A GPU-accelerated DOT scheme for the coupled 2D shallow water and Exner equations

Abstract. This work presents SERGHEI-SWE-RE, a performance-portable, parallel model that couples a fully dynamic two-dimensional Shallow Water Equation (SWE) solver with a three-dimensional Richards Equation (RE) solver within the Kokkos framework to simulate surface–subsurface flow exchange. The model features a modular architecture with sequential coupling strategy, supporting both synchronous and asynchronous executions of surface and subsurface modules. The SERGHEI-SWE-RE model is validated against five benchmark problems incorporating stationary and fluctuating free-surface tests, a tilted v-catchment, a lateral-flow slope without ponding, and a heterogeneous superslab. The results demonstrate good agreement with established models. Asynchronous coupling reduces wall-clock time by up to about 75 % in the superslab case while preserving simulation accuracy. Strong and weak scaling tests on multiple Intel Xeon CPUs and NVIDIA GPUs reveal robust portability, with near-ideal RE scaling and less-satisfactory SWE scaling at high GPU counts, suggesting future improvements on differentiated meshes or more advanced domain decomposition strategies. Overall, the results presented establish SERGHEI-SWE-RE as an efficient, flexible and scalable model for integrated surface-subsurface flow simulations.

Flood and flash flood events can generate severe hydraulic actions on civil structures, requiring modeling strategies able to link flow features to structural damage. This paper proposes a two-scale numerical framework based on advanced finite element modeling to assess the vulnerability of structures subjected to inundation and flood-driven impact. At the macroscale, the flood propagation and the interaction with the built environment are simulated through the depth-averaged Shallow Water Equations, adopting a time-explicit interface treatment to capture the evolution of the free surface. The macroscale model provides time-dependent water depth and flow velocity along the external surfaces of the structure, which are then used to derive hydrostatic and hydrodynamic actions, also in comparison with code-based formulations. At the mesoscale, these actions are transferred to a detailed structural model to investigate the nonlinear mechanical response of the building. Structural components are described through a coupled damage–plasticity constitutive law, enabling the prediction of stiffness degradation, cracking-driven damage patterns, and the identification of the most critical structural zones under flood loading. The proposed workflow is finally applied to a real structure located in the municipality of Cosenza (Italy), demonstrating the capability of the approach to combine hydraulic intensity measures with physics-based structural damage assessment, supporting scenario analyses and risk mitigation evaluations.

The Traveling Wave Problem for the Shallow Water Equations: Well-posedness and the Limits of Vanishing Viscosity and Surface Tension

Hybrid nature-based and engineered solutions for wave resonance reduction

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

On the application of the two-time stepping Euler forward Runge-Kutta schemes to the shallow water equations: Global truncation error, numerical viscosity, consistency, energy conservation, inertial stability and phase error

A theoretical study is made of steady, subcritical (Froude number $F \lt 1$ ) two-dimensional free-surface flow due to a uniform stream flowing over smooth, locally confined bottom topography of large horizontal extent ( $L \gg 1$ ) and finite peak height ( $\varepsilon = O(1)$ ). In earlier work, this flow was analysed based on the nonlinear shallow-water equations which neglect the effects of dispersion altogether. This so-called hydraulic theory predicts a steady disturbance confined in the vicinity of the topography if $\varepsilon$ is below a critical value $\varepsilon _{\textit{crit}}(F)$ . The present asymptotic analysis of the full potential flow equations focuses on how dispersive effects (controlled by $\mu = 1/L \ll 1$ ) influence this steady state, particularly in regard to a steady short-scale radiating wave downstream that is ignored by hydraulic theory. Utilizing exponential asymptotics, it is shown that as $\varepsilon$ is increased this dispersive wave, whose amplitude is formally exponentially small with respect to $\mu$ , grows sharply and ultimately it becomes comparable with the hydraulic wave disturbance when $\varepsilon$ approaches $\varepsilon _{\textit{crit}}$ . Thus, the nonlinear shallow-water equations break down in the vicinity of $\varepsilon _{\textit{crit}}$ regardless of $\mu \ll 1$ . The asymptotic results are supported by numerical solutions of the full potential flow theory, which also reveal a limiting $\varepsilon$ , $\varepsilon _{\textit{lim}} \approx \varepsilon _{{ crit}}$ , above which steady wave responses cannot be computed. For $\varepsilon$ just below $\varepsilon _{\textit{lim}}$ , the downstream wave resembles a steep steady Stokes periodic wave, while for $\varepsilon$ slightly above $\varepsilon _{\textit{lim}}$ unsteady computations suggest that the downstream disturbance steepens and breaks.

Abstract Studying exoplanet flow and variability requires solving atmospheric dynamics equations accurately. Here we use the shallow-water equations to evaluate and employ Dedalus 3 , a spectral method-based software package for solving differential equations. A well-known jet instability test is used for the evaluation; then, the package is used to investigate the nonlinear evolution of observed, Jupiter’s zonal (east–west) jets; finally, the package is used to compare hot-Jupiter flows with different initial conditions. Our results indicate that Dedalus 3 can be a useful tool for investigating planetary flow dynamics, but careful testing and execution are necessary for each problem.

We present direct numerical simulations of planar intrusions from a constant source into a linearly stratified ambient fluid for Reynolds numbers between $200$ and $5000$ , inlet widths $W\geqslant 2.2 \sqrt {Q/N}$ and ambient layer thicknesses $H_a$ between $W$ and $20\sqrt {Q/N}$ , where $Q$ is the supply rate (area per unit time) and $N$ is the buoyancy frequency. Across this broad parameter space, the intrusions form a universal self-similar shape with a constant thickness of approximately $2.2\sqrt {Q/N}$ at the source tapering towards a tip that propagates at a constant speed of approximately $0.7\sqrt {NQ}$ . This broad-scale structure does not change regardless of whether the intrusions are subcritical or supercritical relative to internal waves. The perturbations to the ambient resulting from the intrusive flow appear as a near-universal uplift/depression of isopycnals immediately above/below the intrusion, upstream blocking ahead of the intrusion and, for subcritical intrusions, columnar disturbances. A moderate-amplitude wave train is also formed on the surface of subcritical intrusions. This appears at approximately the mid-length of the intrusion, with the waves propagating towards the tip. We also compare our results with the solution of the Mei shallow-water model. The comparison is poor and we rederive the model, carefully examining the underlying assumptions against the simulation data. The only assumption that is violated is that the ambient density is unperturbed. We present extensions to the model allowing for a density perturbation based on (i) simple data fits and (ii) a solution to the Dubreil-Jacotin–Long equation for shallow ambients. These significantly improve the predictive ability of the model for this geometry.

Modeling the interaction between storms such as tropical cyclones over the ocean is highly relevant in the fields of ocean, coastal, and offshore engineering, as it enables the forecasting of loads and risks, as well as the mitigation of damages caused by these extreme events in coastal areas. The present study aims to investigate the water elevation caused by the formation of tropical cyclones at sea, using the Finite Element Method to solve the Shallow Water Equations. The wind generated by the cyclone was simulated using an analytical model that accounts for meteorological parameters and their interaction with water bodies, modeled as tangential stresses applied to the free surface. After verifying the shallow water flow solver with a benchmark problem, simulations of wind-induced flows were performed considering two different domains: one with constant bathymetry and another representing a linearly varying coastal region. The effects of model parameters describing the tropical cyclone were investigated and discussed. As a result, it was found that the most significant factors influencing water elevation are the radius at which the cyclone's maximum wind speed occurs and the pressure difference between the storm center and its periphery, leading to considerable water elevation in the most severe cases. In contrast, the translational speed of the cyclone had little impact on water elevation, whereas smaller wind incidence angles produced higher elevations along the coastline. Finally, an additional investigation is presented, considering a more complex domain configuration to illustrate the potential applications of the proposed model.

Abstract This study around shallow water wave dynamics, deriving a KP equation for weakly nonlinear flows and evaluating the transverse stability of its soliton solutions. Using multi-scale asymptotics, the work rigorously derives a KP equation from the incompressible, inviscid, weakly nonlinear shallow-water model. Perturbation and eigenfunction analyses yield dispersion relations and stability criteria for solitary waves under transverse disturbances. The findings confirm lowest-order soliton stability and show consistency with related plasma and shallow-water studies, demonstrating the universality of the theoretical framework. The study advances understanding of nonlinear shallow-water dynamics and supplies theoretical support for wave prediction and engineering applications.

Abstract This paper presents a high-precision differentiable numerical framework based on Automatic Differentiation (AD) for solving nonlinear differential equations in ocean dynamics. Using the Korteweg-de Vries (KdV) and shallow water equations as examples, we systematically evaluate AD’s performance in numerical solution accuracy, parameter inversion, and spatiotemporal evolution simulation. Results demonstrate that the AD approach achieves high accuracy in approximating soliton solutions and successfully recovers key parameters from noisy synthetic data. Furthermore, the framework effectively simulates complex wave dynamics including generation, propagation, and dissipation under reflective boundary conditions. These findings highlight AD’s potential as a robust and accurate tool for high-fidelity ocean modeling and parameter estimation, paving the way for enhanced interpretability and predictive capability in marine science.

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

Simulations of beryllium castellation gap bridging during vertical displacement events

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 This paper is concerned with the low Mach and Rossby number limits of 3D compressible rotating Euler equations with ill‐prepared initial data in the whole space. More precisely, the initial data is the sum of a 3D part and a 2D part. With the help of a suitable intermediate system, we perform this singular limit rigorously with the target system being a 2D QG‐type. This particularly gives an affirmative answer to the question raised by Ngo and Scrobogna. As a by‐product, our proof gives a rigorous justification of the convergence from the 2D inviscid rotating shallow water equations to the 2D QG equations in the whole space.