Search Results (403)

The shallow water equations (SWEs) model fluid flow in rivers, coasts, and tsunamis. Their nonlinearity challenges analytical solutions. We present a numerical algorithm combining the finite integration method with Chebyshev polynomial expansion (FIM-CPE) to solve one- and two-dimensional SWEs. The method transforms partial differential equations into integral equations, approximates spatial terms via Chebyshev polynomials, and uses forward differences for time discretization. Validated on stationary lakes, dam breaks, and Gaussian pulses, the scheme achieved errors below ${10}^{-12}$ for water height and velocity, while conserving mass with volume deviations under ${10}^{-5}$. Comparisons showed superior shock-capturing versus finite difference methods. For two-dimensional cases, it accurately resolved wave interactions over complex topographies. Though limited to wet beds and small-scale two-dimensional problems, the method provides a robust simulation tool. Full article

In 2071, the Hydraulic community will commemorate the second centenary of the Baré de Saint-Venant equations, also known as the Shallow Water Equations (SWE). These equations are fundamental to the study of open-channel flow. As non-linear partial differential equations, their solutions were largely unattainable until the development of computers and numerical methods. Following 1960, various numerical schemes emerged, with Preissmann’s scheme becoming the most widely employed in many software applications. In the 1990s, some researchers identified a significant limitation in existing software and codes: the inability to simulate transcritical flow. At that time, Preissmann’s scheme was the dominant method employed in hydraulics tools, leading the research community to conclude that this scheme could not handle transcritical flow due to suspected instability. In response to this concern, several researchers suggested modifications to Preissmann’s scheme to enable the simulation of transcritical flow. This paper will demonstrate that these accusations against the Preissmann scheme are unfounded and that the proposed improvements are unnecessary. The observed instability is not due to the numerical method itself, but rather a mathematical instability inherent to the SWE, which can lead to ill-posed conditions if a specific derived condition is not met. In the context of a friction slope formula based on Manning or Chézy types, the condition for ill-posedness of the 1D shallow water equations simplifies to the Vedernikov number condition, which is necessary for roll waves to develop in uniform flow. This derived condition is also relevant for the formation of roll waves in unsteady flow when the 1D shallow water equations become ill-posed. Full article

The Korteweg–de Vries (KdV) equation is a nonlinear evolution equation that reflects a wide variety of dispersive wave occurrences with limited amplitude. It has also been used to describe a range of major physical phenomena, such as shallow water waves that interact weakly and nonlinearly, acoustic waves on a crystal lattice, lengthy internal waves in density-graded oceans, and ion acoustic waves in plasma. The KdV equation is one of the most well-known soliton models, and it provides a good platform for further research into other equations. The KdV equation has several forms. The aim of this study is to introduce and investigate a (2+1)-dimensional generalized fifth-order KdV equation with power law nonlinearity (gFKdVp). The research methodology employed is the Lie group analysis. Using the point symmetries of the gFKdVp equation, we transform this equation into several nonlinear ordinary differential equations (ODEs), which we solve by employing different strategies that include Kudryashov’s method, the $(G^{\prime }/G)$ expansion method, and the power series expansion method. To demonstrate the physical behavior of the equation, 3D, density, and 2D graphs of the obtained solutions are presented. Finally, utilizing the multiplier technique and Ibragimov’s method, we derive conserved vectors of the gFKdVp equation. These include the conservation of energy and momentum. Thus, the major conclusion of the study is that analytic solutions and conservation laws of the gFKdVp equation are determined. Full article

Various nature-based solutions (NbS)—such as constructed wetlands, drainage ditches, and vegetated buffer strips—have recently demonstrated strong potential for mitigating pollutant transport in open channels and river systems. Numerical modeling is a widely adopted and effective approach for assessing the performance of these interventions. This study presents the first development of a two-dimensional (2D) meshless advection–diffusion model based on an Eulerian smoothed particle hydrodynamics (SPH) framework, specifically designed to simulate passive pollutant transport in open channel flows. The proposed model marks a pioneering application of the ESPH technique to environmental pollutant transport problems. It couples the 2D depth-averaged shallow water equations with an advection–diffusion equation to represent both fluid motion and pollutant concentration dynamics. A uniform particle arrangement ensures that each fluid particle interacts symmetrically with eight neighboring particles for flux computation. To represent the pollutant transport process, the dispersion coefficient is defined as the sum of molecular and turbulent diffusion components. The turbulent diffusion coefficient is calculated using a prescribed turbulent Schmidt number and the eddy viscosity obtained from a Smagorinsky-type mixing-length turbulence model. Three analytical case studies, including one-dimensional transcritical open channel flow, 2D isotropic and anisotropic diffusion in still water, and advection–diffusion in a 2D uniform flow, are employed to verify the model’s accuracy and convergence. The model demonstrates first-order convergence, with relative root mean square errors (RRMSEs) of approximately 0.2% for water depth and velocity, and 0.1–0.5% for concentration. Additionally, the model is applied to a laboratory experiment involving 2D pollutant dispersion in a 90° junction channel. The simulated results show good agreement with measured velocity and concentration distributions. These findings indicate that the developed model is a reliable and effective tool for evaluating the performance of NbS in mitigating pollutant transport in open channels and river systems. Full article

This study presents a one-dimensional solver of the shallow water equations designed for the wet-bed step Riemann problem. Nonlinear mass and momentum equations incorporating shock and rarefaction waves in a straight one-dimensional channel are expressed as a pair of equations that depend solely on local depth values either side of the step. These unified equations are uniquely designed for the four conditions involving shock and rarefaction waves that can occur in the Step Riemann Problem. The Levenberg–Marquardt method is used to solve these simplified nonlinear equations. Four verification tests are considered for shallow free surface flow in a wet-bed channel with a step. These cases involve two rarefactions, opposing shock-like hydraulic bores, and a rarefaction and shock-like bore. The numerical predictions are in close agreement with existing theory, demonstrating that the method is very effective at solving the wet-bed step Riemann problem. Full article

The primary aim of this research article is to investigate the soliton dynamics of the M-truncated derivative nonlinear Kodama equation, which is useful for optical solitons on nonlinear media, shallow water waves over complex media, nonlocal internal waves, and fractional viscoelastic wave propagation. We utilized two recently developed analytical techniques, the generalized Arnous method and the generalized Kudryashov method. First, the nonlinear Kodama equation is transformed into a nonlinear ordinary differential equation using the homogeneous balance principle and a traveling wave transformation. Next, various types of soliton solutions are constructed through the application of these effective methods. Finally, to visualize the behavior of the obtained solutions, three-dimensional, two-dimensional, and contour plots are generated using Maple (2023) mathematical software. Full article

Extreme rainstorms are difficult to predict and often result in catchment-scale rainfall flooding, leading to substantial economic losses globally. Enhancing the numerical computational efficiency of flood models is essential for improving flood forecasting capabilities. This study presents an integrated hydrological–hydrodynamic model accelerated using GPU (Graphics Processing Unit) technology to perform high-efficiency and high-precision rainfall flood simulations at the catchment scale. The model couples hydrological and hydrodynamic processes by solving the fully two-dimensional shallow water equations (2D SWEs), incorporating GPU-accelerated parallel computing. The model achieves accelerated rainstorm flooding simulations through its implementation on GPUs with parallel computing technology, significantly enhancing its computational efficiency and maintaining its numerical stability. Validations are conducted using an idealized V-shaped catchment and an experimental benchmark, followed by application to a small catchment on the Chinese Loess Plateau. The computational experiments reveal a strong positive correlation between grid cell numbers and GPU acceleration efficiency. The results also demonstrate that the proposed model offers better computational accuracy and acceleration performance than the single-GPU model. This GPU-accelerated hydrological–hydrodynamic modeling framework enables rapid, high-fidelity rainfall flood simulations and provides critical support for timely and effective flood emergency decision making. Full article

In this article, we present the extended simple equations method (SEsM) for finding exact solutions to systems of fractional nonlinear partial differential equations (FNPDEs). The expansions made to the original SEsM algorithm are implemented in several directions: (1) In constructing analytical solutions: exact solutions to FNPDE systems are presented by simple or complex composite functions, including combinations of solutions to two or more different simple equations with distinct independent variables (corresponding to different wave velocities); (2) in selecting appropriate fractional derivatives and appropriate wave transformations: the choice of the type of fractional derivatives for each system of FNPDEs depends on the physical nature of the modeled real process. Based on this choice, the range of applicable wave transformations that are used to reduce FNPDEs to nonlinear ODEs has been expanded. It includes not only various forms of fractional traveling wave transformations but also standard traveling wave transformations. Based on these methodological enhancements, a generalized SEsM algorithm has been developed to derive exact solutions of systems of FNPDEs. This algorithm provides multiple options at each step, enabling the user to select the most appropriate variant depending on the expected wave dynamics in the modeled physical context. Two specific variants of the generalized SEsM algorithm have been applied to obtain exact solutions to two time-fractional shallow-water-like systems. For generating these exact solutions, it is assumed that each system variable in the studied models exhibits multi-wave behavior, which is expressed as a superposition of two waves propagating at different velocities. As a result, numerous novel multi-wave solutions are derived, involving combinations of hyperbolic-like, elliptic-like, and trigonometric-like functions. The obtained analytical solutions can provide valuable qualitative insights into complex wave dynamics in generalized spatio-temporal dynamical systems, with relevance to areas such as ocean current modeling, multiphase fluid dynamics and geophysical fluid modeling. Full article

The remote sensing inversion of internal solitary waves (ISWs) enables the retrieval of ISW parameters and facilitates the analysis of their spatial variability. In this study, we utilize continuous optical imagery from the FY-4B satellite to extract real-time ISW propagation speeds throughout their evolution from generation to shoaling. ISW parameters are retrieved in the northern South China Sea based on the quantitative relationship between sea surface current divergence and ISW surface features in optical imagery. The inversion method employs a fully nonlinear equation with continuous stratification to account for the strongly nonlinear nature of ISWs and uses the propagation speed extracted from continuous imagery as a constraint to determine a unique solution. The results show that as ISWs propagate from deep to shallow waters in the northern South China Sea, their statistically averaged amplitude initially increases and then decreases, while their propagation speed continuously decreases with decreasing depth. The inversion results are consistent with previous in situ observations. Furthermore, a three-day consecutive remote sensing tracking analysis of the same ISW revealed that the spatial variation in its parameters aligned well with the abovementioned statistical results. The findings provide an effective inversion approach and supporting datasets for extensive ISW monitoring. Full article

Pipe-jacking construction technology is favored in urban construction due to its advantages of high safety and being a non-excavation technique. However, instability of the tunnel face often occurs due to unfavorable conditions, such as pipe jacking across the river channel, shallow soil cover, and improper control of the support pressure. In this study, we made a use of the limit balance method and mud–water balance theory. At this moment of passive damage and active destruction occurring at the pipe-jacking tunnel face, the general mathematical expressions of the tunnel-face support pressure (with lower limit value P_min and upper limit value P_max) are derived. In the non-river impact area and river impact area, the optimal value P_o of support pressure at the tunnel face is thus derived. Then, based on the Y25-Y26 pipe-jacking project across the Chu River channel in Hefei North District, a numerical simulation method is used to support further discussion. The results indicate that, when the river overburden is 3 m, the ultimate support pressure calculated by means of numerical simulation is 881.786 kN, and the optimal support ratio $\mathsf{\lambda }$ is taken in the interval of 1.0~1.5. Secondly, the upper limit value P_max, lower limit value P_min, and optimum value P_o calculated using the theoretical equations are 2669.977 kN, 309.910 kN, and 1044.870 kN, respectively. These results leads us to recommend setting the support pressure of the tunnel face in a reasonable range between the upper limit value P_max and the lower limit value P_min, to ensure that the tunnel-face support pressure and resistance during pipe jacking always remain in a balanced state. The relevant research results from this study provide an important technical guarantee for the successful implementation of the examined project and, at the same time, can serve as a reference example for similar projects. Full article

This study presents, for the first time, a detailed linear stability analysis (LSA) of bedform evolution under low-flow conditions using a one-dimensional vertically averaged and moment (1D-VAM) approach. The analysis focuses exclusively on bedload transport. The classical Saint-Venant shallow water equations are extended to incorporate non-hydrostatic pressure terms and a modified moment-based Chézy resistance formulation is adopted that links bed shear stress to both the depth-averaged velocity and its first moment (near-bed velocity). Applying a small-amplitude perturbation analysis to an initially flat bed, while neglecting suspended load and bed slope effects, reveals two distinct modes of morphological instability under low-Froude-number conditions. The first mode, associated with ripple formation, features short wavelengths independent of flow depth, following the relation F² = 1/(kh), and varies systematically with both the Froude and Shields numbers. The second mode corresponds to dune formation, emerging within a dimensionless wavenumber range of 0.17 to 0.9 as roughness increases and the dimensionless Chézy coefficient $C^{\ast }$ decreases from 20 to 10. The resulting predictions of the dominant wavenumbers agree well with recent experimental observations. Critically, the model naturally produces a phase lag between sediment transport and bedform geometry without empirical lag terms. The 1D-VAM framework with Exner equation offers a physically consistent and computationally efficient tool for predicting bedform instabilities in erodible channels. This study advances the capability of conventional depth-averaged models to simulate complex bedform evolution processes. Full article

This work explores modern mathematical avenues as part of fractional calculus research. We apply fractional dispersion relations to the fractional wave equation to numerically examine various formulations of the generalized fractional wave equation. The research explores Drinfeld–Sokolov–Wilson and shallow water equations as fundamental differential equations forming the basis of wave theory studies. This work presents effective methods to obtain the numerical solution of the fractional-order FDSW and FSW coupled system equations. The analysis employs Caputo fractional derivatives during studies of fractional orders. This study develops the new iterative transform technique (NITM) and homotopy perturbation transform method (HPTM) using Elzaki transform (ET) with a new iteration method and a homotopy perturbation method. The proposed techniques generate approximation solutions that adopt an infinite fractional series with fractional order solutions converging towards analytic integer solutions. The proposed method demonstrates its precision through tabular simulations of computed approximations and their absolute error values while representing results with 2D and 3D graphics. The paper presents the physical analysis of solution dynamics across diverse $ϵ$ ranges during a suitable time frame. The developed computational techniques yield numerical and graphical output, which are compared to analytic results to verify the solution convergence. The computational algorithms have proven their high accuracy, flexibility, effectiveness, and simplicity in evaluating fractional-order mathematical models. Full article

In this paper, we consider a modified Benjamin–Bona–Mahony (BBM) equation, which, for example, arises in shallow-water models. We discuss the well-posedness of the Dirichlet initial-boundary-value problem for the BBM equation. Our focus is on identifying a time-dependent source based on integral observation. First, we reformulate this inverse problem as an equivalent direct (forward) problem for a nonlinear loaded pseudoparabolic equation. Next, we develop and implement two efficient numerical methods for solving the resulting loaded equation problem. Finally, we analyze and discuss computational test examples. Full article

Accurately simulating wave breaking is crucial for modeling hydrodynamics over steep coral reef slopes, yet it remains a challenge for Boussinesq-type models like FUNWAVE-TVD. The model’s standard hybrid breaking mechanism, triggered by a fixed free surface elevation-to-depth ratio ($\eta /d>0.8$), often lacks physical sensitivity to local slope and wave conditions prevalent in reef environments and suffers from inaccuracies associated with using η as a direct proxy for wave height (H). This study introduces and validates a novel, enhanced hybrid breaking module within FUNWAVE-TVD, specifically designed to overcome these limitations on steep slopes. The core novelty lies in the synergistic implementation of two key components: (1) replacing the fixed threshold with a dynamic, physically-based criterion derived from the Modified Goda formula (MGO) by Rattanapitikon and Shibayama, which calculates the breaking wave height ($H_b$) based on local depth, slope, and deep-water wavelength; and (2) developing and applying a practical method, using the wave vertical asymmetry relationship proposed by Yu and Li, to dynamically convert the calculated $H_b$ into an equivalent breaking surface elevation threshold (${\eta }_b$). This derived dynamic threshold (${\eta }_b/d$) is then used to trigger the model’s existing switch from Boussinesq to Nonlinear Shallow Water Equations (NSWE), allowing for energy dissipation via shock-capturing while retaining the physical basis of the MGO criterion. The performance of this enhanced module was rigorously evaluated against five laboratory experiments of regular waves breaking on impermeable slopes ranging from mild (1:10) to extremely steep (1:1), contrasting results with the original FUNWAVE-TVD. The modified model demonstrates significantly improved accuracy (model skill increases ranging from 10.16% to 42.49%) compared to the original model for breaking location and wave height prediction on steeper slopes ($m\ge 1:6$). Conversely, tests on the 1:1 slope confirmed the inherent limitations of the MGO criterion itself under surging breaker conditions ($m\ge 1:2.3$), highlighting the applicability range. This work provides a validated methodology for incorporating slope-aware, dynamic breaking criteria effectively into hybrid Boussinesq models, offering a more robust tool for simulating wave processes on steep reef topographies. Full article

Gravity water waves in the shallow-ocean scenario described by generalized Boussinesq–Broer–Kaup–Whitham (gBBKW) equations are discussed. The residual symmetry and Bäcklund transformation associated with the gBBKW equations are systematically constructed. The time and space evolution of wave velocity and height are explored. Additionally, it is demonstrated that the gBBKW equations are solvable through the consistent Riccati expansion method. Leveraging this property, a novel Bäcklund transformation, solitary wave solution, and soliton–cnoidal wave solution are derived. Furthermore, miscellaneous novel solutions of gBBKW equations are obtained using the modified Sardar sub-equation method. The impact of variations in the diffusion power parameter on wave velocity and height is quantitatively analyzed. The exact solutions of gBBKW equations provide precise description of propagation characteristics for a deeper understanding and the prediction of some ocean wave phenomena. Full article