Numerical modelling of one dimensional problems for the shallow water equations on an adaptive moving meshes

This work provides an explicit, accurate, and physically interpretable formulation of the waveguide invariant (WI) for the shallow-water Pekeris model with a finite-impedance seabed, addressing limitations of existing approximations in the low-frequency regime. This is achieved by deriving an approximate closed-form expression for the intermodal WI based on the physically intuitive cycle-distance formula for modal group velocities. Its accuracy is established through comprehensive validation against full-wave KRAKEN simulations, showing close agreement with the benchmarks and a rapid decay of error beginning immediately above the modal cutoff. Three key analytical insights are derived. First, a rigorous lower bound-confirming that finite seabed impedance elevates the WI above its ideal baseline-is formally established and experimentally supported by large WI values from seabed-dominated, low-frequency data. Second, a compact closed-form expression is obtained for the limit as the grazing angle approaches zero, helping to explain the stability of far-field interference structures. Third, a continuous angular-dependent approximation for adjacent-mode WIs is presented. Experimental analysis further defines the framework's operational boundary, confirming its optimal use where seabed effects dominate over water-column stratification. Together, the derived formulation, analytical insights, and experimental evidence constitute a refined framework for understanding modal interference in shallow-water waveguides.

Data-driven approach to shallow water equation in ocean engineering: Multi-soliton solutions, chaos, and sensitivity analysis

We investigate the discrete energy behavior and long-time stability of a second-order Crank–Nicolson mixed finite element discretization for the shallow water equations with nonlinear bottom friction. The method combines a compatible BDM1–DG0 spatial approximation with a skew-symmetric formulation of the advective terms and a midpoint treatment of dissipative source terms. At the fully discrete level, we derive a precise mechanical energy identity showing that the scheme is energy-consistent;the discrete energy satisfies a balance law consisting of a nonnegative frictional dissipation term and a higher-order midpoint defect of the order O(Δt3). Although the method is not unconditionally energy-dissipative, we prove that strict Lyapunov decay holds under a mild CFL-type restriction on the time step. Furthermore, we establish uniform long-time boundedness of the discrete energy and asymptotic recovery of the continuous dissipation law as Δt→0. We also analyze the interaction between nonlinear solver tolerances and energy diagnostics, showing that the observed positive energy increments are controlled, non-accumulating, and intrinsic to the midpoint quadrature structure rather than solver artifacts. The scheme is proven to be precisely well balanced for lake-at-rest equilibria, including nonlinear bottom friction. Comprehensive numerical experiments confirm second-order temporal accuracy, robustness under friction, asymptotic monotonicity under time step refinement, and strict equilibrium preservation. The results provide a rigorous energy-diagnostic framework clarifying when Crank–Nicolson schemes are physically reliable despite the absence of unconditional discrete dissipation.

Abstract To improve the cost-effectiveness of modelling of wave interactions, a “numerical wavetank” is presented whose distinctive novel feature is its ability to couple both deep-water potential-flow and shallow-water models to controllable, prespecified wavemaker motion and beach topography. The coupling is in part obtained via a variational principle approach that guarantees important conservation properties and numerical stability. The model presented is the first fully nonlinear model to couple deep-water (discretised as finite elements) and shallow-water equations (discretised as finite volumes). Resulting simulations of wave generation, propagation and absorption by shallow-water-wave breaking are presented and analysed. A discussion is given on the efficacy of the novel approach.

Coastal erosion is increasingly influenced by anthropogenic alterations to the sediment cycle and morphological transformations. Traditional shallow water models often neglect the mechanical behavior of the seabed and its rheological response to hydrodynamic forcing, limiting their accuracy in forecasting erosion patterns. To address these limitations, this study extends the classical one-dimensional Saint-Venant (shallow water) model by incorporating effects of viscosity, frictional effects, sediment transport and viscoelasticity. The seabed is treated as a Kelvin–Voigt material, characterized by an elastic modulus and a viscous damping coefficient, to account for both immediate and time-dependent mechanical responses. Using the COMSOL Multiphysics platform, the evolution of the water column and seabed was simulated in six idealized case studies under various conditions, including changes in seabed topography and different frictional and dispersive regimes. The results demonstrate the influence of seabed topography, friction Sf, diffusion/dispersion regularization term E, and viscoelastic properties on wave seabed interactions and morphodynamic bed evolution (Exner-type). The inclusion of viscoelastic damping contributes to the stabilization of morphological evolution, mitigating abrupt changes in bathymetry and enhancing the physical realism of the simulations. The whole research aims to improve the prediction capabilities of erosion processes and advance the current modeling tools.

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.

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.

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

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

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

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.