Urban waterlogging presents a significant menace to urban operations and the livelihoods of residents. It is of the utmost necessity to establish an accurate and efficient early warning system. This research focuses on multi-source data fusion and intelligent models, and conducts a comprehensive exploration of the integration of data from meteorology, hydrology, geospatial information, and drainage systems. It processes multi-source data in real time through a distributed computing architecture. By applying methods such as the Horton infiltration formula, the isochron method, the Saint-Venant equations, and the Hazen-Williams formula, precise simulation of surface runoff and monitoring of urban drainage capacity are realized. Furthermore, the waterlogging risk level is dynamically adjusted according to real-time data. The experimental findings suggest that, when compared with AquaTalk, MIKE FLOOD, CAE S.p.A., and FIEDLER, the urban waterlogging early warning model proposed in this paper shows improvements in the accuracy, reliability, timeliness, and spatial precision of early warning. This offers a reference for urban waterlogging prevention and disaster relief.

River hydrodynamics are influenced by numerous factors that traditional models often fail to fully capture. Simulating complex hydrographs can benefit from parsimonious upscaling models, such as fractional derivative equations, that reduce the need to account for all variables. While fractional Saint-Venant equations (SVEs) have been mathematically explored, they lack clear physical interpretation and have not been applied in practical scenarios. This study introduces novel fractional-order Saint-Venant equations (FSVEs) for simulating river flow dynamics, addressing limitations in conventional modeling. Three models—constant, tempered, and variable time-fractional SVEs (CtFSVE, TtFSVE, and VtFSVE)—are developed to capture peak attenuation and tailing more effectively. Numerical experiments indicate that lower time-fractional derivative values enhance retention, producing a lower peak, delayed peak arrival, and pronounced late-time tailing. TtFSVE models transient tailing in hydrographs, VtFSVE captures transient evolution where inflow and outflow differ, and CtFSVE balances accuracy and simplicity with a single added parameter for various hydrographs. In the simulation of real-world hydrograph data, the fractional SVEs show high predictive accuracy. However, they should be regarded as effective proxy models that require parameter calibration and are not yet fully ‘plug-and-play’ predictive models. Comparative analysis with the Long Short-Term Memory (LSTM) machine learning model and distributed domain coupling model (DDCM) shows CtFSVE’s superior performance in capturing complex flow dynamics with minimal data, while field validation demonstrates its accuracy over traditional SVE, underscoring its practicality for complex river networks. The fractional engine shows promise as an effective tool for upscaling surface flow without the prohibitive burden of mapping detailed system heterogeneity.Supplementary InformationThe online version contains supplementary material available at 10.1038/s41598-025-23061-4.

Abstract We use the general framework of summation-by-parts operators to construct conservative, energy-stable, and well-balanced semidiscretizations of two different nonlinear systems of dispersive shallow water equations with varying bathymetry: (i) a variant of the coupled Benjamin-Bona-Mahony (BBM) equations and (ii) a recently proposed model by Svärd and Kalisch (2025) with enhanced dispersive behavior. Both models share the property of being conservative in terms of a nonlinear invariant, often interpreted as energy. This property is preserved exactly in our novel semidiscretizations. To obtain fully-discrete energy-stable schemes, we employ the relaxation method. Our novel methods generalize energy-conserving methods for the BBM-BBM system to variable bathymetries. Compared to the low-order, energy-dissipative finite volume method proposed by Svärd and Kalisch, our schemes are arbitrary high-order accurate, energy-conservative or -stable, can deal with periodic and reflecting boundary conditions, and can be any method within the framework of summation-by-parts operators including finite difference and finite element schemes. We present improved numerical properties of our methods in some test cases.

A well-balanced Finite Difference WENO Scheme for Shallow Water Equations with Moving water Equilibrium

Abstract. The applicability of one-dimensional (zonally invariant) harmonic and trapped wave theories for Inertia-Gravity waves to simulations on the mid-latitude β-plane is examined by comparing the analytical estimates in the geostrophic adjustment and Ekman adjustment problems with numerical simulations of the linearized rotating shallow water equations. The spatial average of the absolute differences between the theoretical solutions and the simulations, ϵ(t), is calculated for values of the domain's north-south extent, L, ranging from L=4 to L=60 (where L is measured in units of the deformation radius). The comparisons show that: (i) though ϵ oscillates with time, its low-pass filter, ϵLP(t), increases with time. (ii) In small domains, ϵLP(t) in harmonic theory is significantly smaller than in trapped wave theory, while the opposite occurs in large domains. (iii) The accuracy of the harmonic wave theory decreases with L for 0<L<20, while for L>20 the trend changes with time. (iv) The accuracy of the trapped wave theory increases with L in the geostrophic adjustment problem, while in the Ekman adjustment problem, its best accuracy is obtained when L≈30. (v) There is a range of L and t values for which no theory provides reasonable approximations, and this range is wider in the Ekman adjustment problem than in the geostrophic adjustment problem. Non-linear simulations of a multilayered stratified ocean show that internal inertia-gravity waves exhibit the same characteristics as the wave solutions of the linearized rotating shallow water equations in a single layer ocean.

This paper focuses on the numerical investigation of the shallow water equations (SWEs) with temperature gradients, commonly referred to as the Ripa system, using the finite volume upwind Conservation Element Solution Element (CE/SE) scheme. The inclusion of source-term on the right-hand side of the model introduces additional complexity, making the equations non-conservative and challenging to solve numerically. The primary objective is to develop a numerical scheme that effective ly handles non-conservative differential terms with accuracy and efficiency. To address these challenges, a finite volume upwind CE/SE technique is proposed. The method is tested on several numerical problems to evaluate its effectiveness, robustness, and stability, particularly in resolving discontinuities and shocks. The results of the proposed scheme are compared with those obtained using the standard CE/SE scheme.

Abstract. This study presents the High Order Prediction Environment (HOPE), an automatically differentiable, non-oscillatory finite-volume dynamical core for shallow water equations on the cubed-sphere grid. HOPE integrates five key features: (1) arbitrary high-order accuracy through genuine two-dimensional reconstruction schemes; (2) essential non-oscillation via adaptive polynomial order reduction in discontinuous regions; (3) exact mass conservation inherited from finite-volume discretization; (4) automatically differentiable and (5) GPU-native scalability through PyTorch-based implementation. Another innovation is the development of a two-way coupled ghost cell interpolation method. This approach incorporates information from adjacent panels on both sides of the boundary, thereby overcoming the integration instability inherent in one-sided ghost cell interpolation approaches when using high-order reconstruction. Leveraging the linear operator nature of this interpolation scheme, we optimized its computation: information exchange across the panel boundary is achieved through a single matrix-vector multiplication instead of iterative coupling, without accuracy loss. Numerical experiments demonstrate the capabilities of HOPE: The 11th-order scheme reduces errors to near double-precision round-off levels in steady-state geostrophic flow tests on coarse grids. Maintenance of Rossby–Haurwitz waves over 100 simulation days without crashing. A cylindrical dam-break test case confirms the genuinely two-dimensional WENO scheme exhibits significantly better isotropy compared to dimension-by-dimension approaches. Moreover, normalized conservation errors in total energy, total potential enstrophy, and total zonal angular momentum significantly reduce with increasing order of the reconstruction scheme. Two implementations are developed: a Fortran version for convergence analysis and a PyTorch version leveraging automatic differentiation and GPU acceleration. The PyTorch implementation maps reconstruction and quadrature operation to 2D convolution and Einstein summation respectively, achieving about 2× speedup on single NVIDIA RTX3090 GPU versus Dual Intel E5-2699v4 CPUs execution. This design enables seamless coupling with neural network parameterizations, positioning HOPE as a foundational tool for next-generation differentiable atmosphere models.

A Well-balanced Point-Average-Moment PolynomiAl-Interpreted (PAMPA) Method for Shallow Water Equations with Horizontal Temperature Gradients on Triangular Meshes

We present a new solution to the nonlinear shallow water equations (NSWEs) and show that it accurately predicts the swash flow due to obliquely approaching bores in large-scale wave basin experiments. The solution is based on an application of Snell’s law of refraction in settings where the bore approach angle $\theta$ is small. We therefore use the weakly two-dimensional NSWEs (Ryrie 1983 J. Fluid Mech. 129 , 193), where the cross-shore dynamics are independent of, and act as a forcing to, the alongshore dynamics. Using a known solution to the cross-shore dynamics (Antuono 2010 J. Fluid Mech. 658 , 166), we solve for the alongshore flow using the method of characteristics and show that it differs from previous solutions. Since the cross-shore solution assumes a constant forward-moving characteristic variable, $\alpha$ , we call our solution the ‘small- $\theta$ , constant- $\alpha$ ’ solution. We test our solution in large-scale experiments with data from 16 wave cases, including both normally and obliquely incident waves generated using the wall reflection method. We measure water depths and fluid velocities using in situ sensors within the surf and swash zones, and track shoreline motion using quantitative imaging. The data show that the basic assumptions of the theory (Snell’s law of refraction and constant- $\alpha$ ) are satisfied and that our solution accurately predicts the swash flow. In particular, the data agrees well with our expression for the time-averaged alongshore velocity, which is expected to improve predictions of alongshore transport at coastlines.

Smoothed particle hydrodynamics (SPH) has been widely adopted to solve both Navier–Stokes (NS) and shallow water equations (SWEs). This study provides a systematic comparison of NS-SPH and SWE-SPH across dam-break flows, fluid–structure interactions, and field-scale floods, using multiple hydrodynamic indicators including pressure, water level, velocity, structural forces, and inundation extent. The results yield several novel findings. We first identify a previously unreported phenomenon in dam-break simulations, in which the wavefront predicted by SWE-SPH propagates faster than that by NS-SPH in the initial phase but it is subsequently overtaken. This behaviour can be explained by the influence of a non-hydrostatic pressure neglected in the SWE formulation. NS-SPH shows better capability in resolving three-dimensional processes such as splashing and peak impact loads on structures, while SWE-SPH, despite its hydrostatic assumption, remains robust for large-scale floods. It requires nearly 80% less computation and shows low sensitivity to particle spacing. Importantly, we reveal that model accuracy depends not only on the underlying assumptions but also on the interplay between particle resolution and problem scale; in large-scale cases, SWE-SPH may outperform NS-SPH when vertical resolution is insufficient. These findings provide new insights for selection of SPH-based models according to problem scale, computational resources, and the required level of hydrodynamic details.

The dynamics of an electrified liquid surface are investigated using a shallow water model that is self-consistently coupled with an electrostatic solver. To account for the spatiotemporal variation of a curved liquid surface, the electric field on curved surfaces is calculated by solving a two-dimensional electrostatic equation using the weighted least squares (WLSQ) interpolation method. First, the WLSQ implementation is verified with analytical theory obtained from a Laplace solution assuming a half-cylinder liquid surface profile. Then, the coupled electrified shallow water model is used to study the instability of liquid surface perturbation as a function of the potential drop between an electrode and conducting liquid, surface tension, and gravity. We present a linear dispersion theory of liquid surface instability for long-wavelength perturbations, including the effects of liquid viscosity. Furthermore, the effects of multiple sinusoidal surface perturbations on the electrified liquid surface instability are investigated.

A robust structure-preserving surface reconstruction scheme for two-layer shallow water equations based on a relaxation model and an extension on adaptive moving triangles

This work presents a method to incorporate vertical velocity into a two-dimensional depth-averaged Shallow Water Equation (2DH SWE) model, thereby improving the calculation of particle trajectories in a Lagrangian Particle Tracking (LPT) framework. The resulting formulation couples Eulerian and Lagrangian approaches. The vertical velocity is also used to modify the dispersion terms in the LPT model. The proposed approximation is first validated—without particle transport—by comparison with Hyperbolic–Elliptic and Hyperbolic-Relaxed Non-Hydrostatic Pressure (NHP) models. The differences between models are minor, confirming the suitability of the vertical velocity approximation for shallow flow problems. Subsequently, the method is applied to particle transport scenarios, demonstrating that including vertical velocity yields more realistic particle trajectories in complex flow situations. • An efficient approach for vertical velocity in 2D shallow models is proposed. • Vertical velocity approach improves Lagrangian particle transport in shallow flows. • Validated against non-hydrostatic models with good agreement in steady cases. • Reduces particle buildup and improves vertical transport realism in 2D flows.

Abstract This paper presents a new hybrid deterministic‐stochastic river morphodynamics numerical modeling approach that integrates a hydrodynamic model (deterministic) with a bed morphodynamic model (deterministic) and a bank erosion model (Markovian stochastic). The hydrodynamic model solves the Shallow Water Equations and the standard ‐ model for turbulence closure. Bedload transport is estimated using the Meyer‐Peter and Müller formula, and bed evolution is solved using the Exner equation. The Markovian stochastic bank erosion model uses a new method to evaluate bank erosion risk. The approach was applied to a meander bend cutoff event in the Maiqu River on the Tibetan Plateau. Sixteen different bank‐material critical shear stress cases were considered, representing highly erodible banks ( Pa) to resistant banks ( Pa). Ten statistical realizations were performed for each case with different bank‐material erodibility to obtain ensemble‐averaged results. The flow field and bed evolution in the cutoff channel suggest that the model can successfully simulate bank erosion processes during cutoff channel evolution, and bank topographic irregularities are reasonably captured. A newly introduced calibration parameter is the ratio of mesh size to the coupling period between the bank erosion model and the hydrodynamic and morphodynamic model. Unlike the erosion rate calibration parameters used by traditional bank erosion models, the present model requires estimating the size of a typical slump failure to determine the size of the computational mesh. This new modeling approach enhanced existing tools for fluvial geomorphologists and river engineers focused on bank erosion with real‐world complex geometric features under intricate hydraulic conditions.

This study presents a mathematical model to investigate the dynamics of gravity currents in channels with vegetation and varying cross section (CS) geometries. The model, based on a system of shallow-water equations, incorporates a vegetation-induced drag force of the form Fd∼|u|λ (where u is the current speed and λ is constant) allowing for the analysis of flow behavior under different configurations, including fixed volume (lock-release) and constant influx conditions. We focused on power-law CSs defined by f(z)=zγ and explored the influence of both the channel geometry and the drag parameter, λ, through analytical similarity solutions and numerical computations. Our findings reveal a complex interplay between drag and channel shape. For fixed-volume flows, the propagation distance of the current increases with γ, while for certain constant-influx cases, the trend is inverted, with the fastest propagation occurring in rectangular channels (γ=0). The propagation rate for the constant-height influx case and the Boussinesq lock-exchange problem was found to be dependent solely on the drag parameter λ and independent of the channel's geometry. Furthermore, we demonstrate that flows with non-constant drag can be accurately approximated using an averaged drag value. Ultimately, this study establishes a quantitative relationship that is essential for accurately modeling and controlling the propagation of density-driven flows in complex environments where both channel geometry and vegetation-induced drag are variable.

This study presents numerical simulations of the shallow water equations (SWEs) for the Bay of Bengal (BoB) using the Lagrange–Galerkin method (LGM) on a triangular mesh, with the transmission boundary conditions (TBCs). To examine the positional sensitivity of the transmission boundaries, the simulations are conducted with TBCs imposed at two distinct locations within the Bay of Bengal domain. The computed total mass and -norm of the surface elevation demonstrate that the transmission boundary condition performs efficiently and exhibits minimal dependence on its placement. These results indicate that the TBC is well-suited for modeling open-sea boundaries, ensuring smooth wave propagation without artificial reflection. The study serves as a foundational step toward developing an accurate and stable storm surge prediction framework for the Bay of Bengal using the Lagrange–Galerkin approach.

To improve the accuracy of second-order cell-centered finite volume method in near-boundary regions for solving the two-dimensional shallow water equations, a numerical scheme with globally second-order accuracy was proposed. Having the primary objective to overcome the challenge of accuracy degradation in near-boundary regions and to develop a robust numerical framework combining high-order accuracy with strict conservation, the key research objectives had been as follows: Firstly, a physical variable reconstruction method combining a vertex-based nonlinear weighted reconstruction scheme and a monotonic upwind total variation diminishing scheme for conservation laws was proposed. While the overall computational efficiency was maintained, linear-exact reconstruction in near-boundary regions was achieved. The variable reconstruction in interior regions was integrated to achieve global second-order accuracy. Subsequently, a flux boundary condition treatment method based on uniform flow was proposed. Conservative allocation of hydraulic parameters was achieved, and flow stability in inflow regions was enhanced. Finally, a series of numerical test cases were provided to validate the performance of the proposed method in solving the shallow water equations in terms of high-order accuracy, exact conservation properties, and shock-capturing capabilities. The superiority of the method was further demonstrated under high-speed flow conditions. The high-precision numerical model developed in this study holds significant value for enhancing the predictive capability of simulations for natural disasters such as flood propagation and tsunami warning. Its robust boundary treatment methods also provide a reliable tool for simulating free-surface flows in complex environments, offering broad prospects for engineering applications.

This paper presents a mathematical analysis of a coupled system for modeling sediment transport and dune evolution in coastal environments. The model combines the Shallow Water Equations (SWE) with the Long-Term Dune Dynamics (LTDD) equation, incorporating viscosity and modified pressure terms. Using methods of mathematical analysis, we prove the existence and uniqueness of solutions for the system. We also provide estimates for its dimensionless form and conclude by presenting a homogenized version of the combined model.

Abstract. This paper presents a Lagrangian model for particle transport driven by a two-dimensional (2D) shallow water model, assuming that the particles have negligible mass and volume, are located at the free surface, and without interactions between them. Particle motion is based on advection and turbulent diffusion, which is added using a random-walk model. The equations for particle advective transport are solved using the flow velocity provided by a 2D shallow water solver and an online first-order Euler method, an online fourth-order Runge–Kutta method and an offline fourth-order Runge–Kutta method. The primary objective of this work is to present the capabilities of the new Lagrangian particle transport (LPT) model, while also providing an analysis of the accuracy and computational efficiency of the numerical schemes and their implementation for particle transport. To verify the accuracy and computational cost, several test cases inspired by laboratory set-ups are simulated. In this analysis, the Euler online method provides the best compromise between accuracy and computational efficiency. Finally, a localized precipitation event in the Arnás catchment is simulated to test the model's capability to represent particle transport in overland flow over irregular topography.

River migration and anthropogenic controls on hydrological processes may play important roles in estuarine system transformations and nutrient diffusion. We used a two-dimensional shallow water equation hydrodynamic water quality model to simulate total nitrogen (TN) transport under the situations of river migration and the “Water-Sediment Regulation Scheme” (WSRS). The results showed the following: (1) River migration changed the diffusion direction of high-TN-concentration water in the YRE from the east–west diffusion in 2009 to the north–south diffusion in 2019. (2) In the years the WSRS was active, the maximum diffusion distance of high-concentration-TN water is basically the same as that of the plume edge. In 2009 and 2019, it was 30 km in the southeast of the estuary and 26.5 km in the north. Concentrations of 0.5 mg/L and 1.05 mg/L in 2009 and 2019 can be used as the threshold for judging the farthest distance of diffusion. (3) In the years without the WSRS, the TN concentration in the YRE from June to July was generally lower than the same period in 2019, and the northward diffusion distance of high-concentration-TN water in 2017 was only 10% of that during the WSRS in 2019. (4) Runoff determines the diffusion range of TN in the YRE. The average runoff during the WSRS in 2019 was 6.88 times that of the same period in 2017, and the high concentration diffusion distance of TN in 2019 was 10 times that of 2017. Changes in estuary morphology determine the diffusion direction of nutrients. The results of this paper are helpful to further understand the nutrient diffusion law of estuaries and coasts under the influence of different factors, and to provide reference for the protection of water quality safety.