatment of multiple sediment types in morphodynamics. The governing equations include the two-dimensional shallow water equations for hydrodynamics, an Exner-type equation for morphodynamics, a two-dimensional transport equation for the suspended sediments, and a set of empirical equations for entrainment and deposition. Multilayer sedimentary beds are formed of different erodible soils with sediment properties and new exchange conditions between the bed layers are developed for the model. The coupled equations yield a hyperbolic system of balance laws with source terms. As a numerical solver for the system, we implement a fast finite volume characteristics method. The numerical fluxes are reconstructed using the method of characteristics which employs projection techniques. The proposed finite volume solver is simple to implement, satisfies the conservation property and can be used for two-dimensional sediment transport problems in non-homogeneous isotropic beds without need of complicated three-dimensional equations. To assess the performance of the proposed models, we present numerical results for a wide variety of shallow water flows over sedimentary layers. Comparisons to experimental data for dam-break problems over movable beds are also included in this study.

Two-dimensional flow movement in the area of protective regulatory structures

The article discusses the results of numerical studies of the flow movement with a sharp change in the parameters of the channel. Basically, the results of the study using the system of two-dimensional equations of hydrodynamics-Saint-Venant are analyzed. The divergent form of two-dimensional equations describing the movement of a water stream at a site of regulation of a channel by protective and regulatory dams is given. The influence of the length step on the results of numerical experiments is investigated numerically. Graphs of the time variation of the longitudinal velocity component behind the sudden double expansion of the channel are compiled. The flow was unsteady all the time and had the character of stationary pulsations, and the finer the grid, the richer the spectrum of these pulsations. It was noted that in numerical calculations, the time step in the calculations was always much less than the minimum pulsation period, therefore, these pulsations were not associated with difference oscillations that can arise when approximating by central differences. It is concluded that, according to the authors from the following and the present work, they collectively show that the pulsations on different grids differ significantly, the average values of the velocities are close, and thereby the solution for the average values is well converged, this shows that the pulsations are a property source equations of Saint-Venant. The applicability of the numerical model, consisting of two-dimensional shallow water equations, the vector equation of momentum conservation and the scalar equation of mass conservation, in description the flow with the presence of circulation zones, which is typical when water flows are constrained by protective-regulatory structures. In this case, the solution pulsates around a certain average value, and the average length of the circulation zone behind the sudden expansion of the open flow is in good agreement with the laboratory experiments of G.L. Mazhbits.

Hydrodynamic model of Daya Bay based on finite element method

The current problems are hot spots in recent years. With the rapid development of modern computer technology, numerical simulation technology has become an important means of scientific research, and the correlation numerical solution of two-dimensional shallow water equation is endless. In this paper, the two-dimensional shallow water equation was discretized by the finite element weighted lumped mass method, and the time is discretized by the forward Euler scheme, then the planar two-dimensional hydrodynamic model was established. The water boundary condition was defined by harmonic analysis method. The two-dimensional hydrodynamic model was applied to Daya Bay area and verified according to the measured meteorological data. Therefore, the hydrodynamic model established in this paper can be used to study the actual water flow in Daya Bay area.

Short-Term and Long-Term Solutions of Two-Dimensional Shallow Water Equations Using the Modified Decomposition Method

In this paper, two modifications of the decomposition method (DM) including its multistage form (say here MSDM) and its combination with the Pade approximants (say here DMPA) are proposed and applied for the short-term and long-term solutions of two-dimensional shallow water equations (SWEs) including friction effects. Due to nonlinear behavior of the model, the DM is used to facilitate the computations on the nonlinear terms. However, the MS technique is used to obtain a solution for longer periods of time and to prevent accuracy issues associated with the standard DM at large times. The results of the proposed method are compared with those obtained using the PAs as an after treatment technique which also yields an accurate solution in our work. It has been shown that the MS technique can efficiently be used in the solution of the nonlinear coupled SWEs and would extend the domain of validity of the solution obtained via the standard DM. However, the necessity of predicting the long-term behavior of flow in some applications makes the PAs superior to the MS technique with respect to accuracy and CPU time. Numerical and graphical results of the flow characteristics for a given application of two-dimensional flood wave over two types of mild and steep slopes are presented to elucidate the efficiency of the proposed modifications in the study of the flow behavior. Moreover, a convergence analysis and the time evolution of the residuals as an index of the solution error are also presented in tabular and graphical forms.

Group classification of the two-dimensional shallow water equations with the beta-plane approximation of coriolis parameter in Lagrangian coordinates

Two-dimensional shallow water equations with uneven bottom and a Coriolis parameter f=f0+βy, (β ≠ 0) in mass Lagrangian coordinates are studied in this paper. The equations describing these flows are reduced to two Euler–Lagrange equations. The paper provides a complete group classification of the equations and applications of Noether’s theorem for constructing conservation laws.

A New Parallel Framework of SPH-SWE for Dam Break Simulation Based on OpenMP

Due to its Lagrangian nature, Smoothed Particle Hydrodynamics (SPH) has been used to solve a variety of fluid-dynamic processes with highly nonlinear deformation such as debris flows, wave breaking and impact, multi-phase mixing processes, jet impact, flooding and tsunami inundation, and fluid–structure interactions. In this study, the SPH method is applied to solve the two-dimensional Shallow Water Equations (SWEs), and the solution proposed was validated against two open-source case studies of a 2-D dry-bed dam break with particle splitting and a 2-D dam break with a rectangular obstacle downstream. In addition to the improvement and optimization of the existing algorithm, the CPU-OpenMP parallel computing was also implemented, and it was proven that the CPU-OpenMP parallel computing enhanced the performance for solving the SPH-SWE model, after testing it against three large sets of particles involved in the computational process. The free surface and velocities of the experimental flows were simulated accurately by the numerical model proposed, showing the ability of the SPH model to predict the behavior of debris flows induced by dam-breaks. This validation of the model is crucial to confirm its use in predicting landslides’ behavior in field case studies so that it will be possible to reduce the damage that they cause. All the changes made in the SPH-SWEs method are made open-source in this paper so that more researchers can benefit from the results of this research and understand the characteristics and advantages of the solution proposed.

Numerical entropy production as smoothness indicator for shallow water equations

The numerical entropy production (NEP) for shallow water equations (SWE) is discussed and implemented as a smoothness indicator. We consider SWE in three different dimensions, namely, one-dimensional, one-and-a-half-dimensional, and two-dimensional SWE. An existing numerical entropy scheme is reviewed and an alternative scheme is provided. We prove the properties of these two numerical entropy schemes relating to the entropy steady state and consistency with the entropy equality on smooth regions. Simulation results show that both schemes produce NEP with the same behaviour for detecting discontinuities of solutions and perform similarly as smoothness indicators. An implementation of the NEP for an adaptive numerical method is also demonstrated.
 
 doi:10.1017/S1446181119000154

Complete group classification of the two-Dimensional shallow water equations with constant coriolis parameter in Lagrangian coordinates

The two-dimensional shallow water equations with constant Coriolis parameter and variable topography bottom in mass Lagrangian coordinates are studied in this paper. The equations describing these flows are reduced to two Euler-Lagrange equations. Group classification of these equations with respect to the function describing the topography of the bottom is performed in the paper. For some particular functions (circular and plane), transformations mapping the two-dimensional shallow water equations with constant Coriolis parameter into the gas dynamics equations of a polytropic gas with polytropic exponent γ=2 are found.