ABSTRACT Algeria, like many Mediterranean regions, is highly vulnerable to flooding, which often occurs in catastrophic forms. The control and management of such events represent major challenges to both economic and social development. Since 2012, the rehabilitation of 8 km of the Saïda Wadi in northwestern Algeria has played a significant role in controlling recurrent floods. This study aims to analyze and compare hydraulic conditions before and after the rehabilitation of the Saïda Wadi. Flood simulation is essential for predicting wave arrival times and delineating flood-prone areas, particularly in ungauged catchments. HEC-RAS 2D, coupled with a Geographic Information System (GIS), is applied to model unsteady flows governed by the Saint-Venant equations. The results demonstrate a notable improvement following the rehabilitation works, with reductions of 29.4% in water depth, 8.97% in floodplain extent, and 43.9% in maximum flow velocity.

In this paper, a well-balanced and positivity-preserving scheme for the nonconservative two-layer shallow water equations is developed in the framework of the finite volume method. To address the challenges posed by wet–dry fronts, the focus of our study is on reconstructing them for each layer to ensure a well-balanced property. To this end, a new numerical discretization and a special wet–dry front reconstruction are proposed. In addition, the draining time method is employed to ensure the positivity of the water depth. We prove that the proposed scheme is both well-balanced in steady-state solutions and positivity-preserving. Finally, numerical experiments demonstrate the robustness of the scheme.

Abstract We build a multilayer magnetohydrodynamic shallow-water model to study the thickness and shape of the solar tachocline. This allows us to include characteristics of both the overshoot and the radiative parts of the tachocline. The equations derived include equilibrium in latitude among Coriolis, pressure gradient, and magnetic curvature stresses for each layer, and magnetohydrostatic equilibrium in the radial direction. In each layer, the total mass is conserved; mass is redistributed for different amplitudes and latitude positions of toroidal bands, thus producing variations in tachocline shape and thickness with solar cycle phases. While we solve here for equilibrium of two layers, the equations can be readily generalized for additional layers. In pure hydrodynamic tachocline with no differential rotation, thickness and shape are independent of latitude. With differential rotation and/or magnetic fields, the tachocline is, in general, oblate in equatorial regions but prolate in polar latitudes. A local bump occurs at the poleward side of tachocline toroidal band. Hence, depending on latitude-location and amplitude of magnetic band as function of solar cycle, the local bump drifts equatorward trailing the magnetic field. Oblateness and prolateness are much larger in the overshoot than in the radiative layer, due to its lower effective gravity. Our results can provide guidance for interpreting helioseismic estimates of variations in tachocline shape and thickness in latitude, including upper limits to banded toroidal field amplitudes.

Hydropower is one of the most commercially developed renewable energy sources globally. This study assessed the hydropower generation potential of the Pwalugu River section of the White Volta River in Ghana using the depth-averaged shallow-water equations (SWEs) hydrodynamic model. Various topographic models of the seabed were engineered to assess the topographic response to hydrodynamic flow, including the formation of whirlpools. The modelled equations were discretised using the Lax-Wendroff iteration scheme, and Python 3.07 was employed to implement the algorithm. The computed hydropower associated with the flowing fluid showed a significant difference before and after interacting with the engineered bottom-topographic structures. The topographic response to the flow led to increased mass flow rate, instigated by the developed flowing whirlpool, which served as a dynamic energy storage system for the flow channel. The topographic model with two mounts arranged along the river channel could produce power within the range of 1.8 MW - 2.9 MW. These were observed at locations x = 240 m, x = 481 m, and x = 962 m from the source of disturbances. The study, therefore, showed that the Pwalugu River section of White Volta River has the potential of generating hydropower if turbines are sited at these locations to enhance power generation capacity for the electrification of rural communities and the utlisation of the spillage from the Bagre dam in Burkina Faso, which causes perennial flooding in low-lying communities along the White Volta and the Black Volta in Ghana.

ABSTRACT Urban pluvial flooding driven by localized extreme rainfall increasingly exceeds the capacity of metropolitan drainage systems. Manhole surcharge overflow interacting with surface runoff produces complex inundation dynamics that are difficult to capture in real time. High-fidelity 1D–2D coupled numerical models are too computationally expensive for operational deployment, whereas purely data-driven deep-learning surrogates, although fast, do not enforce conservation laws or provide physically interpretable behaviour. We propose a modular coupled physics-informed neural network (MC-PINN) that bridges this gap. MC-PINN consists of a 1D PINN for sewer network flow governed by the Saint-Venant equations and a 2D PINN for surface flow governed by shallow-water equations, coupled through manhole overflow acting as a boundary condition. Mass continuity is enforced at the interface and momentum is strongly constrained via physics-based residual losses, enabling a hybrid surrogate with near real-time inference potential. To demonstrate feasibility, we present a manufactured-solution proof-of-concept in which a 1D advection PINN and a 2D diffusion PINN are coupled via a shared interface signal. The example shows that MC-PINN can jointly approximate both PDE fields and a consistent interface, supporting the mathematical viability of the modular coupling.

A conservative staggered discontinuous Galerkin method on triangular grids for the shallow water equations

This paper follows on from previous work on the development of a shallow-water model to simulate partial wetting phenomena at large scale [1]. The originality of this model lies in the use of a scalar quantity (called the color function), advected by the liquid film speed, to locate the position of the contact line between the film and the wall. The advantage of this function is that it allows to easily calculate (via its gradient) the macroscopic resultant of the capillary forces acting in the vicinity of the contact line, without the need to introduce a disjoining pressure model as in [2, 3, 4] that is necessarily associated with a regularization parameter h * for numerical purpose. Using the hyperbolic structure of the system of equations, we show that we can build an HLLC-type solver that preserves the positivity of the film thickness and the maximum principle on the color function under a CFL-like condition. Finally, a series of 1D and 2D numerical results demonstrating the robustness and accuracy of the proposed numerical method is presented.

ABSTRACT Computational analysis of river flow remains a cornerstone of mathematical hydrological research, particularly in ecologically sensitive and morphologically complex environments. This study introduces an integrated workflow that combines aerial image‐based obstacle detection in the riverbed with two‐dimensional (2D) depth‐averaged simulations to assess meso‐scale flow structures in morphologically complex lowland rivers. It presents a novel framework for modelling shallow rivers that is able to capture meso‐ to micro‐scale flow dynamics influenced by instream boulders, spatially heterogenous vegetation types, and bottom roughness. In situ measurements from four lowland river reaches in Lithuania were used to parameterize and calibrate flow resistance based on two boulder types and three distinct vegetation cover classes. A finite element (FE) approach was applied to solve the shallow water equations (SWE) with site‐specific roughness coefficients. The model simulations were validated against observed velocity data, with relative errors of less than 20% at most observation points. The proposed approach demonstrated strong potential for accurate reproduction of fine‐scale hydrodynamic behaviour in shallow lowland rivers, offering a flexible environment and scalable platform for future integration with automated AI‐based remote sensing data. The precision and adaptability of the model make it a valuable tool for ecological assessments, flood risk studies, and restoration planning.

Icing on aircraft engine inlet components can impair performance and safety. This study numerically simulates the icing process on a full-scale engine inlet strut under real operating conditions. A modified Shallow-Water Icing Model (SWIM) and an automatic icing type-detection algorithm were employed to predict water film flow and icing phase changes, calculated with a self-developed program. The models' accuracy was validated through comparison of simulated three-dimensional ice shapes on a NACA0012 airfoil with experimental results. Based on this validation, we conducted a numerical analysis of the unsteady icing process of the strut, focusing on the effects of inflow temperature, velocity, and other factors on icing. The results demonstrate that ice horn formation significantly influences ice shape development, with the maximum local water collection efficiency shifting from the stagnation point to the ice horns over time, thereby accentuating the dual-horn characteristic. Inflow velocity impacts icing differently depending on temperature. At -10°C, low velocities produce dual-horn ice, while high velocities yield streamlined ice due to aerodynamic heating, reducing ice thickness at the stagnation point by 18.5%. At -20°C, low velocities result in streamlined ice, whereas high velocities promote dual-horn ice due to higher airflow recovery temperatures, leading to a 121.9% increase in ice thickness at the stagnation point.

Multi-model physics informed neural networks to the shallow water equations for cosine bell advection

A structure-preserving nonstaggered central scheme for shallow water equations with wet–dry fronts and Coriolis force on triangles

We present a novel approach to handle open boundary conditions for a Boussinesq-type wave model coupled with the nonlinear shallow water equations. Traditional methods for managing open boundaries — such as sponge layers and source functions — are computationally intensive and require ad hoc calibration. To address this, we reformulate the Boussinesq equations as a system of conservation laws with nonlocal flux and a rapidly decaying source term. This reformulation is adapted to generate waves at the boundary of the numerical domain, from surface elevation data in situations where both incoming and outgoing waves are present. The proposed numerical scheme employs a MacCormack prediction–correction strategy combined with finite volume and finite difference methods, preserving key physical properties and ensuring stability. Comparison with laboratory experiments demonstrates that our approach avoids boundary reflection issues. In particular, it is able to accurately reproduce infragravity waves associated with a random wave field propagating over a sloping beach. This work opens important perspectives for improving phase-resolving coastal wave models, with the aim of forecasting complex random wave conditions in natural environments. • Wave generation and absorption is hard to manage in Boussinesq-type models. • Sponge layers and source function methods are computationally costly and need tuning. • We propose a rigorous, efficient boundary treatment with no calibration required. • Comparisons with experiments show accurate handling of incoming and outgoing waves.

High order asymptotic preserving well-balanced finite difference WENO schemes for all-speed rotating shallow water equations in the quasi-geostrophic limit

Spectral instability of peakons for a class of cubic quasilinear shallow-water equations

This work presents an analytical study of several partial differential equations commonly used to model physical phenomena such as heat diffusion, wave propagation, and fluid flow. Emphasis is placed on the use of trigonometric functions to derive exact or synthetic solutions. The heat equations are then is examined using Fourier series and a complex-variable approach. The linearized Saint-Venant equations are then analyzed to describe shallow water wave propagation. The Burgers, in both inviscid and viscous forms, is used to illustrate nonlinear effects, damping, and shock formation. Finally, the Korteweg-de Vrie equation is discussed through its soliton solution, highlighting the balance between nonlinearity and dispersion. These results underline the importance of analytical and trigonometric methods in the modeling of thermal and hydraulic phenomena.

Abstract Targeting the issues of insufficient predictive ability and inefficient computation in two‐dimensional shallow water equations (2D‐SWEs), this study deeply couples the mesh and hydrodynamic boundary, constructing multiple 2D hydrodynamic models (run 2,640 times). This study proposes and validates, for the first time, a hydrodynamic boundary classification framework (strongly and weakly constrained boundary) based on constraint strength, and systematically quantifies the uncertainty and computational performance of various meshes under different boundaries. Two reasons for insufficient predictive ability were identified: improper boundary setting and mesh selection. Through numerical analysis and theoretical derivation, it was demonstrated that appropriate boundary and mesh choices can reduce the uncertainty of 2D‐SWEs. Calculation results indicate that the strongly constrained boundary (water level) significantly reduces model errors; the Unstructured Quadrilateral Mesh (UQM) demonstrates excellent computational robustness, with cumulative deviations in simulated water levels reduced by 30 ∼ 90% compared to the Unstructured Triangular Mesh (UTM). Additionally, the impact of hydrodynamic boundary types on computational efficiency varies with changes in mesh density, type, topography, and other parameters, but the impact of boundary type on computational efficiency does not exceed 4%. UQM improves computational efficiency by 55% ∼ 130% compared to UTM. Additionally, this study identifies the “impossible triangle” region in quadrilateral meshes, which constrains the generation of high‐quality meshes. Taking into account the different grid computational performance, flux propagation characteristics, grid quality, and the convenience of large‐scale applications, it is recommended to primarily use UQM in river channels and UTM in floodplains.

The application of two-dimensional (2D) hydrologic and hydraulic modeling tools is increasing for overland flow simulation, as they represent spatial changes in depth, velocity, and flow conditions more accurately. Recently, the US Army Corps HEC-HMS (Hydrologic Engineering Center Hydrologic Modeling System) added the capability to import an unstructured 2D mesh, which enables the routing of excess precipitation across the mesh, as a fully distributed hydrological model. In HEC-HMS, the 2D diffusion-wave component functions as a hydrologic transform representing overland flow routing. In contrast, HEC-RAS 2D (Hydrologic Engineering Center-River Analysis System), initially applied to river flow simulation, can apply either the 2D shallow-water equations or the 2D diffusion-wave option. Similarly to HEC-HMS, HEC-RAS also includes rain-on-grid (RoG) capability and infiltration algorithms, and in this fashion has some hydrological modeling capabilities. Still, while HEC-HMS is capable of representing extended-period hydrological simulations, HEC-RAS hydrological capabilities are limited to event-based simulations, as there are no provisions to represent abstractions such as evapotranspiration or groundwater/baseflow contributions together. Studies performing a direct comparison between the HEC-HMS RoG and HEC-RAS RoG approaches for representing urban hydrology remain scarce. This study aims to fill that gap by assessing their performance in Moore’s Mill Creek Watershed, in Lee County, Alabama, with a focus on continuous rainfall-runoff modeling. Both models run on the same unstructured mesh and use identical rainfall, terrain, land-use, and soil data. Model simulations are compared over an extended period to evaluate simulated depth, velocity, and flow hydrographs against field observations. The comparison shows HEC-HMS’s superior performance for extended simulation and provides practical guidance on parameter alignment, data needs, and tool selection.

Traditionally, the shallow-water equations have been formulated and developed within the Eulerian framework for studying shallow-water wave problems. In this paper, we present a Lagrangian-based approach based on Hamilton’s variational principle to derive an extended displacement shallow-water equation (EDSWE). Using elliptic functions, we obtain exact traveling wave solutions of the resulting EDSWE. The conditions for the formation of various wave types—including cnoidal waves, looped waves, and peaked waves—are systematically analyzed and summarized. The proposed displacement method, grounded in the Lagrangian description, provides an analytical framework for hydrodynamic problems and can be applied to symplectic formulations in fluid mechanics.

Baroclinic wave resonance, particularly Rossby waves, has attracted great interest in ocean and atmospheric physics since the 1970s. Research on Rossby wave resonance covers a wide variety of phenomena that can be unified when focusing on quasi-stationary Rossby waves traveling at the interface of two stratified fluids. This assumes a clear differentiation of the pycnocline, where the density varies strongly vertically. In the atmosphere, such stationary Rossby waves are observable at the tropopause, at the interface between the polar jet and the ascending air column at the meeting of the polar and Ferrel cell circulation, or between the subtropical jet and the descending air column at the meeting of the Ferrel and Hadley cell circulation. The movement of these air columns varies according to the declination of the sun. In oceans, quasi-stationary Rossby waves are observable in the tropics, at mid-latitudes, and around the subtropical gyres (i.e., the gyral Rossby waves GRWs) due to the buoyant properties of warm waters originating from tropical oceans, transported to high latitudes by western boundary currents. The thermocline oscillation results from solar irradiance variations induced by the sun’s declination, as well as solar and orbital cycles. It is governed by the forced, linear, inviscid shallow water equations on the β-plane (or β-cone for GRWs), namely the momentum, continuity, and potential vorticity equations. The coupling of multi-frequency wave systems occurs in exchange zones. The quasi-stationary Rossby waves and the associated zonal/polar and meridional/radial geostrophic currents modify the geostrophy of the basin. Here, it is shown that the ubiquity of resonant forcing in (sub)harmonic modes of Rossby waves in stratified media results from two properties: (1) the natural period of Rossby wave systems tunes to the forcing period, (2) the restoring forces between the different multi-frequency Rossby waves assimilated to inertial Caldirola–Kanai (CK) oscillators are all the stronger when the imbalance between the Coriolis force and the horizontal pressure gradients in the exchange zones is significant. According to the CK equations, this resonance mode ensures the sustainability of the wave systems despite the variability of the forcing periods. The resonant forcing of quasi-stationary Rossby waves is at the origin of climate variations, as well-known as El Niño, glacial–interglacial cycles or extreme events generated by cold drops or, conversely, heat waves. This approach attempts to provide some new avenues for addressing climate and weather issues.

We investigate the Lagrange formulation for the one-dimensional Saint Venant–Exner system. The system describes shallow-water equations with a bed evolution, for which the bedload sediment flux depends on the velocity, Qt,x=Agum,m≥1. In terms of the Lagrange variables, the nonlinear hyperbolic system is reduced to one master third-order nonlinear partial differential equation. We employ Lie’s theory and find the Lie symmetry algebra of this equation. It was found that for an arbitrary parameter m, the master equation possesses four Lie symmetries. However, for m=3, there exists an additional symmetry vector. We calculate a one-dimensional optimal system for the Lie algebra of the equation. We apply the latter for the derivation of invariant functions. The invariants are used to reduce the number of the independent variables and write the master equation into an ordinary differential equation. The latter provides similarity solutions. Finally, we show that the traveling-wave reductions lead to nonlinear maximally symmetric equations which can be linearized. The analytic solution in this case is expressed in closed-form algebraic form.