Two-dimensional Shallow Water Equations Research Articles

An Improved PINN Algorithm for Shallow Water Equations Driven by Deep Learning

Solving shallow water equations is crucial in science and engineering for understanding and predicting natural phenomena. To address the limitations of Physics-Informed Neural Network (PINN) in solving shallow water equations, we propose an improved PINN algorithm integrated with a deep learning framework. This algorithm introduces a regularization term as a penalty in the loss function, based on the PINN and Long Short-Term Memory (LSTM) models, and incorporates an attention mechanism to solve the original equation across the entire domain. Simulation experiments were conducted on one-dimensional and two-dimensional shallow water equations. The results indicate that, compared to the classical PINN algorithm, the improved algorithm shows significant advantages in handling discontinuities, such as sparse waves, in one-dimensional problems. It accurately captures sparse waves and avoids smoothing effects. In two-dimensional problems, the improved algorithm demonstrates good symmetry and effectively reduces non-physical oscillations. It also shows significant advantages in capturing details and handling complex phenomena, offering higher reliability and accuracy. The improved PINNs algorithm, which combines neural networks with physical mechanisms, can provide robust solutions and effectively avoid some of the shortcomings of classical PINNs methods. It also possesses high resolution and strong generalization capabilities, enabling accurate predictions at any given moment.

Improved local time-stepping schemes for storm surge modeling on unstructured grids

This paper presents improved explicit local time-stepping (LTS) schemes of both first and second order accuracy for storm surge modeling. The two-dimensional shallow water equations are numerically solved on unstructured triangular meshes using finite volume method with Roe’s approximate Riemann solver. The LTS algorithms are designed based on explicit Euler and strong stability preserving Runge–Kutta time integration methods. A single-layer interface prediction–correction scheme is adopted to combine coarse and fine time discretization, further enhancing the stability of the LTS schemes, particularly at higher LTS levels and during long time simulations. An ideal numerical test validates the efficiency of the improved LTS models, revealing their capability to improve computational speed while preserving conservation properties and reducing accuracy loss as LTS levels increase. We further apply the LTS models to cross-scale simulations of storm surges in the Northwest Pacific. Results show that compared to the global time-stepping (GTS) models, the LTS models significantly boost computational speed by up to 37%, all while delivering equally reliable computational outcomes. With expanding high-resolution coastal data and the need for high-resolution modeling, the improved LTS models show great potential for cross-scale storm surge modeling.

Application of 1D model for overland flow simulations on 2D complex domains

High computational demand limits the applications of two-dimensional (2D) shallow water equation models for high resolution overland flow simulations, while one-dimensional (1D) models can achieve higher computing efficiency. This study applied a 1D hydrodynamic model to surrogate 2D models for overland flow simulations on complex 2D domains. With one dimension reduced, the surrogate model simulation would have acceptable accuracy with much higher computing efficiency. The surrogating is fulfilled through mimicking 2D models in mesh generations, so that the 1D channel network is generated in such a way that it geometrically covers the whole domain without overlapping and intersection, and hydrologically follows the steepest slopes. Several benchmark cases on 2D domains in both laboratory and field scales with complex geometry, where no 1D models have been ever applied, are used to compare the 1D and 2D model simulations. The comparisons demonstrate that the 1D model does have potentials to efficiently simulate overland flow on 2D complex domains with accuracy comparable to 2D models.

A parallel porosity model for large-scale modelling of river floods in urban areas

With the purpose of modelling at a large-scale river floods affecting urban areas, this work presents a numerical model solving the two-dimensional Shallow Water Equations with porosity. The distribution of buildings within urban fabrics is accounted for by means of a spatially-distributed field of the storage porosity parameter. The finite volume scheme conserves mass and it ensures the well-balancing property over complex topography. A structured multi-resolution grid is adopted for the domain discretization and the parallelization on Graphic Processing Units (GPUs) enables the reduction of the computational costs. Model testing is performed against experimental data and applications over real domains as well. The results provided by the proposed model agree with those obtained by resolving building footprints on a high-resolution mesh, meanwhile reducing the runtimes up to 6 times.

Influences of coriolis force and friction on fluid dynamics in specific paraboloid basins

Natural water features, such as rivers, lakes, bays, and straits, exhibit diverse shapes and dynamics. Among these, paraboloid basins, resembling the contours of lakes or closed bays, offer a relatively simple yet intriguing case. This study delves into the dynamics of water waves within two distinct paraboloid basin variations: the parabolic canal and the circular paraboloid. In this investigation, we analyze shoreline movement, considering influential factors such as bottom friction and the Coriolis effect through a mathematical model. The foundation of our analysis lies in the two-dimensional shallow water equations, with the Thacker assumption serving as the basis for deriving an analytical solution. To enhance our understanding, we employ a numerical solution, employing the finite volume method on a staggered grid, to simulate wave behavior within these basin variations. For validation, we compare these numerical findings with our analytical solution. Furthermore, this study conducts a sensitivity analysis of the Coriolis parameter and bottom friction parameter under various conditions. Through this exploration, we gain valuable insights into the interplay of these critical factors in shaping the oscillation of water within paraboloid basins, enriching our understanding of coastal and basin dynamics.

OPTIMIZATION OF DESIGN OF THE FISH-SPAWNING PASS OF THE BAGAEVSKY HYDROELECTRIC COMPLEX USING NUMERICAL 2D MODELING

Numerical hydrodynamic modeling of the flow of water and sediment through the fish-spawning pass of the Bagaevsky hydroelectric complex on the river Don is considered. The calculations use two-dimensional shallow water and sediment transport equations by the STREAM 2D CUDA software complex. Recommendations are made to change the longitudinal profile of the bottom and the configuration of the inlet section, as well as to increase the coefficient of roughness of the bottom and sides of the channel to ensure fish protection requirements.

Symmetry Analysis of the Two-Dimensional Stationary Gas Dynamics Equations in Lagrangian Coordinates

This article analyzes the symmetry of two-dimensional stationary gas dynamics equations in Lagrangian coordinates, including the search for equivalence transformations, the group classification of equations, the derivation of group foliations, and the construction of conservation laws. The consideration of equations in Lagrangian coordinates significantly simplifies the procedure for obtaining conservation laws, which are derived using the Noether theorem. The final part of the work is devoted to group foliations of the gas dynamics equations, including for the nonstationary isentropic case. The group foliations approach is usually employed for equations that admit infinite-dimensional groups of transformations (which is exactly the case for the gas dynamics equations in Lagrangian coordinates) and may make it possible to simplify their further analysis. The results obtained in this regard generalize previously known results for the two-dimensional shallow water equations in Lagrangian coordinates.

A novel efficient and robust treatment of the friction source term in 2D shallow water inundation models

The technological advancements of the last few decades have fostered the use of two-dimensional (2D) Shallow water Equations (SWE) flood simulations not only in academic research but also in practical real-world applications and territorial planning. The evaluation of flow resistance due to friction is crucial as it plays a relevant role in a variety of flow conditions. However, problems arise in dam-break and overland flow applications involving very small water depth because the time scale connected to the friction term may be much smaller than the hydrodynamic time scale. To cope with this issue, known as source term stiffness, an implicit treatment of the friction term is frequently adopted in Finite Volume (FV) numerical schemes, but there are cases (null roughness coefficient or null flow velocity) where a division by zero may occur during calculations, leading to a crash of the algorithm. Herein, we propose a reformulation of the implicit friction term that is general, more stable and computationally efficient (with a speed-up of approximately 3 times for the routine running the friction computation) than currently available approaches. The novel implicit method allows to manage special conditions with null roughness or null discharge, without resorting to ad-hoc thresholds, and can be easily implemented in existing schemes with pointwise friction treatment. The novel friction approach is implemented in a purposely simple FV numerical scheme (hydrostatic reconstruction to account for the bed slope terms, first-order accuracy in time and space) - to better focus on the friction term treatment - and it is validated against analytical, synthetic, laboratory and real-world case studies, showing promising capabilities.

Operational Mode for Water–Sediment Regulation in Plain-Type Sand-Laden Reservoirs: A Case Study of the Haibowan Reservoir

Excessive sedimentation in sand-laden rivers significantly hinders the normal operation and overall effectiveness of reservoirs. This is observed particularly in plain-type sand-laden reservoirs where weak hydraulic conditions in the reservoir area contribute to sediment deposition. Water–sediment regulation is essential in reducing sedimentation and prolonging the lifespan of such reservoirs. Taking the plain-type reservoir of the primary stream of the Yellow River—Haibowan Reservoir as an example, based on a two-dimensional shallow water equation, a two-dimensional mathematical model of water and sediment in the reservoir area is established, the corresponding boundary conditions are improved, and the reliability of the model is verified. Under the premise of ensuring the flood control safety of the reservoir dam, considering the impact on the power generation of the reservoir and the downstream river, and maintaining the long-term large effective storage capacity of the reservoir as the basic principle, the water–sediment regulation scheme is proposed. A two-dimensional kinetic model of sediment transport and representative water–sediment series are employed to simulate the erosion and sedimentation processes in the reservoir under different schemes. The optimisation and comparison of multiple schemes reveal that the sediment-flushing scheme, which lowers the water level to 1072 m above sea level when the inflow discharge is 1500–2760 m3/s and inflow sediment concentration is 5–10 kg/m3, can yield an effective reservoir capacity of 97 million m3 and an average annual hydropower generation of 389.5 million kWh after 15 years of operation. This scheme outperforms the existing sediment-flushing scheme and is recommended as an optimised approach for water–sediment regulation in reservoirs. The results of this study provide technical support for the application of water–sediment regulation in the Haibowan Reservoir and may be useful in the exploration and management of water–sediment regulation for plain-type reservoirs on sand-laden rivers.

Surrogate-Assisted Evolutionary Algorithm for the Calibration of Distributed Hydrological Models Based on Two-Dimensional Shallow Water Equations

Distributed hydrological models based on shallow water equations have gained popularity in recent years for the simulation of storm events, due to their robust and physically based routing of surface runoff through the whole catchment, including hill slopes and water streams. However, significant challenges arise in their calibration due to their relatively high computational cost and the extensive parameter space. This study presents a surrogate-assisted evolutionary algorithm (SA-EA) for the calibration of a distributed hydrological model based on 2D shallow water equations. A surrogate model is used to reduce the computational cost of the calibration process by creating a simulation of the solution space, while an evolutionary algorithm guides the search for suitable parameter sets within the simulated space. The proposed methodology is evaluated in four rainfall events located in the northwest of Spain: one synthetic storm and three real storms in the Mandeo River basin. The results show that the SA-EA accelerates convergence and obtains superior fit values when compared to a conventional global calibration technique, reducing the execution time by up to six times and achieving between 98% and 100% accuracy in identifying behavioral parameter sets after four generations of the SA-EA. The proposed methodology offers an efficient solution for the calibration of complex hydrological models, delivering improved computational efficiency and robust performance.

Investigating the bearing performance of the foundation under the combined effects of flood scouring and soaking

Bearing capacity degradation of foundations under the impact of the flood is one of the major reasons responsible for the collapse and damage to the rural buildings, posing a serious threat to the local village societies. Based on a case study of a rural building foundation had been destroyed by flooding. This paper investigated the deterioration process of rural building foundations under the combined effect of dynamic scouring and static soaking caused by flooding. Using the two-dimensional shallow water equation, erosion depth was calculated for different flood velocities. Then, the bearing capacity degradation under the combined scouring-soaking effect was analyzed using the finite element method. Finally, investigating the influence of inflow direction and building group masking on the foundation's bearing capacity. The results indicate that under the combined effect, the bearing capacity of village building foundations decreases by 47.88%, with scouring slightly more impactful than soaking. Inflow angle has minimal effect on bearing performance, while the masking effect of the building group provides better protection for the foundation of rear buildings.

A robust unstructured finite volume scheme for urban flash flood simulation: Duhok city as a case study

ABSTRACT A robust well-balanced finite volume scheme is proposed for simulating the flash floods due to rainfall-runoff process. The solver’s implementation on unstructured triangular meshes is based on a Godunov-type second-order upwind finite volume formulation, and Roe’s average scheme is used to determine the fluxes. The bed slope, friction and rainfall source terms are discretized using an upwind approach. The suggested algorithm uses the slope limiter and Hancock’s predictor-corrector method in order to secure numerical stability. The proposed model overwhelms the drawbacks of many commercial codes that use the finite difference, finite elements and finite volume in their scheme, especially in dealing with rapid change in topography. The prepared scheme solves the two-dimensional shallow water equations over a real topography of Hishkaro basin located in Duhok city in Kurdistan of Iraq where the variation of bed elevation reaches 717 m. The scheme is validated against real observed data in Duhok city provided by the Sewage directory. The data of five observation points over the study area were used to validate the code. The results showed that the max water level reaches 3.67, 2.12, 1.20, 0.67 and 0.3 m at observed points 3, 2, 4, 1 and 5, respectively.

Equivalence of Systems of Two-Dimensional Shallow-Water Equations over Horizontal and Sloping Bottom

Equivalence of Systems of Two-Dimensional Shallow-Water Equations over Horizontal and Sloping Bottom

A well-balanced positivity-preserving multidimensional central scheme for shallow water equations

Recently, an efficient and multidimensional central scheme has been proposed by applying the limiting process diagonally (Yan et al. (2023) [25]). The aforementioned approach is applied to solve the two-dimensional shallow water equations with non-flat bottom topography in this study. The nonnegativity of the updated water depth is preserved under a reasonable CFL condition. The well-balancedness is achieved by simply applying central difference discretization to the source term using the surface gradient method. The conservation of water mass is obtained by changing the scheme into conservation form. All the key features of the present scheme have been rigorously proven. Several numerical experiments are provided to illustrate the robustness of the proposed scheme, including simulations of shallow water equations with Manning's bottom friction and Coriolis forces.

Hydraulic analyses using two-dimensional shallow water equations for functional evaluation of the Yamadazeki barrage in the Chikugo river, Japan

Hydraulic analyses using two-dimensional shallow water equations for functional evaluation of the Yamadazeki barrage in the Chikugo river, Japan

A novel technique for implementing the finite element method in a shallow water equation

We presented a novel approach to investigate the two-dimensional shallow water equation in its primitive form. Its employs the P1NC−P1 element pair to simulate various cases: standing waves, dam-break planar, and wave absorbing with embedded radiation boundary conditions. Unlike the conventional method, we approximate the free surface variable using a conformal basis P1 whereas the velocity potential is approximated using a non-conformal basis, P1NC. Thus, for each case, the weak form needs to be reformulated as well as the discrete form. The resulting scheme is a first-order ordinary differential system and solved by Crank Nicholson. The mass matrix in the momentum equation contains the multiplication between the two bases, which computed by the mass lumping. So, our method is explicit, flexible and easy to implement. Validation using standing waves demonstrated first-order accuracy, free from numerical damping and convergent to the analytical solution. Dam-break simulation result shown an agreement with ANUGA software. Our scheme's flexibility is demonstrated when it can mimic wave absorbing simulation employing embedded radiation boundary conditions. The reflection at the boundary seems small enough, thus can be neglected. All these findings have shown the robustness and capability of our scheme to predict accurate results for various shallow water flow problems.•A novel technique for solving 2D SWE in primitive form•It is explicit, flexible, easy to implement, accurate, and robust•Our approach is suitable for coastal/oceanographic simulations

Coping with geometric discontinuities in porosity-based shallow water models

The use of classic two-dimensional (2D) shallow water equations (SWE) for flooding simulation in complex urban environments is computationally expensive, due to the need of refined meshes for the representation of obstacles and building. Aiming to reduce the computational burden, a class of sub-grid SWE models, where small-scale building features are preserved on relatively coarse meshes by means of macroscale porosity parameters, has been recently introduced in the literature. Among the other porosity-based models, the single porosity (SP) model is relevant because the corresponding one-dimensional (1D) Riemann problem is the building block for the construction of many porosity-based numerical schemes. Like the Riemann problem connected to mathematical models such as the SWE with variable bed elevation and the 1D Euler equations in contracting pipes, the SP Riemann problem may exhibit multiple solutions for certain initial conditions. This ambiguity can be solved by restoring the microscale information of the 2D SWE model that is lost at the SP macroscale. In the present paper, we disambiguate the solutions' multiplicity by systematically comparing the solution of the SP Riemann problem at local porosity discontinuities with the corresponding 2D SWE numerical solutions in contracting channels. An additional result of this comparison is that the SP Riemann problem should incorporate an adequate amount of head loss when strongly supercritical flows past sudden porosity reductions occur. An approximate Riemann solver, able to pick the physically congruent solution among the alternatives and equipped with the required head loss amount, shows promising results when implemented in a 1D single porosity finite volume scheme.

Numerical three-dimensional modeling of earthen dam piping failure

A physically-based numerical three-dimensional earthen dam piping failure model is developed for homogeneous and zoned soil dams. This model is an erosion model, coupled with force/moment equilibrium analyses. Orifice flow and two-dimensional (2D) shallow water equations (SWE) are solved to simulate dam break flows at different breaching stages. Erosion rates of different soils with different construction compaction efforts are calculated using corresponding erosion formulae. The dam's real shape, soil properties, and surrounding area are programmed. Large outer 2D-SWE grids are used to control upstream and downstream hydraulic conditions and control the boundary conditions of orifice flow, and inner 2D-SWE flow is used to scour soil and perform force/moment equilibrium analyses. This model is validated using the European Commission IMPACT (Investigation of Extreme Flood Processes and Uncertainty) Test #5 in Norway, Teton Dam failure in Idaho, USA, and Quail Creek Dike failure in Utah, USA. All calculated peak outflows are within 10% errors of observed values. Simulation results show that, for a V-shaped dam like Teton Dam, a piping breach location at the abutment tends to result in a smaller peak breach outflow than the piping breach location at the dam's center; and if Teton Dam had broken from its center for internal erosion, a peak outflow of 117 851 m3/s, which is 81% larger than the peak outflow of 65 120 m3/s released from its right abutment, would have been released from Teton Dam. A lower piping inlet elevation tends to cause a faster/earlier piping breach than a higher piping inlet elevation.

Experience in Numerical Hydrodynamic Simulation of Long River Reaches

The article formulates the main requirements to numerical algorithms for hydrodynamic 2D-modeling of long and very long river segments with lengths of up to several thousand kilometers. The main feature is the use of adaptive unstructured mesh along with algorithms that give correct values of water surface elevations on coarse mesh, taking into account abrupt changes of bed elevations. A hydrodynamic model of a segment of the Amur R. with a total length of 3 thous. km is presented, which is based on a numerical solution of two-dimensional shallow-water equations (Saint-Venant) by a new high-accuracy algorithm, taking into account road and protection structures in the floodplain. The stages of model construction and verification are described, and the results of calculations for an extraordinary flood in 2013 and a high flood in 2020 are given. Water levels (with estimates of their errors) and the rates of water discharge at gages are given along with flow fields and the inundation zones of floodplain areas.

NUMERICAL STUDY OF ICE AND CHANNEL PROCESSES ON THE BRIDGE SUPPORTS ACROSS THE KUIBYSHEV RESERVOIR (KAZAN)

Numerical hydrodynamic modeling of water movement at discharges of 1% and 10% supply is considered in order to study the effect of ice effects and bottom deformations on the bridge supports across the reservoir of the Kuibyshev hydroelectric complex, and to predict the general erosion of the channel. The calculations use two-dimensional shallow water and sediment transport equations implemented in the STREAM 2D CUDA software package.