Shallow Water Equations Research Articles

Data-driven and knowledge-guided denoising diffusion probabilistic model for runoff uncertainty prediction

The existing medium and long-term runoff prediction methods, which are based on data-driven and knowledge-guided methods, are associated with inherent limitations, and chaotic phenomena in runoff prediction models often leads to oscillation in the prediction error, affecting the robustness of the prediction. A knowledge-guided denoising diffusion probabilistic model (DK-RDDPM) that introduces physical theory to guide constraint quantification and obtain effective runoff uncertainty prediction results is therefore proposed in this study. The main advantage of this model is that the physical randomness in the runoff prediction process can be captured and combined with the Saint-Venant process to guide model optimization and realize more accurate medium- and long-term runoff prediction. The main contributions of this study are the establishment of a dynamic runoff probabilistic prediction model with stochastic quantification characteristics that includes the prediction uncertainty over time, and modelling of the physical constraint boundary of runoff prediction from the perspective of partial differentiation. The effectiveness of the DK-RDDPM was verified by predicting runoff in the Qijiang and Tunxi Basins in China. The results show that: 1) Encoding the physical random uncertainty operator in runoff prediction into the network of the denoising diffusion probabilistic model (DDPM) effectively captures the physically complex implicit randomness of the process, thus reducing the error that results from randomness in runoff prediction. 2) The constraint matrix that is formed using the Saint-Venant equation and the prediction matrix are layered and projected, with the fluctuation range of the constraints in each step adjusted in the optimization direction within a certain random threshold range. 3) The DK-RDDPM shows superior performance to the benchmark models, even under the influence of different noise interference factors.

A multilayer shallow water model for polydisperse reactive sedimentation

A three-dimensional model of polydisperse reactive sedimentation is developed by means of a multilayer shallow water approach. The model consists of a variety of solid particles of different sizes and densities, and substrates diluted in water, which produce biochemical reactions while the sedimentation process occurs. Based on the Masliyah–Lockett–Bassoon settling velocity, compressibility of the sediment and viscosity of the mixture, the system of governing equations is composed by non-homogeneous transport equations, coupled to a momentum equation describing the mass-average velocity. Besides, the free-surface depicted by the total height of the fluid column is incorporated and fully determined through the multilayer approach. A finite volume numerical scheme on Cartesian grids is proposed to approximate the model equations. Numerical simulations of the denitrification process exemplify the performance of the numerical scheme and model under different scenarios and bottom topographies.

Simulation of large plausible tsunami scenarios associated with the 2019 Durres (Albania) earthquake source and adjacent seismogenic zones

We present an analysis of the hazards of potential earthquake-generated tsunamis along the Albanian–Adriatic coast. The study adopts a case study approach to model plausible tsunamigenic events associated with the 2019 Mw 6.4 Durres (Albania) earthquake source zone. The approach combines current findings on regional tectonics and scenario-based calculations of potential tsunami impacts. The study’s goal is to analyse the propagation of tsunami waves generated by identified seismogenic sources (namely ALCS002 [Lushnje] and ALCS018 [Shijak]) and determine the tsunami risk assessment for Durres City on the Albanian–Adriatic coast. The sources can generate earthquakes with maximum moment magnitudes of Mw 7.5 and Mw 6.8, which are likely to trigger tsunamis that could cause significant impacts in the region. The modelling is performed deterministically with the NAMI DANCE numerical code, including scenarios associated with the largest plausible earthquake. The model integrates bathymetry and topography datasets of large and medium resolutions. Each tsunami scenario simulation is based on the solution of the non-linear shallow water equations used to generate maximum positive wave amplitudes (water elevation), travel time, and tsunami inundation maps. In Durres City, modelling indicates that medium-sized waves could reach up to 2.5 m inland, posing a significant danger to the city’s low-lying areas. The most substantial tsunami waves are expected to impact the area within the first 10 to 20 min. Combining inundation maps and information on exposed assets allows for identifying areas where damages can be expected. In terms of human impact, a preliminary analysis shows that the study area is prone to tsunami threat, with more than 138,000 inhabitants living in vulnerable urban areas of Durres City by 2036. The model’s capacity to capture details related to the presence of buildings is limited due to constraints posed by the resolution of bathymetry and topography datasets available during this study. If refined with high-resolution bathymetry and topography datasets, our results can be considered a backbone for exposure and resilience assessment features to be integrated into preparedness or new urban development plans.

A Localized Particle Filtering Approach to Advance Flood Frequency Estimation at Large Scale Using Satellite Synthetic Aperture Radar Image Collection and Hydrodynamic Modelling

This study describes a method that combines synthetic aperture radar (SAR) data with shallow-water modeling to estimate flood hazards at a local level. The method uses particle filtering to integrate flood probability maps derived from SAR imagery with simulated flood maps for various flood return periods within specific river sub-catchments. We tested this method in a section of the Severn River basin in the UK. Our research involves 11 SAR flood observations from ENVISAT ASAR images, an ensemble of 15 particles representing various pre-computed flood scenarios, and 4 masks of spatial units corresponding to different river segmentations. Empirical results yield maps of maximum flood extent with associated return periods, reflecting the local characteristics of the river. The results are validated through a quantitative comparison approach, demonstrating that our method improves the accuracy of flood extent and scenario estimation. This provides spatially distributed return periods in sub-catchments, making flood hazard monitoring effective at a local scale.

Numerical simulation study on the effect of underground drainage pipe network in typical urban flood

Rapid urbanization and climate change have heightened urban flood risks. An in-depth examination of the underground drainage pipe network’s effect on urban flood inundation can enhance urban stormwater management and mitigate disaster risks. This study presents a full hydrodynamic urban flood model (FHUM) coupling the pipe transport module of the Storm Water Management Model (SWMM) with a shallow water model. Based on unstructured grid cells, runoff yield and confluence calculations are carried out to effectively separate the underground drainage pipe network from other factors. The model is applied to Minzhi Area and Haidian Island to quantitatively assess the influence of the underground drainage system on urban flooding. Results indicate that FHUM performs well, accurately reflecting the process from rainfall to inundation. The underground drainage pipe network reduces surface inundation in the study area and improves the city’s stormwater drainage capacity. With increasing rainstorm intensity, the impact of the pipe network does not exhibit a consistent trend. During heavy rainstorms, the pipe network consistently plays a significant and stable role, particularly in reducing high inundation depths. However, attention should be paid to the potential flood risk transfer associated with using pipe networks in urban flood management. This work provides a scientific basis for urban stormwater management, holding significant importance in improving flood control and disaster reduction capabilities.

Flooding study of the Loira River (Galicia, Spain): Importance of pre-evaluation in land management

Floods pose recurrent threats to numerous countries, carrying substantial social, economic and at times, catastrophic implications, especially in extreme scenarios. In addition, coastal areas are strongly threatened by extreme weather events and climate change impacts. Given these pressing challenges, it becomes imperative to conduct thorough assessments of future flood impacts, particularly in areas of significant environmental and socio-economic importance. This study addresses this need by focusing on the potential consequences of flooding in the Loira River, situated in the municipality of Marín, Pontevedra, Northwestern Spain. For this purpose, the free software IBER has been used, these is a free software that solves the 2D depth-averaged shallow water equations (SWEs) by using a finite volume scheme, with the domain being discretized with both structured and unstructured triangular or quadrilateral elements. The hydrological model has been established from the maximum flow for different return periods (T10, T100 and T500). Field observations, including channel widths, bridge configurations, and various parameters, further inform our analysis. Our findings reveal critical insights, including a maximum height differential of 0.48 m, particularly significant at the upstream bridge, exacerbating flooding risks in adjacent plains. Moreover, velocities along riverbanks reach hazardous levels, notably during the 500-year return period, necessitating urgent protective measures for local residents. The hydrological model also indicates that small-scale forestation has no significant effect on flood prevention. This research underscores the utility of advanced hydrological modelling in informing risk management strategies and underscores the necessity of pre-emptive hydrological analyses to mitigate future flood risks effectively.

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.

A super-convergent quasi-discontinuous Galerkin method for approximating inertia-gravity and Rossby waves in geophysical flows

The numerical approximation of the shallow-water equations, which support geophysical wave propagation, namely the fast external Poincaré and Kelvin waves and the slow large-scale planetary Rossby waves, is a delicate and difficult problem. Indeed, the coupling between the momentum and the continuity equations may lead to the presence of erratic or spurious solutions for these waves, e.g. the spurious pressure and inertial modes. In addition, the presence of spurious branches and the emergence of spectral gaps in the dispersion relation at specific wavenumbers may lead to anomalous dissipation/dispersion in the representation of both fast and slow waves. The aim of the present study is to propose a class of possible discretization schemes, via the discontinuous Galerkin method, that is not affected by the above-mentioned problems. A Fourier/stability analysis of the 2D shallow-water model is performed by analysing the finite volume method and the linear discontinuous P1DG and non conforming P1NC Galerkin discretizations. These are stabilized by means of a family of numerical fluxes via the Polynomial Viscosity Matrix approach. A long time stability result is proven for all schemes and fluxes examined here. Further, a super-convergent result is demonstrated for the P1NC discrete frequencies compared to the P1DG ones, except for the slow mode in the Roe flux case. Indeed, we show that the Roe flux yields spurious frequencies and sub-optimal rates of convergence for the slow mode, for both the P1DG and P1NC methods. Finally, numerical solutions of linear and non-linear test problems confirm the theoretical results and reveal the computational cost efficiency of the P1NC approach.

Implementation of a GPU-enhanced multiclass soil erosion model based on the 2D shallow water equations in the software Iber

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.

A study of self-adjointness, Lie analysis, wave structures, and conservation laws of the completely generalized shallow water equation

This article explores the analysis of the completely generalized Hirota–Satsuma–Ito equation through Lie symmetry analysis. The equation under consideration represents a more comprehensive form of the (2+1)-dimensional HSI equation, encompassing four additional second-order derivative terms: Δ3Hϖϖ\\documentclass[12pt]{minimal} \\usepackage{amsmath} \\usepackage{wasysym} \\usepackage{amsfonts} \\usepackage{amssymb} \\usepackage{amsbsy} \\usepackage{mathrsfs} \\usepackage{upgreek} \\setlength{\\oddsidemargin}{-69pt} \\begin{document}$$\\Delta _{3}H_{\\varpi \\varpi }$$\\end{document}, Δ4Hϖι\\documentclass[12pt]{minimal} \\usepackage{amsmath} \\usepackage{wasysym} \\usepackage{amsfonts} \\usepackage{amssymb} \\usepackage{amsbsy} \\usepackage{mathrsfs} \\usepackage{upgreek} \\setlength{\\oddsidemargin}{-69pt} \\begin{document}$$\\Delta _{4}H_{\\varpi \\iota }$$\\end{document}, Δ3Hϖϖ\\documentclass[12pt]{minimal} \\usepackage{amsmath} \\usepackage{wasysym} \\usepackage{amsfonts} \\usepackage{amssymb} \\usepackage{amsbsy} \\usepackage{mathrsfs} \\usepackage{upgreek} \\setlength{\\oddsidemargin}{-69pt} \\begin{document}$$\\Delta _{3}H_{\\varpi \\varpi }$$\\end{document}, Δ4Hϖι\\documentclass[12pt]{minimal} \\usepackage{amsmath} \\usepackage{wasysym} \\usepackage{amsfonts} \\usepackage{amssymb} \\usepackage{amsbsy} \\usepackage{mathrsfs} \\usepackage{upgreek} \\setlength{\\oddsidemargin}{-69pt} \\begin{document}$$\\Delta _{4}H_{\\varpi \\iota }$$\\end{document}, and Δ6Hιι\\documentclass[12pt]{minimal} \\usepackage{amsmath} \\usepackage{wasysym} \\usepackage{amsfonts} \\usepackage{amssymb} \\usepackage{amsbsy} \\usepackage{mathrsfs} \\usepackage{upgreek} \\setlength{\\oddsidemargin}{-69pt} \\begin{document}$$\\Delta _{6}H_{\\iota \\iota }$$\\end{document}, emerging from the inclusion of second-order dissipative-type elements. We calculate the infinitesimal generators and determine the symmetry group for each generator using the Lie group invariance condition. Employing the conjugacy classes of the Abelian algebra, we transform the considered equation into an ordinary differential equation through similarity reduction. Subsequently, we solve these ordinary differential equations to derive closed-form solutions for the completely generalized Hirota–Satsuma–Ito equation under certain conditions. For other scenarios, we utilize the extended direct algebraic method to obtain soliton solutions. Furthermore, we rigorously calculated the conserved quantities corresponding to each symmetry generator, the conservation laws of the model are established using the multiplier approach. Additionally, we present the graphical representation of selected solutions for specific values of the physical parameters of the equation under scrutiny.

Development of a Semi-Analytical Dynamic Force Model

A moving water mass generates force which is exerted on its moving path. Cyclone generated storm surge or earthquake generated tsunami are specific examples of moving water mass generated force along the coasts. In addition to human lives, these moving water masses cause severe damages to the coastal infrastructure due to tremendous force exerted on these structures. To assess the damage on these infrastructures, an essential parameter is the resultant force exerted on these structures. To evaluate the damages, there is hardly any quantitative method available to compute this force. In this paper we have developed a semi-analytical model, named as Dynamic Force Model (DFM), by using Variational Iteration Method to compute this force. We have derived the governing equation on the basis of Saint-Venant equations which are basically 1D shallow water equations derived from the Navier-Stokes equations. DFM is verified, calibrated, validated, and applied in Bangladesh coastal zone to compute dynamic thrust force due to tropical cyclone SIDR.

A Positivity-Preserving and Well-Balanced High Order Compact Finite Difference Scheme for Shallow Water Equations

A Positivity-Preserving and Well-Balanced High Order Compact Finite Difference Scheme for Shallow Water Equations

Potential Sunda Strait tsunami hazard due to the current deformation of Anak Krakatau

Abstract The tsunami Sunda Strait event at 2018 is a prove that volcanic activity may trigger a devastating and unpredictable tsunami. Several findings indicate that this catasrophe generated by flank collapse caused by the instability of Gunung Anak Krakatau (AK). Other processes of volcanic activity that cause tsunamis include pyroclastic flows, subaerial and submarine landslides, underwater explosions, blasts, and caldera collapse. In the 1883, Krakatau Mount produced the largest tsunami event, and it is apparent that AK is one of threats in Sunda Strait that may potentially produce tsunami in the future since it is still active until now. Some latest research shows there are two submarine landslide threats near Gunung Anak Krakatau; in the northeast part of AK and an elliptical landslide source in the west part of AK with the estimated landslide volume about 0.014 km3 and 0.6 km3, respectively. Those threats will be simulated by shallow water equation model to obtain the tsunami wave height near the source and at crucial areas along Sunda Strait as well as its tsunami time arrival. The first scenario is sourced from a new deposit from the 2018 eruption, and the latter is a change of slopes bathymetry due to volcanic or seismic activity. It is observed that the highest elevation of tsunami from the first scenario reached only 2.5 cm at Tanjung Lesung, Banten and 80 cm at the nearest island, Panjang Island. The second scenario has the height of 50 cm at Labuan, Banten and 70 cm at Panjang Island. The tsunami time travel at the surrounding islands is ranging between 0.6 - 5.8 minutes (scenario 1) and 0.7-4.9 minutes (scenario 2). Both scenario reached the inexpensive device of sea level monitoring at the east side of Rakata after 5 minutes and the height is 30 cm and 1.6 cm respectively for scenario 1 and 2.

Zeitlin Truncation of a Shallow Water Quasi‐Geostrophic Model for Planetary Flow

AbstractIn this work, we consider a Shallow‐Water Quasi Geostrophic equation on the sphere, as a model for global large‐scale atmospheric dynamics. This equation, previously studied by Verkley (2009, https://doi.org/10.1175/2008jas2837.1) and Schubert et al. (2009, https://doi.org/10.3894/james.2009.1.2), possesses a rich geometric structure, called Lie‐Poisson, and admits an infinite number of conserved quantities, called Casimirs. In this paper, we develop a Casimir preserving numerical method for long‐time simulations of this equation. The method develops in two steps: first, we construct an N‐dimensional Lie‐Poisson system that converges to the continuous one in the limit N → ∞; second, we integrate in time the finite‐dimensional system using an isospectral time integrator, developed by Modin and Viviani (2020, https://doi.org/10.1017/jfm.2019.944). We demonstrate the efficacy of this computational method by simulating a flow on the entire sphere for different values of the Lamb parameter. We particularly focus on rotation‐induced effects, such as the formation of jets. In agreement with shallow water models of the atmosphere, we observe the formation of robust latitudinal jets and a decrease in the zonal wind amplitude with latitude. Furthermore, spectra of the kinetic energy are computed as a point of reference for future studies.

Entropy-Conservative Discontinuous Galerkin Methods for the Shallow Water Equations with Uncertainty

In this paper, we develop an entropy-conservative discontinuous Galerkin (DG) method for the shallow water (SW) equation with random inputs. One of the most popular methods for uncertainty quantification is the generalized Polynomial Chaos (gPC) approach which we consider in the following manuscript. We apply the stochastic Galerkin (SG) method to the stochastic SW equations. Using the SG approach in the stochastic hyperbolic SW system yields a purely deterministic system that is not necessarily hyperbolic anymore. The lack of the hyperbolicity leads to ill-posedness and stability issues in numerical simulations. By transforming the system using Roe variables, the hyperbolicity can be ensured and an entropy-entropy flux pair is known from a recent investigation by Gerster and Herty (Commun. Comput. Phys. 27(3): 639–671, 2020). We use this pair and determine a corresponding entropy flux potential. Then, we construct entropy conservative numerical two-point fluxes for this augmented system. By applying these new numerical fluxes in a nodal DG spectral element method (DGSEM) with flux differencing ansatz, we obtain a provable entropy conservative (dissipative) scheme. In numerical experiments, we validate our theoretical findings.

A one-dimensional augmented Shallow Water Equations system for channels of arbitrary cross-section

This work provides a new formulation of the one-dimensional augmented Shallow Water Equations system for open channels and rivers with arbitrarily shaped cross sections, suitable for numerical integration when discontinuous geometry is encountered. The additional variable considered can be the bottom elevation, a reference width, a shape coefficient, or a vector containing these or other geometric parameters. The appropriate numerical method, which is well suited to coupling with the mathematical one, is a path-conservative method, capable of reconstructing the behaviour of physical and geometrical variables at the cell boundaries, where the discrete solution of hyperbolic systems of equations is discontinuous. A nonlinear path suitable for the shallow water context is adopted. The resulting model is shown to be well-balanced and accurate to the second order and is further validated against analytical solutions related to channels with power-law cross-sections, specifically for dam break patterns over a variable-width channel and the run-up dynamics of long water waves over sloping bays.

Estimation of soil infiltration coefficient in the furrow irrigation using the combination of the hydrodynamics model and Richard's equation

Present study optimizes the coefficients of the Kostiakov-Lewis infiltration Eq. and estimates water infiltration in the soil with the help of simulation of the advancing phase using the numerical solution of the Saint-Venant equations. In order to validate the numerical model, two series of field data measured in two regions of Iran were used and the results of the simulation were compared with actual field results. Also, using the Richards' Eq., the HYDRUS-1D software, the Hydrus-2D software, and a combination of the Hydrodynamics model with each of the mentioned methods the infiltration simulation and prediction of coefficients of the Kostiakov-Lewis infiltration Eq. were provided. The criteria for the accuracy of the methods were considered the comparison between the infiltration volumes of the water (calculated from each method) and the actual infiltration volume of the water in Furrow irrigation. The results illustrated that the combination of the Hydrodynamics model with each of the HYDRUS-1D, HYDRUS-2D software and the numerical solution of the Richards' Eq. increased the accuracy of the estimation process of the coefficients of the Kostiakov-Lewis's infiltration equation and the infiltration volume. So, the best and worst methods for simulating water infiltration through the soil has been introduced based on the less error between numerical and observed values of the infiltration volume as the combined method of the Hydrodynamics- HYDRUS 2D with NRMSE = 17.62% and numerical solution of the Richard's Eq. with NRMSE = 36.02%, respectively. The results illustrated that the error of the Hydrodynamics- HYDRUS-2D compared with the numerical solution of the Richard's Eq. (t(10) = −3.018) was significantly less value.

Bathymetry reconstruction from experimental data using PDE-constrained optimisation

Knowledge of the bottom topography, also called bathymetry, of rivers, seas or the ocean is important for many areas of maritime science and civil engineering. While direct measurements are possible, they are time consuming, expensive and inaccurate. Therefore, many approaches have been proposed how to infer the bathymetry from measurements of surface waves. Mathematically, this is an inverse problem where an unknown system state needs to be reconstructed from observations with a suitable model for the flow as constraint. In many cases, the shallow water equations can be used to describe the flow. While theoretical studies of the efficacy of such a PDE-constrained optimisation approach for bathymetry reconstruction exist, there seem to be few publications that study its application to data obtained from real-world measurements. This paper shows that the approach can, at least qualitatively, reconstruct a Gaussian-shaped bathymetry in a wave flume from measurements of the free surface level at up to three points. Achieved normalised root mean square errors (NRMSE) are in line with other approaches.

Parametric analysis of the performance of the SPH solutions of unsteady free-surface flow

ABSTRACT In this article, the performances of the SPH method to solve Shallow Water Equations SWEs with three investigation parameters were studied, such as the type of kernel functions, namely: cubic spline, Gaussian and quintic spline kernels, the number of particles used and the stabilization terms injected, specifically: Lax Friedrichs flux, artificial viscosity and two shocks Riemann solver. Three benchmarking tests make the subject of unsteady free surface flow in this study. It is 1D typical dam-break on wet and dry bottom; 2D partial dam-break on a dry floodplain and 2D partial dam-break on channel with 90° bend. The analysis of the different errors between the numerical and analytical solutions and/or the experimental data shows that the SPH method gives reliable values with the selected optimal parameters which are the cubic kernel function and the artificial viscosity term. The increase in the number of particles increases the precision but also the calculation time.

Numerical investigation for water flow in an irregular channel using Saint-Venant equations

Given recent flood events in Indonesia, understanding the hydraulic dynamics in open channels, mirroring real-world drainage and river scenarios, has become paramount. To address this, our research employs a numerical model to simulate water flow in both regular and irregular open channels. Specifically, we utilize the Saint-Venant Equations to explore fluid flow evolution in irregular channels. This model is solved numerically using a staggered finite volume method. We complement our research with laboratory experiments and validate our numerical model against the data collected. Furthermore, we cross-reference our numerical results with those from HEC-RAS to examine the robustness and accuracy of our model in predicting water levels and velocities under varying conditions. To deepen our understanding, we conduct sensitivity analyses that offer valuable insights into the core factors influencing water levels and velocity. This knowledge provides a foundation for practitioners to normalize rivers.