Nonlinear Shallow Water Equations Research Articles

An operational discontinuous Galerkin shallow water model for coastal flood assessment

Hydrodynamic modeling for coastal flooding risk assessment is a highly relevant topic. Many operational tools available for this purpose use numerical techniques and implementation paradigms that reach their limits when confronted with modern requirements in terms of resolution and performances. In this work, we present a novel operational tool for coastal hazards predictions, currently employed by the BRGM agency (the French Geological Survey) to carry out its flooding hazard exposure studies and coastal risk prevention plans on International and French territories. The model, called UHAINA (wave in the Basque language), is based on an arbitrary high-order discontinuous Galerkin discretization of the nonlinear shallow water equations with SSP Runge–Kutta time stepping on unstructured triangular grids. It is built upon the finite element library AeroSol, which provides a modern C++ software architecture and high scalability, making it suitable for HPC applications. The paper provides a detailed development of the mathematical and numerical framework of the model, focusing on two key-ingredients : (i) a pragmatic P0 treatment of the solution in partially dry cells which garantees efficiently well-balancedness, positivity and mass conservation at any polynomial order; (ii) an artificial viscosity method based on the physical dissipation of the system of equations providing nonlinear stability for non-smooth solutions. A set of numerical validations on accademic benchmarks is performed to highlight the efficiency of these approaches. Finally, UHAINA is applied on a real operational case of study, demonstrating very satisfactory results.

Entropy stable discontinuous Galerkin methods for the shallow water equations with subcell positivity preservation

AbstractHigh order schemes are known to be unstable in the presence of shock discontinuities or under‐resolved solution features, and have traditionally required additional filtering, limiting, or artificial viscosity to avoid solution blow up. Entropy stable schemes address this instability by ensuring that physically relevant solutions satisfy a semi‐discrete entropy inequality independently of discretization parameters. However, additional measures must be taken to ensure that solutions satisfy physical constraints such as positivity. In this work, we present a high order entropy stable discontinuous Galerkin (ESDG) method for the nonlinear shallow water equations (SWE) on two‐dimensional (2D) triangular meshes which preserves the positivity of the water heights. The scheme combines a low order positivity preserving method with a high order entropy stable method using convex limiting. This method is entropy stable and well‐balanced for fitted meshes with continuous bathymetry profiles.

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.

Simulation of Sloped-Bed Tuned Liquid Dampers Using a Nonlinear Shallow Water Model

Simulation of Sloped-Bed Tuned Liquid Dampers Using a Nonlinear Shallow Water Model

High order well-balanced asymptotic preserving IMEX RKDG schemes for the two-dimensional nonlinear shallow water equations

In this paper, a high order well-balanced asymptotic preserving scheme is presented for the two-dimensional nonlinear shallow water equations over variable bottom topography in all Froude number regimes. To obtain the well-balanced property, the system is first reformulated as a new form by introducing an auxiliary parameter. The flux is then split into a linear stiff part to be treated implicitly and a nonlinear non-stiff part to be treated explicitly, and the source term is treated explicitly. An implicit-explicit Runge-Kutta discontinuous Galerkin scheme is designed for solving the equations. The proposed scheme can be proved to be well-balanced, asymptotic preserving and asymptotically accurate. Finally, several numerical tests are carried out to validate the performance of our proposed scheme.

Numerical Tools For Wave Overtopping At Rubble Mound Breakwaters With Submerged Berms

In the era of climate change, the adaptation of existing coastal structures as a response to potential increasing loads, has become a trending topic in coastal engineering. The combination of sea level rise, storm surge, tides and waves is going to dramatically increase the wave overtopping. In this context, this research is focused on the evaluation of wave overtopping for conventional rubble mound breakwaters, modified by the introduction of a submerged berm. Recent studies illustrated how the introduction of emerged berms on the seaward slope reduces the wave overtopping. The goal then becomes to turn the spotlight on the submerged berms, whose effectiveness has yet to be attested, although it has proven useful in reducing wave loads on the armour layer and seabed pressure under the structure. Therefore, the reliability of two different numerical models in detecting overtopping phenomenon have been assessed. IHFOAM and SWASH have been used (see Figure 1). The first one can solve both Reynolds Averaged Navier Stokes and the Volume Averaged Reynolds Averaged Navier–Stokes equations. The second one solves the non-linear shallow water equations with a non-hydrostatic pressure term, representing a simplified form of the Navier-Stokes equations, with associated limited computational cost. The numerical models have been validated based on experimental tests carried out in a wave flume at the University of L’Aquila.

Single porosity model: Exploring the spatial resolution limits in complex urban patterns

When modeling large-scale urban floods, the Porosity Nonlinear Shallow Water Equations (PNSWE) emerge as an interesting and consistent subgrid approach to reduce computational time while preserving the solution structure, allowing for computational efficiency at the cost of some loss in the accuracy of the results. Porosity accounts for changes in storage and exchanges due to obstacles in urban areas, and it introduces an extra source term proportional to the porosity gradient into the momentum equations. However, no systematic analyses on the effects of grid size have been performed in real domains, when this model is used to represent fine-scale topographic information at a coarser scale. In this study, we analyze how accuracy is affected by gradually increasing grid resolution in a generalized porosity approach computed at the cell-level. The Single Porosity model (SP) in Cartesian coordinates is employed to simulate a real-world urban flooding event, with resolution transitioning from fine- to macro-scale. At an intermediate scale, the meso-scale, where cell size approximates street width and computational time is significantly reduced, the model captures main preferential flow paths by means of the porosity gradient within the urban area. Good agreement with refined classical model solutions is observed at this scale for flood extension and hazard maps, providing valuable information for early-warning systems. Numerical results underscore the importance of porosity models in rapidly assessing flow properties during an event, enhancing real-time decision-making with reliable information.

Machine learning emulation of high resolution inundation maps

SUMMARY Estimating coastal tsunami impact for early-warning or long-term hazard analysis requires the calculation of inundation metrics such as flow-depth or momentum flux. Both applications require the simulation of large numbers of scenarios to capture both the aleatory variability and the epistemic tsunami uncertainty. A computationally demanding step in simulating inundation is solving the non-linear shallow water (NLSW) equations on meshes with sufficiently high resolution to represent the local elevation accurately enough to capture the physics governing the flow. This computational expense is particularly challenging in the context of Tsunami Early Warning where strict time constraints apply. A machine learning (ML) model that predicts inundation maps from offshore simulation results with acceptable accuracy, trained on an acceptably small training set of full simulations, could replace the computationally expensive NLSW part of the simulations for vast numbers of scenarios and predict inundation rapidly and with reduced computational demands. We consider the application of an encoder–decoder based neural network to predict high-resolution inundation maps based only on more cheaply calculated simulated time-series at a limited number of offshore locations. The network needs to be trained using input offshore time-series and the corresponding inundation maps from previously calculated full simulations. We develop and evaluate the ML model on a comprehensive set of inundation simulations for the coast of eastern Sicily for tens of thousands of subduction earthquake sources in the Mediterranean Sea. We find good performance for this case study even using relatively small training sets (order of hundreds) provided that appropriate choices are made in the specification of model parameters, the specification of the loss function and the selection of training events. The uncertainty in the prediction for any given location decreases with the number of training events that inundate that location, with a good range of flow depths needed for accurate predictions. This means that care is needed to ensure that rarer high-inundation scenarios are well-represented in the training sets. The importance of applying regularization techniques increases as the size of the training sets decreases. The computational gain of the proposed methodology depends on the number of complete simulations needed to train the neural network, ranging between 164 and 4196 scenarios in this study. The cost of training the network is small in comparison with the cost of the numerical simulations and, for an ensemble of around 28 000 scenarios, this represents a 6- to 170-fold reduction in computing costs.

2DH numerical model of nonlinear wave propagation based on a hybrid finite-volume and finite-difference scheme

A two-dimensional hybrid (2DH) numerical model suitable for nonlinear wave propagation from deep to shallow waters is developed based on a hybrid finite-volume and finite-difference scheme. To utilize this method conveniently, the selected governing equations are derived again in a conservative form. The fourth-order monotonic upstream-centered scheme for conservation laws–total variation diminishing scheme with a Riemann solver is employed to calculate the numerical flux. The finite-difference scheme is employed to discretize the other spatial derivatives, and the third-order strong stability-preserving (SSP) Runge–Kutta scheme is used for time-stepping. Wave breaking is simulated by locally switching the governing equations to nonlinear shallow-water equations when the Froude number exceeds a certain threshold. A comparison of the numerical results and analytical or experimental data showed that the numerical model performs well in simulating nearshore wave propagation.

On the data assimilation of initial distribution for 2-dimensionalshallow-water equation model

Abstract The utilization of data assimilation (DA) techniques is prevalent in marine meteorology for the purpose of estimating the complete state of the system. This is done to address the practical limitations associated with measurement data, which can only be specified at finite number of discrete points within a limited domain. We develop an efficient DA algorithm to reconstruct the initial state of the shallow-water equations (SWE) within a 2-dimensional rectangular domain using sparse spatial measurement data. Our algorithm takes into account both the complete Coriolis force and the ocean bottom topography in the SWE model, resulting in accurate recovery of the initial status. After establishing the uniqueness of the solution to the nonlinear SWE with appropriate boundary conditions, we proceed to establish the conservation laws for the suitably defined energy quantity for this traveling wave system. This generalization of the known conservation laws for the simplified SWE system, which ignores the Coriolis force and topography, allows us to reveal the influence of nonconstant sea floor topography on wave propagation. In order to restore the initial state through the minimization of a cost functional using DA techniques, we proceed by deriving the adjoint problem for our iteration process. Additionally, we establish a discrete scheme for the governing equations in the Arakawa C-grid framework, from which we rigorously derive the error associated with energy conservation in discrete form. The numerical implementations are also provided to validate our proposed scheme through the verification of energy conservation and the reconstruction effect of the initial state for various configurations.

Optimization of stepped revetment configuration on minimizing wave run-up and overtopping

A stepped revetment has proven to be able to attenuate wave run-up height and wave overtopping discharge more effectively than a dike or a smooth-slope revetment does. However, the investigation into the optimal design of a stepped revetment, tailored to different wave and coastal conditions, is notably lacking. This paper introduces a computational model developed from the non-linear shallow water equations and the Genetic Algorithm. The aim is to optimize the width of a stepped revetment's step and crest, considering wave run-up and overtopping as key parameters. Multiple benchmark tests were conducted to validate the model's accuracy in estimating wave run-up and overtopping for specific setups. The results demonstrate a strong correlation between the numerical findings and experimental data. The Genetic Algorithm was then employed to predict the optimal step width for minimizing wave run-up height and determining the optimal step and crest widths to reduce wave overtopping discharge to an acceptable level. The findings reveal that for a gently sloping stepped revetment, the optimal step widths, based on wave run-up, remain relatively consistent across different wave heights. In contrast, when optimizing for wave overtopping, the optimal step width tends to fluctuate in most cases, although it shows some consistency in certain scenarios.

Simulation of the Mediterranean tsunami generated by the Mw 6.0 event offshore Bejaia (Algeria) on 18 March 2021

SUMMARY On 18 March 2021 an earthquake of magnitude Mw = 6.0 occurred offshore the Algerian coasts and generated a tsunami with offshore amplitudes smaller than a few millimetres crossing the western Mediterranean Sea. The objective of this study is threefold: first, to determine whether seismic sources calculated in the context of tsunami early warning are relevant; secondly, to determine whether tsunami simulations are able to reproduce tide-gauge observations and thirdly, to define the sensitivity of simulations to the grid resolutions and tsunami parameters. In the Mediterranean Sea, a very small number of available coastal tide gauges recorded the tsunami. Among them, a few French tide gauge stations recorded water waves with amplitudes smaller than a few centimetres and with periods ranging from 5 to 20 min associated to harbour or bay resonances. Numerical simulations of the tsunami are performed by the operational code Taitoko for seven different source fault models. Three of them allow for a rapid source detection and characterization in the framework of tsunami warning at CENALT (Centre National d'Alerte aux Tsunamis, France). The integrated code Taitoko uses a system of multiple nested grids. Standard Boussinesq equations are solved in the Mediterranean grid, whereas non-linear shallow water equations are solved in coastal and harbour grids with 25 and 5 m resolutions, respectively. Whatever the fault model, the observed time-series of water heights are reproduced satisfactorily both in phase and amplitude by the model at Nice and Monaco but poorly at Port Mahon (Minorca) and Toulon.

Tsunami hazard evaluation of river embankment structures incorporating their vulnerability to seismic strong motion

Development of coastal areas in Japan for various land uses since the 1960s has contributed to industrial upgrades and improved the efficiency of transportation networks. However, there are concerns about the vulnerability of developments on alluvial plains and reclaimed lands to geological events, like ground subsidence due to liquefaction during large earthquakes. Realistic assessment of earthquake and tsunami hazards and evaluation of possible countermeasures require accurate estimation of the amount of subsidence that can be expected from liquefaction at coastal and riverside sites supporting various structures. In this study, to evaluate the amount a river embankment structure might be expected to settle as a result of strong motion from an assumed Nankai Trough great earthquake, we conducted a numerical simulation using the soil–water coupled finite deformation analysis code GEOASIA. We then investigated the effect of the estimated embankment subsidence on tsunami inundation, which was simulated by using nonlinear shallow-water equations and a grid spacing as fine as 3.3 m. The influence of urban structures on the inundated area was taken into account by using a structure-embedded elevation model (SEM). The results showed that subsidence of river embankments and the collapse of parapet walls on top of them would increase both the depth and area of inundation caused by a tsunami triggered by a Nankai Trough scenario earthquake. Our findings underscore the importance of evaluating not only earthquake resistance but also vulnerability of coastal and riverside structures to strong motion in tsunami hazard analyses. Furthermore, the importance of tsunami inundation analysis using a SEM for predicting the behavior of tsunami flotsam in urban areas was demonstrated.

Aceh's tsunami wave evolution and its interaction with hybrid protection structure

The 2004 Aceh tsunami tragedy was one of the most catastrophic occurrences, resulting in damage and severe casualties in multiple countries. This study proposes a hybrid coastal protection system made up of mangroves, a sea dike, a trench, or a combination of the three structures to prevent similar devastation in the future. This system is expected to reduce the tsunami wave height, thus lowering their potential damage. The tsunami wave propagation is reproduced using a nonlinear shallow water equation model. To construct a numerical scheme, a staggered grid finite volume method is implemented. This scheme is then validated using several benchmark tests. Once validated, the computational results are compared to experimental data collected at the Laboratory of Coastal Dynamics in Yogyakarta, Indonesia. Both the physical and numerical models use the downscaled Aceh tsunami waves and real bathymetry. Several scenarios of structures combination are presented with the aim to determine the most effective combination. A sensitivity analysis is also conducted to support the results.

The highly nonlinear shallow water equation: local well-posedness, wave breaking data and non-existence of sech2\\documentclass[12pt]{minimal} \\usepackage{amsmath} \\usepackage{wasysym} \\usepackage{amsfonts} \\usepackage{amssymb} \\usepackage{amsbsy} \\usepackage{mathrsfs} \\usepackage{upgreek} \\setlength{\\oddsidemargin}{-69pt} \\begin{document}$$^2$$\\end{document} solutions

In the context of the initial data and an amplitude parameter ε\\documentclass[12pt]{minimal} \\usepackage{amsmath} \\usepackage{wasysym} \\usepackage{amsfonts} \\usepackage{amssymb} \\usepackage{amsbsy} \\usepackage{mathrsfs} \\usepackage{upgreek} \\setlength{\\oddsidemargin}{-69pt} \\begin{document}$$\\varepsilon $$\\end{document}, we establish a local existence result for a highly nonlinear shallow water equation on the real line. This result holds in the space Hk\\documentclass[12pt]{minimal} \\usepackage{amsmath} \\usepackage{wasysym} \\usepackage{amsfonts} \\usepackage{amssymb} \\usepackage{amsbsy} \\usepackage{mathrsfs} \\usepackage{upgreek} \\setlength{\\oddsidemargin}{-69pt} \\begin{document}$$H^k$$\\end{document} as long as k>5/2\\documentclass[12pt]{minimal} \\usepackage{amsmath} \\usepackage{wasysym} \\usepackage{amsfonts} \\usepackage{amssymb} \\usepackage{amsbsy} \\usepackage{mathrsfs} \\usepackage{upgreek} \\setlength{\\oddsidemargin}{-69pt} \\begin{document}$$k>5/2$$\\end{document}. Additionally, we illustrate that the threshold time for the occurrence of wave breaking in the surging type is on the order of ε-1,\\documentclass[12pt]{minimal} \\usepackage{amsmath} \\usepackage{wasysym} \\usepackage{amsfonts} \\usepackage{amssymb} \\usepackage{amsbsy} \\usepackage{mathrsfs} \\usepackage{upgreek} \\setlength{\\oddsidemargin}{-69pt} \\begin{document}$$\\varepsilon ^{-1},$$\\end{document} while plunging breakers do not manifest. Lastly, in accordance with ODE theory, it is demonstrated that there are no exact solitary wave solutions in the form of sech and sech2\\documentclass[12pt]{minimal} \\usepackage{amsmath} \\usepackage{wasysym} \\usepackage{amsfonts} \\usepackage{amssymb} \\usepackage{amsbsy} \\usepackage{mathrsfs} \\usepackage{upgreek} \\setlength{\\oddsidemargin}{-69pt} \\begin{document}$$sech^2$$\\end{document}.

Two efficient numerical techniques for solutions of fractional shallow water equation

In this paper, the nonlinear shallow water equation appears in the mathematical modeling of tsunami wave propagation along the coastline of an ocean is solved numerically. The model investigates fractional derivatives and is presented through a system of nonlinear partial differential equations. The variational iteration method (VIM) and the residual power series method (RPSM) have been applied to the fractional system of equations for different parameters. In addition, tsunami wave velocity and run-up height with respect to coast slope and sea depth are analyzed for different time and order in detail. A detailed error analysis is presented for tsunami velocity and tsunami wave propagation. The results obtained are compared to other existing results to show the efficiency and reliability of the proposed methods.

Stratified inclined duct: two-layer hydraulics and instabilities

The stratified inclined duct (SID) sustains an exchange flow in a long, gently sloping duct as a model for continuously forced density-stratified flows such as those found in estuaries. Experiments have shown that the emergence of interfacial waves and their transition to turbulence as the tilt angle is increased appears to be linked to a threshold in the exchange flow rate given by inviscid two-layer hydraulics. We uncover these hydraulic mechanisms by (i) using recent direct numerical simulations (DNS) providing full flow data in the key flow regimes (Zhu et al., J. Fluid Mech., vol. 969, 2023, A20), (ii) averaging these DNS into two layers, and (iii) using an inviscid two-layer shallow-water and instability theory to diagnose interfacial wave behaviour and provide physical insight. The laminar flow is subcritical and stable throughout the duct and hydraulically controlled at the ends of the duct. As the tilt is increased, the flow becomes supercritical everywhere and unstable to long waves. An internal jump featuring stationary waves first appears near the centre of the duct, then leads to larger-amplitude travelling waves, and to stronger jumps, wave breaking and intermittent turbulence at the largest tilt angle. Long waves described by the (nonlinear) shallow-water equation are interpreted locally as linear waves on a two-layer parallel base flow described by the Taylor–Goldstein equation. This link helps us to interpret long-wave instability and contrast it with short-wave (e.g. Kelvin–Helmholtz) instability. Our results suggest a transition to turbulence in SID through long-wave instability relying on vertical confinement by the top and bottom walls.

A numerical methodology for estimating site-specific cascading earthquake and tsunami dynamic loading on critical infrastructure

A numerical methodology for deriving cascading earthquake and tsunami loading patterns for use in critical infrastructure design is proposed. Earthquake source parameters are first used to generate synthetic ground motion acceleration time series at the site of a critical infrastructure using an open-source stochastic extended finite-source simulation algorithm. Using the same earthquake source information, the tsunami phases of generation, propagation, and inundation are modeled using a coupling approach for determining tsunami wave forces. The coupling approach combines (1) a code based on non-linear shallow water equations solved by the Finite Volume method, which is used to capture the tsunami flow characteristics from generation through the inundation phase until relatively close to the structure, coupled with (2) an open-source computational fluid simulation code, which relies on solving the Navier–Stokes equations using the Smoothed Particle Hydrodynamics method. An appendix presents the validation of the numerical methodology through correlation of the numerical simulations with field data recorded during the 2011 Japan Earthquake and Tsunami. The methodology is then employed to characterize the ground shaking and tsunami loading patterns at the site of the container terminal of the Sines deep-water seaport in Portugal, when subjected to the historic 1755 Great Lisbon Earthquake and Tsunami plausible scenarios. For the site, the generated ground motion acceleration time series show peak accelerations exceeding Eurocode 8 design values. Tsunami flow depths and velocities were computed in horizontal and vertical directions and could be used in the assessment of the existing infrastructure or the design of the future expansion of the port.

Modelling study of the spread of liquid fuel spill fires using shallow water equations

In oil spill–fire accidents, the spread of burning fuel from petroleum product containers may increase accident losses and trigger further accidents. Studying the spreading process of spill fires is important to prevent accidents and mitigate losses. Herein, a spill-fire spread model was proposed to predict the spread rate and time varying burning area of liquid fuel spill fires. The model was developed using one-dimensional nonlinear shallow water equations. Fuel consumption was used as the source term in the equations. To ensure numerical stability and accuracy, the third-order TVD Runge–Kutta method and the second-order MUSCL scheme were used for time advancement and numerical space reconstruction, respectively. Numerical flux was calculated by using the HLL approximate Riemann solver. The algorithm of the shallow water equations was modified in accordance with the spread characteristics of spill fires. Using the developed spread model, the dynamic burning area and dynamic fuel thickness of a spill fire were calculated. The simulation results were verified using the experimental results on continuously released n-heptane spill fires, and the two results were in good agreement.

An integrated numerical model for landslide-generated waves: a case study of the Gongjiafang landslide, Three Gorges Reservoir, China

ABSTRACT Landslide is a frequent geological hazard in the Three Gorges reservoir area of China. Impulse waves could be created and cause substantial damage if the landslide collides with a water body. A nonlinear shallow water equation in combination with a progressive landslide model is proposed to simulate this catastrophic disaster. The proposed approach is distinguished by rapid calculation, less computational burden, and high practical value. By simulating the 2008 Three Gorges Reservoir Gongjiafang landslide and the subsequent impulse waves, we illustrated and validated this methodology. The initiation and motion processes of the landslide, as well as the generation, propagation, and run-up processes of impulse waves, were thoroughly performed. The simulation results were in good agreement with the field survey data, demonstrating the viability of the proposed approach for simulating the impulse waves triggered by landslide events. This numerical approach is beneficial for engineers to perform a preliminary estimation of landslide-generated waves and further facilitate the prevention and mitigation of geological hazards.