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In this work, we study the generalized shallow water wave equation to obtain novel solitary wave solutions. The application of this nonlinear model can be found in tidal waves, weather simulations, tsunami prediction, river and irrigation flows. To obtain the new exact solutions of the considered model, we have applied a novel analytical technique namely ( G ′ G ′ + G + A) -expansion method. Using the aforementioned method and computational software, we have obtained different kinds of periodic and singular solitary wave solutions of the generalized shallow water wave equation. The obtained solutions are exponential function and trigonometric function solutions. Using 2D and 3D plots of the wave solutions, the dynamic behaviors of the developed solutions are displayed. The retrieved solutions validated the effectiveness and robustness of the proposed technique. [ABSTRACT FROM AUTHOR]

Physics-informed neural networks (PINNs) have recently emerged as a promising alternative to traditional numerical methods for solving partial differential equations (PDEs) in fluid dynamics. By using PDE-derived loss functions and auto-differentiation, PINNs can recover solutions without requiring costly simulation data, spatial gridding, or time discretization. However, PINNs often exhibit slow or incomplete convergence, depending on the architecture, optimization algorithms, and complexity of the PDEs. To address these difficulties, a variety of novel and repurposed techniques have been introduced to improve convergence. Despite these efforts, their effectiveness is difficult to assess due to the wide range of problems and network architectures. As a novel test case for PINNs, we propose one-dimensional shallow water equations with closed boundaries, where the solutions exhibit repeated boundary wave reflections. After carefully constructing a reference solution, we evaluate the performance of PINNs across different architectures, optimizers, and special training techniques. Despite the simplicity of the problem for classical methods, PINNs only achieve accurate results after prohibitively long training times. While some techniques provide modest improvements in stability and accuracy, this problem remains an open challenge for PINNs, suggesting that it could serve as a valuable testbed for future research on PINN training techniques and optimization strategies. [ABSTRACT FROM AUTHOR]

We consider the stochastic control problem of the shallow lake and continue the work of Kossioris, Loulakis, and Souganidis. First, we generalise the characterisation of the value function as the viscosity solution of a well-posed problem to include more general recycling rates. Then, we prove approximate optimality under bounded controls and we establish quantitative estimates. Finally, we implement a convergent and stable numerical scheme for the computation of the value function to investigate properties of the optimally controlled stochastic shallow lake. This approach permits to derive tails asymptotics for the invariant distribution and to extend results of Grass, Kiseleva and Wagener. ] beyond the small noise limit. [ABSTRACT FROM AUTHOR]

In this work, we develop a new framework to deal numerically with discontinuous solutions in nonconservative hyperbolic systems. First an extension of the MOOD methodology to nonconservative systems based on Taylor expansions is presented. This extension combined with an in‐cell discontinuous reconstruction operator are the key points to develop a new family of high‐order methods that are able to capture exactly isolated shocks. Several test cases are proposed to validate these methods for the Modified Shallow Water equations and the Two‐Layer Shallow Water system. [ABSTRACT FROM AUTHOR]

To study the water management of natural rivers, two-dimensional (2D) hydrodynamic models have become essential tools. The performance of five different hydrodynamic modelling packages, SRH 2D (developed by U.S. Bureau of Reclamation), Delft 3D (developed by Deltares), FESWMS (developed by Federal Highway Administration), MIKE 21 FM (developed by DHI), and Hec Ras 2D (developed by US Army Corps of Engineers) was evaluated in this study. Numerical solutions to shallow water equations were obtained using finite elements, finite differences, and finite volumes. This work aims to provide the most accurate and objective information that can help select two-dimensional models for different annual scenarios. This study focused on comparing annual flow cases in-channel rather than flood flow. A study area on the Nile River was selected for this study. A performance assessment has been conducted based on the calibration of the model, the representation of the channel flow, and the computing efficiency of the model. The results show that Delft 3D and SRH 2D models were the most compatible with the measured data. Delft 3D model outperforms SRH 2D in computational efficiency up to 66% times faster. [ABSTRACT FROM AUTHOR]

Keywords: Regularity criterion; bootstrap argument; Green's function; global existence; decay rate of solution.GraphBy Jie Qi and Weike WangReported by Author; Author [Extracted from the article]

A large Courant–Friedrichs–Lewy (CFL) algorithm is presented for the explicit, finite volume solution of hyperbolic systems of conservation laws, with a focus on the shallow water equations. The Riemann problems used in the flux computation are determined using averaging kernels that extend over several computational cells. The usual CFL stability constraint is replaced with a constraint involving the kernel support size. This makes the method unconditionally stable with respect to the size of the computational cells, allowing the computational mesh to be refined locally to an arbitrary degree without altering solution stability. The practical implementation of the method is detailed for the shallow water equations with topographical source term. Computational examples report applications of the method to the linear advection, Burgers and shallow water equations. In the case of sharp bottom discontinuities, the need for improved, well‐balanced discretisations of the geometric source term is acknowledged. [ABSTRACT FROM AUTHOR]

To develop a numerical method for global geophysical fluids, we usually need to choose a spherical grid and numerical approximations to represent the partial derivative equations. Some alternatives include the use of finite differences or finite volumes with latitude–longitude or reduced grids. Each of these cases has some advantages and also some limitations. This paper presents a comparison between two methods and describes a composite model using them side by side. The first is a well-known method for latitude–longitude grids and was used from 75 ∘ S until 75 ∘ N. The second is a recently developed scheme for reduced grids and was used only in the polar regions. The similarity between the two methods allows the use of small adaptations in their approximations to obtain consistency and mass conservation also in the transition between the two regions. The composite model combines advantages of the other two schemes and has a smaller computational cost. Numerical tests indicated order 2 of convergence, prevention of grid-imprinting errors, and avoidance of nonlinear instability. This model has numerical properties that may lead to efficient implementations with massive parallel computation. [ABSTRACT FROM AUTHOR]

The second-order discontinuous Galerkin (DG2) solver of the shallow water equations in LISFLOOD-FP 8.0 is well-suited for predicting small-scale transients that emerge in rapid, multiscale floods caused by impact events like tsunamis. However, this DG2 solver can only be used for simulations on a uniform grid where it may yield inefficient runtimes even when using its graphics processing unit (GPU) parallelised version (GPU-DG2). To maximise runtime reduction, the LISFLOOD-FP 8.2 version integrates GPU parallelised dynamic (in time) grid resolution adaptivity of multiwavelets (MW) with the DG2 solver (GPU-MWDG2). The GPU-MWDG2 solver requires selecting a maximum refinement level, L , based on size and resolution of the Digital Elevation Model (DEM) and an error threshold, ε ≤ 10-3, to preserve similar accuracy as a GPU-DG2 simulation on a uniform grid. The accuracy and efficiency of dynamic GPU-MWDG2 adaptivity is assessed for four tsunami-induced flooding test cases involving increasingly complex tsunamis: from single-wave impact events to wave trains. At ε = 10-3, the GPU-MWDG2 simulations yield predictions similar to the GPU-DG2 simulations but using ε = 10-4 can improve the accuracy in velocity-related predictions. In terms of efficiency, the GPU-MWDG2 simulations show progressively larger speedups over the GPU-DG2 simulations from L ≥ 10, which become significant (≥ 3.3- and 4.5-fold at ε = 10-4 and 10-3 , respectively) for simulating a single-wave impact event. The LISFLOOD-FP 8.2 code is open source, DOI: 10.5281/zenodo.4073010 , as well as the simulation data and the input files and scripts to reproduce them, DOI: 10.5281/zenodo.13909072 , with additional documentation at https://www.seamlesswave.com/Adaptive (last accessed: 9 October 2024). [ABSTRACT FROM AUTHOR]

A one-dimensional numerical overland flow model based on the cascade plane theory was developed to estimate rainfall-induced runoff and soil erosion on converging and diverging plane surfaces. The model includes three components: (i) soil infiltration using Horton's infiltration equation, (ii) overland flow using the kinematic wave approximation of the one-dimensional Saint-Venant shallow water equations for a cascade of planes, and (iii) soil erosion based on the sediment transport continuity equation. The model's performance was evaluated by comparing numerical results with laboratory data from experiments using a rainfall simulator and a soil flume. Four independent experiments were conducted on converging and diverging surfaces under varying slope and rainfall conditions. Overall, the numerically simulated hydrographs and sediment graphs closely matched the laboratory results, showing the efficiency of the model for the tested controlled laboratory conditions. The model was then used to numerically explore the impact of different plane soil surface geometries on runoff and soil loss. Seven geometries were studied: one rectangular, three diverging, and three converging. A constant soil surface area, the rainfall intensity, and the slope gradient were maintained in all simulations. Results showed that increasing convergence angles led to a higher peak and total soil loss, while decreasing divergence angles reduced them. [ABSTRACT FROM AUTHOR]

The article explores the spatial Whitham equation as a model for water wave evolution, comparing it to other equations like the temporal Korteweg–de Vries and Whitham equations. It demonstrates that the spatial Whitham equation accurately predicts various water wave phenomena and compares numerical simulations with laboratory experiments on waves of depression and elevation in shallow and deep water. While the spatial Whitham equation provides accurate predictions for most wave phenomena, it faces challenges in accurately modeling wave packets on deep water. The text also delves into conserved quantities and stability properties of the spatial Whitham equation. [Extracted from the article]

The article "Evolution of water wave packets by wind in shallow water" explores the use of the Korteweg– de Vries (KdV) equation to study wind-driven water wave packets in shallow water. It discusses the impact of forcing/friction terms on wave growth and decay, utilizing the Whitham modulation theory to predict these rates. Numerical simulations validate the theoretical predictions, highlighting the formation of solitary waves and the emergence of a soliton gas in various scenarios. The study provides insights into the evolution of water wave packets by wind in shallow water, shedding light on the complex dynamics involved. [Extracted from the article]

The article in the Journal of Fluid Mechanics examines the attenuation of surface waves passing through irregular floating ice and bathymetry. It corrects previous models that overestimated wave decay and introduces a revised approach supported by numerical simulations. The study focuses on wave attenuation through fragmented floating ice and random ice thickness, shedding light on the relationship between wave frequency and attenuation observed in field data. The research contributes to a deeper understanding of wave behavior in diverse marine environments, including the Arctic and Antarctic regions. [Extracted from the article]

The depth-averaged models such as those based on the shallow water equations (SWEs) are commonly used to simulate the large-scale flows with engineering importance. The smoothed particle hydrodynamics (SPH) approach has been documented to solve the SWEs due to its mesh-free superiority in treating the free surfaces and wet-dry boundaries. However, nearly all SWE-SPH models were developed without a turbulent model, which seriously limited the model applications where the flows are complex and where the turbulent parameters are explicitly needed. For the first time, this paper includes a depth-averaged turbulent k ̂ - ε ̂ model in the SWE-SPH solver, making the model more capable of treating the turbulent flows in the practical field. For comparison purpose, a sub-particle-scale turbulent model widely adopted in three-dimensional (3D) SPH was also included in the present SWE-SPH scheme. To evaluate the performance of the two proposed turbulent SWE-SPH models, various open channel flows of increasing complexity were simulated, and the SPH computations were compared with the reported data in the literature. Through the analysis of results for a rough riverbed, L-shaped and sudden expansion channels, it is demonstrated that the present turbulent SWE-SPH models are equipped with good robustness and accuracy in capturing the shallow water turbulent dynamics, with the potential to be used in practical river and coastal flows. In summary, there are two distinct novelties in the proposed work. First, the mesh-free numerical modeling technique SPH is used to solve the shallow water equations, which enable the model to work in large engineering field through simple and effective tracking of free surfaces and wet-dry boundaries. Second, the proposed research expands the shallow water SPH modeling technique by including robust turbulence simulation capacity. The newly developed model can address more challenging engineering scenarios such as the sediment and pollutant transports when the flow turbulence plays an important role and where the turbulent parameters are explicitly required in the relevant transport equations. [ABSTRACT FROM AUTHOR]

In this work, a fractional consistent Riccati expansion (FCRE) method is proposed to seek soliton and soliton-cnoidal solutions for fractional nonlinear evolutional equations. The method is illustrated by the time-fractional extended shallow water wave equation in the (2 + 1)-dimension, which includes a lot of KdV-type equations as particular cases, such as the KdV equation, potential KdV equation, Boiti–Leon–Manna–Pempinelli (BLMP) equation, and so on. A rich variety of exact solutions, including soliton solutions, soliton-cnoidal solutions, and three-wave interaction solutions, have been obtained. Comparing with the fractional sub-equation method, G ′ / G -expansion method, and exp-function method, the proposed method gives new results. The method presented here can also be applied to other fractional nonlinear evolutional equations. [ABSTRACT FROM AUTHOR]

This paper introduces a unified model for layered rotating shallow‐water (RSW) and quasi‐geostrophic (QG) equations, based on the intrinsic relationship between these two sets of equations. We propose a novel formulation of the QG equations as a projection of the RSW equations. This formulation uses the same prognostic variables as RSW, namely velocity and layer thickness, thereby restoring the proximity of these two sets of equations. It provides direct access to the ageostrophic velocities embedded within the geostrophic velocities resolved by the QG equations. This approach facilitates the study of differences between QG and RSW using a consistent numerical discretization. We demonstrate the effectiveness of this formulation through examples including vortex shear instability, double‐gyre circulation, and a simplified North Atlantic configuration. Plain Language Summary: In this paper, we present a straightforward way to connect two important sets of ocean equations: the layered rotating shallow‐water (RSW) equations and the quasi‐geostrophic (QG) equations. We consider a unified method to formulate the QG equations as a projection of the RSW equations. This method uses the same variables as the RSW equations, making it easier to understand how these two sets of equations relate to each other. Our approach also provides direct access to important velocity information that is usually hidden in the QG equations. This allows us to study the differences between the QG and RSW equations using the same numerical techniques. We show the benefits of our model with two examples: a vortex shear instability and a double‐gyre configuration. This work provides a useful tool for the understanding of ocean dynamics. Developing a unified numerical framework for nested models can greatly simplify oceanographic modeling and enhance accuracy. Key Points: Introduces QG equations as a projection of rotating shallow‐water (RSW) equations using the same prognostic variablesAllows consistent nested numerical methods for both QG and RSW equationsSuccessfully tested on vortex shear instability, double‐gyre circulation, and a simplified North Atlantic configuration [ABSTRACT FROM AUTHOR]

In order to study the supercritical flow in a curved channel of a rectangular cross section, the classical shallow water equations in a cylindrical coordinate system based on the mass and momentum laws that take into account the friction and bottom slope are used. The obtained mathematical model forms nonlinear partial differential equations of first-order. For simplification, a linearization of the partial differential equations (PDEs) set is performed using small perturbation approach valid for weak bends (axial curvature radius extremely larger than the channel width). The governing equations with well-posed initial and boundary conditions were solved for a rectangular bend channel flow by applying the method of characteristics that is capable of transforming the hyperbolic partial differential equations to a system of ordinary differential equations (ODEs). The proposed model is tested and validated by comparing the results with broad available experimental data reported in the literature, and particular attention was paid to the wave maximum and its location. Comparisons indicate a reasonable agreement between the results obtained for the maximum flow depth along the outer channel wall. However, the model prediction is only reliable for a small relative curvature. Despite the model limitations, the results show the reliability and accuracy of the proposed approach for practical design purposes. [ABSTRACT FROM AUTHOR]

The Saint-Venant-Exner system is widely used in industrial codes to model the transport of bed sediments. In practice, most of the methods used for its numerical resolution suffer from significant stability problems due to the fact that they are mainly guaranteed by separation techniques which allow weak coupling between hydraulic and morphodynamic software. The search for an adequate alternative method to this problem is a focus towards which several approaches converge. Recently, many authors have proposed acceptable methods but they are not so trivial to implement in an industrial context and some do not take into account all aspects such as mobile steady states. In this work, we derive a well-balanced scheme that takes into account all stable states, to approach weak solutions of the sediment transport model in shallow waters. We demonstrate that it captures exactly all regular steady solutions with non-evanescent velocities, retains water level positivity and is able to degenerate to a classical shallow water model when the sedimentary transport flow is null. A number of numerical test cases are also presented to illustrate these properties. [ABSTRACT FROM AUTHOR]

Predicting flood wave propagation from reservoir failures is critical to practical flood hazard assessment and risk management. Flood waves are sensitive to topography, channel geometry, structures, and natural features along floodplain paths. Thus, the accuracy of flood wave modelling depends on how precisely those features are represented. This study introduces an enhancing approach to flood wave modelling by accurately representing three-dimensional objects in floodplains using the structure-from-motion (SfM). This method uses an unmanned aerial vehicle to capture topographic complexities and account for ground objects that impact flood propagation. Using the three-dimensional volume of fluid numerical approach significantly improves an enhanced representation of turbulent flow dynamics and computational efficiency, especially in handling large topography datasets. Reproductions from this enhanced three-dimensional approach were validated against recent reservoir failure observations and contrasted with traditional two-dimensional models. The results revealed that the suggested three-dimensional methodology achieved a significant 84.4% reproducibility when juxtaposed with actual inundation traces. It was 35.5%p more accurate than the two-dimensional diffusion wave equation (DWE) and 17.1%p more than the shallow water equation (SWE) methods in predicting flood waves. This suggests that the reproducibility of the DWE and SWE decreases compared to the three-dimensional approach when considering more complex floodplains. These results demonstrate that three-dimensional flood wave analysis with the SfM methodology is optimal for effectively minimising topographic and flood wave reproduction errors across extensive areas. This dual reduction in errors significantly enhances the reliability of flood hazard assessments and improves risk management by providing more precise and realistic predictions of flood waves. [ABSTRACT FROM AUTHOR]

Coastal restoration projects are significantly important in coastal ecosystems as wetland losses accelerate. This study investigates tidal hydrodynamics and the potential impacts of a waterway opening on an existing roadway to restore and revitalise salt marshes in a small estuarine system in Virginia, USA. A depth-integrated, discontinuous Galerkin shallow-water equations model (DG-SWEM) is applied for astronomic tide simulation. The model employs a high-resolution unstructured mesh with a minimum element size of less than one meter and resolves complex tidal flows in the entire barrier island system, including the existing culvert gate and canal system. Compared to the existing system, the water exchange increased dramatically (flushing time also dramatically decreased) under the opened scenarios regardless of the opening width (22.9-, 30.5-, and 38.1-m width). By increasing the opening width, peak velocity through the proposed opening decreased 30–40%, and the maximum shear stress was reduced by more than half. The high-resolution model represented complex tidal flows, including eddies, and assisted in striking a balance of water-exchange capability, opening stability, and minimising their potential erosions for the opening design. Besides, erosion and sediment transport potentials for suspended sediments were estimated using bed shear stress and a Lagrangian particle tracking module. Such proxy modelling approach allows for the impact assessment of civil engineering and ecological waterworks in complex and highly damped tidal flow areas and is readily transferrable to other like systems (e.g. causeway construction, causeway cutting, biota passageways, and inlet modification). [ABSTRACT FROM AUTHOR]

Physics‐informed neural networks (PINNs) are gaining attention as an alternative approach to solve scientific problems governed by differential equations. This work aims at assessing the effectiveness of PINNs to solve a set of partial differential equations for which this method has never been considered, namely the augmented shallow water equations (SWEs) with topography. Differently from traditional SWEs, the bed elevation is considered as an additional conserved variable, and therefore one more equation expressing the fixed‐bed condition is included in the system. This approach allows the PINN model to leverage automatic differentiation to compute the bed slopes by learning the topographical information during training. PINNs are here tested for different one‐dimensional cases with non‐flat topography, and results are compared with analytical solutions. Though some limitations can be highlighted, PINNs show a good accuracy for the depth and velocity predictions even in the presence of non‐horizontal bottom. The solution of the augmented system of SWEs can therefore be regarded as a suitable alternative strategy to deal with flows over complex topography using PINNs, also in view of future extensions to realistic problems. Key Points: Physics‐informed neural networks (PINNs) are applied to solve the augmented shallow water equations with topographyApplications to one‐dimensional cases of free‐surface flows over non‐flat bottom show a good solution accuracySolving the augmented system is an alternative way to deal with non‐flat topography [ABSTRACT FROM AUTHOR]

Flash flooding has become more prominent under climate change, threatening people's life and property. Post‐event investigations of recent events emphasize the role of floating debris, such as vehicles, in exacerbating damage. Few modeling methods and tools have been developed to simulate the full‐process dynamics of floating debris driven by large‐scale flood waves in real world. In this work, a fully coupled model is developed for simulating the full‐process interactive movements of vehicles driven by flash flood hydrodynamics, from entrainment, transport to deposition. The proposed coupled modeling system consists of a finite volume shock‐capturing hydrodynamic model solving the 2D shallow water equations and a 3D discrete element method (DEM) model. The proposed two‐way coupling approach estimates the hydrostatic and hydrodynamic forces acting on solid objects using the water depth and velocity predicted by the hydrodynamic model; the resulting counter forces on the fluid flow are then considered by adding extra source terms in the hydrodynamic model. A multi‐sphere method is further embedded in the DEM model to better represent vehicle shapes. New calculation modules are further implemented to represent the vehicle entrainment, contact and stopping motions. The coupled model is applied to reproduce a flash flood event hit Boscastle in the UK in 2004. Over 100 vehicles were moved and carried downstream by the highly transient flood flow. The model well predicts the hydrodynamics, interactive transport process and the final locations of vehicles. The proposed coupled model provides a new tool for simulating large‐scale flash flooding processes, including debris dynamics. Key Points: A new coupled model for simulation of entrainment, transport and deposition of vehicles driven by and interacting with flood hydrodynamicsThe model is used to reproduce a flash flood event that moved over 100 vehicles, with results consistent with post‐event report and surveyIncreasing number of floating vehicles alters flood hydrodynamics and intensifies debris‐debris and debris‐fluid interactions [ABSTRACT FROM AUTHOR]

In this study, the nonlinear two-dimensional Kadomtsev–Petviashvili-modified equal width (KP-MEW) equation is under investigation, which is described as a nonlinear wave model in weakly restoring forces and shallow water waves in the way of ferromagnetic, solitary waves in two dimensions with short amplitude and long wavelength and are also helpful in the investigation of various behaviors in nonlinear sciences. We successfully found the interesting and novel solitary wave results in bright solitons, kink solitons, dark solitons, anti-kink solitons, periodic singular solitons for nonlinear two-dimensional KP-MEW equation on the base of extension of simple equation technique under symbolic computational software Mathematica. The created solitons play an important role in nonlinear engineering and physics such as nonlinear optics, nonlinear dynamics, soliton wave theory, optical fiber and communication system. We are sure that these constructed results are novel and interesting which does not exist in the previous literature works. The graphical representation of constructed results is shown by 2D, 3D and contour graphs. The investigated research work proved that utilized technique is more concise, straightaway and efficient for study of other nonlinear partial differential equations. [ABSTRACT FROM AUTHOR]

The article "Parametric oscillations of the sessile drop" in the Journal of Fluid Mechanics examines the behavior of waves on a sessile drop under substrate vibrations at an angle. Through mathematical modeling and experimental validation, the study explores three types of drop motions and their relationship to contact angles and slanting angles. The research contributes to understanding drop mobility and directional motion induced by vibrations, shedding light on Faraday waves on sessile drops with varying parameters. The study highlights the importance of contact angle, slanting angle, and boundary conditions in drop dynamics, with experimental results largely supporting theoretical predictions. [Extracted from the article]

This work employs the sub-ODE method to compute new soliton solutions for the component generalization of (2 + 1) -dimensional Schwarz–Kortweg–de Vries (SKdV) equation in shallow water waves. The rational soliton solutions (RSS), Jacobian elliptic function (JEF), periodic solutions (PS), hyperbolic function solutions (HFS), positive solitons, dark soliton solutions (DSS) and bright soliton solutions (BSS) are constructed with this technique. Solitons-like solutions, HFS and trigonometric-function solutions (TFS) are also found as the JEF's parameter m approaches values of 1 and 0, respectively. The suggested approach for the nonlinear equations in mathematical physics is concise, simple, effective, and promising. [ABSTRACT FROM AUTHOR]

We establish the full justification of a "Whitham-Green-Naghdi" system modeling the propagation of surface gravity waves with bathymetry in the shallow water regime. It is an asymptotic model of the water waves equations with the same dispersion relation. The model under study is a nonlocal quasilinear symmetrizable hyperbolic system without surface tension. We prove the consistency of the general water waves equations with our system at the order of precision O (μ 2 (ε + β)) , where μ is the shallow water parameter, ε the nonlinearity parameter, and β the topography parameter. Then we prove the long time well-posedness on a time scale O (1 max ⁡ { ε , β }). Lastly, we show the convergence of the solutions of the Whitham-Green-Naghdi system to the ones of the water waves equations on the later time scale. [ABSTRACT FROM AUTHOR]

Shallow-buried cavities and underground structures are inevitably subjected to transient perturbation from earthquakes or engineering activities during their life circle, leading to the occurrence of dynamic failure. The investigation presented by this paper concentrates on the dynamic responses triggered by an aperiodic transient P wave around a circular lined tunnel with an imperfect interface in a half-space and the spring model is employed to describe the interface between rock mass and liner. Resorting to the wave function expansion scheme, the large arc assumption, the Graf's addition theorem and the numerical inversion algorithm of Laplace transformation, the dynamic responses around a circular shallow lined tunnel with an imperfect interface are evaluated theoretically. The good agreement of time–history curves and contour snapshots for stress between the theoretical solutions and the numerical counterparts validates the analytical scheme. The effect of disturbance direction, embedment depth, thickness of liner and interface coefficients on the dynamic responses around the shallow-buried lined tunnel is analyzed. The results indicate that the positions of peak hoop dynamic stress concentration factor (DSCF) are approximately perpendicular to the perturbation direction. The DSCF distributions converge to be stable when the ratio of embedment depth to excavation radius exceeds 10.0. The thickness of the liner mainly affects the dynamic responses around the interface. Increase of elastic coefficient induces a decreasing DSCF in rock mass and an increasing counterpart in the liner. The results may provide the theoretical basis for the transient responses and aseismic design of underground tunnels. [ABSTRACT FROM AUTHOR]

Numerical simulations were generated to investigate the response of a very large floating airport to an airplane takeoff, using the set of nonlinear shallow water equations of velocity potential for water waves interacting with a floating thin plate. We have proposed two methods to reduce persistent airport vibration: reflectance reduction by decreasing the flexural rigidity in airport edge parts and amplification reduction by decreasing the still water depth partially under airport runways. First, when the flexural rigidity is uniformly decreased in an airport edge part, the reflectance of the floating body waves due to a B737 was reduced because of the multiple reflections. However, the wave reflectance for a B747 increased, depending on the conditions. A too-long edge part was not effective in reducing the wave reflectance. Conversely, when the flexural rigidity is linearly decreased in an airport edge part, the wave reflectance was reduced for both airplanes. Second, when the still water depth under an airport runway is partially reduced at the location where floating body waves are amplified, the wave heights of floating body waves tended to decrease as the still water depth in the shallower area decreased. [ABSTRACT FROM AUTHOR]

In this manuscript, we survey a numerical algorithm based on the combination of the homotopy perturbation method and the Sadik transform for solving the time-fractional nonlinear modified shallow water waves (called Kawahara equation) within the frame of the Caputo–Prabhakar (CP) operator. The nonlinear terms are handled with the assistance of the homotopy polynomials. The stability analysis of the implemented method is studied by using S-stable mapping and the Banach contraction principle. Also, we use the fixed-point method to determine the existence and uniqueness of solutions in the given suggested model. Finally, some numerical simulations are illustrated to display the accuracy and efficiency of the present numerical method. Moreover, numerical behaviors are captured to validate the reliability and efficiency of the scheme. [ABSTRACT FROM AUTHOR]

This study proposes a novel framework of the lattice Boltzmann model for multilayer shallow water equations, considering the mass and momentum exchanges between layers (LABMSWE+). Compared with the original LABMSWE model consisting of N two-dimensional lattice Boltzmann method for shallow water equation (LABSWE) models, the new model includes 1+N LABSWE models. The singular LABSWE model with unit relaxation time is introduced to update the total water depth, and thus, the layer water depths can be obtained explicitly through the fixed layer ratios. The N-layer LABSWE models with the multiple-relaxation-time operator evolve the layer velocities. These two modules are coupled by the total water depth and depth-averaged velocities. The constructed model avoids the freely variable layer thicknesses, which is considered as the main source of the instability. In addition, the mass exchanges enable this model to simulate vertical circulation flows, which are beyond the application of the LABMSWE model. Several numerical tests are then conducted to validate the proposed model. The results show that it exactly satisfies the C-property. In addition, the central difference scheme is more stable and accurate than the upwind and nonequilibrium schemes in the computing of the mass exchanges. The numerical results have an excellent agreement with analytical solutions and reference data, while some unstable and nonphysical results are obtained by the original LABMSWE model. Moreover, the computational time is about 40%–60% of that for the MIKE3, a finite volume solver for the three-dimensional shallow water equations by the Danish Hydraulic Institute. [ABSTRACT FROM AUTHOR]

The present work focuses on the numerical approximation of the weak solutions of the shallow water model over a non-flat topography. In particular, we pay close attention to steady solutions with nonzero velocity. The goal of this work is to derive a scheme that exactly preserves these stationary solutions, as well as the commonly preserved lake at rest steady solution. These moving steady states are solution to a nonlinear equation. We emphasize that the method proposed here never requires solving this nonlinear equation; instead, a suitable linearization is derived. To address this issue, we propose an extension of the well-known hydrostatic reconstruction. By appropriately defining the reconstructed states at the interfaces, any numerical flux function, combined with a relevant source term discretization, produces a well-balanced scheme that preserves both moving and non-moving steady solutions. This eliminates the need to construct specific numerical fluxes. Additionally, we prove that the resulting scheme is consistent with the homogeneous system on flat topographies, and that it reduces to the hydrostatic reconstruction when the velocity vanishes. To increase the accuracy of the simulations, we propose a well-balanced high-order procedure, which still does not require solving any nonlinear equation. Several numerical experiments demonstrate the effectiveness of the numerical scheme. [ABSTRACT FROM AUTHOR]

Compound flooding, the concurrence of multiple flooding mechanisms such as storm surge, heavy rainfall, and riverine flooding, poses a significant threat to coastal communities. To mitigate the impacts of compound flooding, forecasts must represent the variability of flooding drivers over a wide range of spatial scales while remaining timely. One approach to develop these forecasts is through subgrid corrections, which utilize information at smaller scales to "correct" water levels and current velocities averaged over the model scale. Recent studies have shown that subgrid models can improve both accuracy and efficiency; however, existing models are not able to account for the dynamic interactions of hydrologic and hydrodynamic drivers and their contributions to flooding along the smallest flow pathways when using a coarse resolution. Here, we have developed a solver called CoaSToRM (Coastal Subgrid Topography Research Model) with subgrid corrections to compute compound flooding in coastal systems resulting from fluvial, pluvial, tidal, and wind-driven processes. A key contribution is the model's ability to enforce all flood drivers and use the subgrid corrections to improve the accuracy of the coarse-resolution simulation. The model is validated for Hurricane Eta 2020 in Tampa Bay, showing improved prediction accuracy with subgrid corrections at 42 locations. Subgrid models with coarse resolutions (R2 = 0.70, 0.73, 0.77 for 3-, 1.5-, 0.75-km grids) outperform standard counterparts (R2 = 0.03, 0.14, 0.26). A 3-km subgrid simulation runs roughly 50 times faster than a 0.75-km subgrid simulation, with similar accuracy. [ABSTRACT FROM AUTHOR]

This work examines tsunami run-up processes within a building array partially sheltered by a high-elevation, finite length, seawall. Work was performed using numerical modeling of 2D shallow water equations with varying seawall length, wave height, time scale, and structure size. Inundation depth and flow velocity were output at many locations of interest throughout the structural array. The most important influence on results when compared to the no-wall case was the sheltering or clearance angle as defined from the end of the wall. Behind the wall, inundation depth and cross-shore velocity declined significantly as sheltering angles increased (more sheltered). Alongshore inundation velocity (parallel to the seawall) increased strongly with increasing sheltering angle. A longer wave time scale resulted in significant increase in inundation depth and velocity especially at locations sheltered by the wall. Larger structure size or blockage ratio also led to increased inundation depth and flow velocity. However, varying wave height gave little change in inundation and velocity patterns. These results will serve to provide a better understanding of the wave inundation within a coastal urban region in a tsunami event when the seawall is breached, as well as the influence of coastal structure size and spacing on the tsunami inundation. [ABSTRACT FROM AUTHOR]

In this study, we propose a modified version of section-averaged Boussinesq equations of Winckler-Liu. The model is reformulated in conservative variables, allowing a decoupling of the Shallow-Water equations and the dispersive problem. An appropriate hybrid finite volume and finite element discretisations are performed and verified with a solitary wave solution derived for the typical case of prismatic channels with a trapezoidal cross-section. For the finite volume step we compare upwind and energy conservative numerical fluxes. The impact of this choice on the long time dynamics for Favre waves is thoroughly investigated. The proposed model and numerical approximations can accurately reproduce the main features of the wave train's free surface. The impact of the numerical dissipation introduced by upwind fluxes is discussed, emphasising the need for precaution in their applications to evaluate quantities of engineering interest such as maximum wave amplitudes. • A reformulation of Winckler and Liu's model is proposed to study wave propagation in channels with arbitrary cross-sections. • The model simplifies dispersive terms integration into existing shallow water solver, enhancing computational flexibility. • Hybrid discretization method, combining finite volume and finite element is proposed. • Two numerical flux formulations and their dissipative behaviours are investigated over long propagation times of Favre waves. • Comparing with Favre waves laboratory experiments, the numerical model captures key propagation phenomena. [ABSTRACT FROM AUTHOR]

Due to the complexity of terrain and climate in the mountain–plain transition zone, it is difficult to simulate and forecast the flow discharge of river basins accurately. The poor regularity of the river thus leads to uncertain flood control scheduling. Meanwhile, reservoirs and flood detention areas are constructed to store and divert water when severe floods threaten the safety of the basin. In order to improve the accuracy of flood forecasts and the effectiveness of flood control, a hydrological and 1D/2D hydrodynamic coupling model was developed to enable the joint computation of multiple objects, including mountainous streams, plains river networks, hydraulic control structures, and flood detention areas. For the hydrological component, the Xin'anjiang model with the Muskingum module is employed to simulate mountainous flow discharge. For the hydrodynamic component, the Saint–Venant equations and shallow water equations are applied to estimate flood processes in rivers and on land surfaces, respectively. The Dongtiaoxi River Basin in Zhejiang Province, China, serves as the case study, where river flow is influenced by both upstream mountainous floods and downstream backwater effects. Using the integrated model, flood routing and scheduling are simulated and visualized. Both the Xin'anjiang model and the 1D hydrodynamic model demonstrate over 80% acceptability in calibration and validation, confirming their robustness and reliability. Meanwhile, inundation in flood detention areas can be effectively estimated by coupling the 1D and 2D hydrodynamic models with a flood diversion scheduling model. The coupled model proves capable of simulating flood routing in complex river basins that include mountains, plains, and hydraulic control structures, accounting for the interactions between hydrological elements. These findings provide a new perspective on flood simulation in other similarly complex river basins. [ABSTRACT FROM AUTHOR]

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. [ABSTRACT FROM AUTHOR]

Flash floods typically occur suddenly within hours of heavy rainfall. Accurate forecasting of flash floods in advance using the two-dimensional (2D) shallow water equations (SWEs) remains a challenge, due to the governing SWEs being difficult-to-solve partial differential equations (PDEs). Aiming at shortening the computational time and gaining more time for issuing early warnings of flash floods, constructing a new relationship between water storage and outflow in the rainfall–runoff process is attempted by assuming the catchment as a water storage system. Through numerical simulations of the diffusion wave (DW) approximation of SWEs, the water storage and discharge are found to be limited to envelope lines, and the discharge/water-depth process lines during water rising and falling showed a grid-shaped distribution. Furthermore, if a catchment is regarded as a semi-open water storage system, then there is a nonlinear relationship between the inside average water depth and the outlet water depth, namely, the water storage ratio curve, which resembles the shape of a plume. In the case of an open channel without considering spatial variability, the water storage ratio curve is limited to three values (i.e., the upper, the steady, and the lower limits), which are found to be independent of meteorological (rainfall intensity), vegetation (Manning's coefficient), and terrain (slope gradient) conditions. Meteorological, vegetation, and terrain conditions only affect the size of the plume without changing its shape. Rainfall, especially weak rain (i.e., when rainfall intensity is less than 5.0 mm h -1), significantly affects the fluctuations of the water storage ratio, which can be divided into three modes: Mode I (inverse S-shape type) during the rainfall beginning stage, Mode II (wave type) during the rainfall duration stage, and Mode III (checkmark type) during rainfall end stage. Results indicate that the determination of the nonlinear relationship of the water storage ratio curve under different geographical scenarios will provide new ideas for simulation and early warning of flash floods. [ABSTRACT FROM AUTHOR]

The Madden–Julian oscillation (MJO) is believed to play a significant role in triggering El Niño–Southern Oscillation (ENSO) events and affect the dynamics of ENSO. In this study, the dynamic forcing effects of MJO on the equatorial oceanic dynamic fields and the onsets of different types of ENSO events are investigated through sensitive experiments using spatiotemporally filtered forcing based on an anomalous shallow water model. The comparisons between observations and model responses provide meaningful insights into the extent of MJO's impacts on sea surface dynamics relative to other atmospheric variabilities. The following conclusions are made. First, the MJO-forced perturbations on zonal currents are stronger and more significant than those on sea surface heights. Second, MJO is essential for improving zonal current simulation in the western-central Pacific and generating activity centers of zonal currents in the eastern Pacific in the model. Third, MJO tends to contribute to the onset of El Niño events rather than La Niña events. Strong intraseasonal oceanic Kelvin waves forced by MJO are confirmed in simulations during the onset stages of the 1997/98 and 2004/05 events. The 120-day running standard deviations of zonal current and sea surface height anomaly series forced by MJO exhibit positive skewness similar to those of the 20–100-day band-passed observational series. Yet, not all the onsets of historical ENSO events are in company with strong MJO-related perturbations. Additionally, the wind stress formula can amplify the responses of zonal current and sea surface height anomalies to synoptic forcings with periods shorter than 20 days through entraining lower-frequency variabilities. Significance Statement: The Madden–Julian oscillation (MJO) is believed to be able to trigger El Niño–Southern Oscillation (ENSO) events and influence our understanding of the fundamental nature of ENSO. In this study, spatiotemporally filtered forcing experiments are implemented on an anomalous shallow water model. The results show that MJO is more important for improving the simulation of surface zonal currents rather than the sea surface heights and tends to contribute to the onset of El Niño events rather than La Niña events through triggering strong intraseasonal oceanic Kelvin waves. [ABSTRACT FROM AUTHOR]

The rotating thermal shallow water (RTSW) equations is a model for studying thin layer fluids with horizontal variations in material properties. A mathematically rigorous framework is developed for deriving new reduced slow dynamics equations of the multiple-scale RTSW equations with both balanced and unbalanced initial data over the spatial periodic domain. It is rigorously proved that the reduced slow dynamics of the RTSW equations is the thermal quasi-geostrophic (thermal QG) equations by proper rescaling. Moreover, by introducing appropriate elliptic equations and using the time-averaging technique, we further obtained the convergence rates of the solutions with unbalanced initial data. [ABSTRACT FROM AUTHOR]

In this article, we propose a new path-conservative discontinuous Galerkin (DG) method to solve non-conservative hyperbolic partial differential equations (PDEs). In particular, the method here applies the one-stage ADER (Arbitrary DERivatives in space and time) approach to fulfill the temporal discretization. In addition, this method uses the differential transformation (DT) procedure rather than the traditional Cauchy–Kowalewski (CK) procedure to achieve the local temporal evolution. Compared with the classical ADER methods, the current method is free of solving generalized Riemann problems at inter-cells. In comparison with the Runge–Kutta DG (RKDG) methods, the proposed method needs less computer storage, thanks to the absence of intermediate stages. In brief, this current method is one-step, one-stage, and fully-discrete. Moreover, this method can easily obtain arbitrary high-order accuracy both in space and in time. Numerical results for one- and two-dimensional shallow water equations (SWEs) show that the method enjoys high-order accuracy and keeps good resolution for discontinuous solutions. [ABSTRACT FROM AUTHOR]

This paper presents simulations of dam-break flows of Herschel–Bulkley viscoplastic fluids over complex topographies using the shallow water equations (SWE). In particular, this study aims to assess the effects of rheological parameters: power-law index (n), consistency index (K), and yield stress ( τ c ), on flow height and velocity over different topographies. Three practical examples of dam-break flow cases are considered: a dam-break on an inclined flat surface, a dam-break over a non-flat topography, and a dam-break over a wet bed (downstream containing an initial fluid level). The effects of bed slope and depth ratios (the ratio between upstream and downstream fluid levels) on flow behaviour are also analyzed. The numerical results are compared with experimental data from the literature and are found to be in good agreement. Results show that for both dry and wet bed conditions, the fluid front position, peak height, and mean velocity decrease when any of the three rheological parameters are increased. However, based on a parametric sensitivity analysis, the power-law index appears to be the dominant factor in dictating fluid behaviour. Moreover, by increasing the bed slope and/or depth ratio, the wave-frontal position moves further downstream. Furthermore, the presence of an obstacle is observed to cause the formation of an upsurge that moves in the upstream direction, which increases by increasing any of the three rheological parameters. This study is useful for an in-depth understanding of the effects of rheology on catastrophic gravity-driven flows of non-Newtonian fluids (like lava or mud flows) for risk assessment and mitigation. [ABSTRACT FROM AUTHOR]

The wave motions of the initial discontinuities in the inviscid shallow water wave model governed by the bad Jaulent-Miodek system are explored by the Whitham modulation theory and dynamical analysis. The $ N $-phase algebro-geometric solutions along with the related genus-$ N $ Whitham equations are derived by algebro-geometric method and Flaschka-Forest-McLaughlin approach, respectively. The complete classification of solutions to the bad Jaulent-Miodek system is carried out by solving the Whitham equations both theoretically and numerically. Some exotic phenomena of nonlinear wave motions such as rarefaction waves and dispersive shock waves are found in the inviscid shallow water. Particularly, the nonlinear wave motions including the two-phase algebro-geometric solutions are found in the inviscid shallow water wave. It is shown that the results from Whitham modulation theory agree very well with the numerical calculations and dynamical analysis. [ABSTRACT FROM AUTHOR]

This study aims to employ the extended direct algebraic method (EDAM) to generate and evaluate soliton solutions to the nonlinear, space-time conformable Estevez Mansfield-Clarkson equation (CEMCE), which is utilized to simulate shallow water waves. The proposed method entails transforming nonlinear fractional partial differential equations (NFPDEs) into nonlinear ordinary differential equations (NODEs) under the assumption of a finite series solution by utilizing Riccati ordinary differential equations. Various mathematical structures/solutions for the current model are derived in the form of rational, exponential, trigonometric, and hyperbolic functions. The wide range of obtained solutions allows for a thorough analysis of their actual wave characteristics. The 3D and 2D graphs are used to illustrate that these behaviors consistently manifest as periodic, dark, and bright kink solitons. Notably, the produced soliton solutions offer new and critical insights into the intricate behaviors of the CEMCE by illuminating the basic mechanics of the wave's interaction and propagation. By analyzing these solutions, academics can better understand the model's behavior in various settings. These solutions shed light on complicated issues such as configuration dispersion in liquid drops and wave behavior in shallow water. [ABSTRACT FROM AUTHOR]

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. [ABSTRACT FROM AUTHOR]

Storm surge is a natural process that causes flood disasters in coastal zones and results in massive casualties and property losses. Therefore, storm surge inundation is of major concern in formulating appropriate strategies for disaster prevention and mitigation. However, traditional storm surge hydrodynamic models have large limits with respect to computational efficiency and stability in practical applications. In this study, a novel storm surge inundation model was developed based on a wetting and drying algorithm established from a simplified shallow-water momentum equation. The wetting and drying algorithm was applied to a rectangular grid that iterates through a cellular automata algorithm to improve computational efficiency. The model, referred to as the Hydrodynamical Cellular Automata Flood Model (HCA-FM), was evaluated by comparing the simulations to regional field observations and to a widely used hydrodynamic numerical model. The comparisons demonstrated that HCA-FM can reproduce the observed inundation distributions and predict results that are consistent with the numerical simulation in terms of the inundation extent and submerged depth with much improved computational efficiency (predicting inundation within a few minutes) and high stability. The results reflect significant advancement of HCA-FM toward efficient predictions of storm surge inundation and applications at large spatial scales. [ABSTRACT FROM AUTHOR]

This article focuses on the design of semi-implicit schemes that are fully well-balanced for the one-dimensional shallow water equations, that is, schemes that preserve all smooth steady states of the system and not just water-at-rest equilibria. The proposed methods outperform standard explicit schemes in the low-Froude regime, where the celerity is much larger than the fluid velocity, eliminating the need for a large number of iterations on large time intervals. In this work, splitting and relaxation techniques are combined in order to obtain fully well-balanced semi-implicit first and second order schemes. In contrast to recent Lagrangian-based approaches, this one allows the preservation of all the steady states while avoiding the complexities associated with Lagrangian formalism. [ABSTRACT FROM AUTHOR]

We present an algorithm to solve the dispersive depth-averaged Serre--Green--Naghdi equations using patch-based adaptive mesh refinement. These equations require adding additional higher derivative terms to the nonlinear shallow water equations. This has been implemented as a new component of the open source GeoClaw software that is widely used for modeling tsunamis, storm surge, and related hazards, improving its accuracy on shorter wavelength phenomena. We use a formulation that requires solving an elliptic system of equations at each time step, making the method implicit. The adaptive algorithm allows different time steps on different refinement levels and solves the implicit equations level by level. Computational examples are presented to illustrate the stability and accuracy on a radially symmetric test case and two realistic tsunami modeling problems, including a hypothetical asteroid impact creating a short wavelength tsunami for which dispersive terms are necessary. [ABSTRACT FROM AUTHOR]