2D Shallow Water Equations Research Articles

Modeling and Simulation of Tsunami Impact: A Short Review of Recent Advances and Future Challenges

Tsunami modeling and simulation has changed in the past few years more than it has in decades, especially with respect to coastal inundation. Among other things, this change is supported by the approaching era of exa-scale computing, whether via GPU or more likely forms of hybrid computing whose presence is growing across the geosciences. For reasons identified in this review, exa-scale computing efforts will impact the on-shore, highly turbulent régime to a higher degree than the 2D shallow water equations used to model tsunami propagation in the open ocean. This short review describes the different approaches to tsunami modeling from generation to impact and underlines the limits of each model based on the flow régime. Moreover, from the perspective of a future comprehensive multi-scale modeling infrastructure to simulate a full tsunami, we underline the current challenges associated with this approach and review the few efforts that are currently underway to achieve this goal. A table of existing tsunami software packages is provided along with an open Github repository to allow developers and model users to update the table with additional models as they are published and help with model discoverability.

The Pi − Finite Element Method for 1D wave simulation using Shallow Water Equations

We study a simple numerical scheme based on a new type of Finite Element Method (FEM) to solve the 1D Shallow Water Equations. In the new scheme, the surface elevation variable is approximated by a linear continuous basis function (P 1) and the velocity potential variable is approximated by the one-dimensional discontinuous linear non-conforming basis function (). Here, we implement the P 1 − finite element pair to solve the 1D Shallow Water Equations on a structured grid, whereas the Runge Kutta method is adopted for time integration. We verified the resulting scheme by conducting several simulations such as a standing wave simulation, and propagation of an initial hump over sloping bathymetry. The resulting scheme free from numerical damping error, conservative and both standing wave and shoaling phenomena are well simulated.

Experimental and numerical model studies on flash flood inundation processes over a typical urban street

Accurate prediction of flood inundation processes in urban areas is challenging, due to the complexity of street layouts and the variety of infrastructures. In this study, based on a laboratory model of urban flooding with a sewer system underneath, a series of laboratory experiments were conducted to investigate the influences of different street layouts and infrastructures on flood inundation processes. Key hydrographs of water depth and flow velocity were recorded at several measuring points to provide comprehensive information about the hydrodynamic characteristics of urban flooding. Furthermore, a 2D shallow water equation numerical model, based on the finite volume method, was also utilized to replicate the experimental scenarios considered. An analysis of the mesh resolution and discharge capacity formulae for street inlets was also performed through a series of numerical tests. The following conclusions are drawn from this study: (i) the sewer system has a strong influence on the flood inundation processes in terms of reducing both the surface water depth and flood wave velocity, as compared with street layouts and other infrastructures; (ii) the results from the numerical simulations agree well with the experimental findings, with the NSE values being greater than 0.7 and the RMSE values less than 3×10−2; (iii) a further increment in the mesh resolution may benefit slightly the model predictions, but at the expense of an increasing computational cost; and (iv) of all the inlet discharge capacity formulae used in this study, the weir and orifice formulae considering the influence of rain boxes were the most appropriate for representing the geometric features of street inlets and showed the best performance in calculating the flow drainage between the surface runoff and the underground sewer system.

A Numerical Model for 1D Bed Morphology Calculations

The present work proposes a new morphodynamic model for flow simulation in open channels. The governing one dimensional (1D) shallow water equations along with the Exner sediment continuity equation are solved here using coupled approach. A TVD based MacCormack predictor corrector method is used here for solution of the coupled governing equations. The developed numerical model is used to replicate several test cases of known experimental and approximate solutions. These cases include bed aggradation due to sediment overloading, bed degradation due to sediment shut-off, dam break flow over erodible bed and movable bump due to subcritical and transcritical flows. The results obtained are found to be excellent and free from oscillations near sharp gradient. The numerical model is also found to be capable of capturing the wave-fronts of both water surface and bed sediment. Quantitative analysis of the results in terms of two statistical parameters establishes the potential of the present model in more complicated practical problems.

Growth-decay model of vegetation based on hydrodynamics and simulation on vegetation evolution in the channel

Hydrodynamic conditions are important controlling factors in the evolution of vegetation and ecosystems, especially aquatic ecosystems. This study established a conceptual vegetation evolution model to explore the succession law of vegetation patches in the effective area of a channel, including the evolution of vegetation coverage (Ce) and the corresponding longitudinal dispersion coefficient (Ke). 2D shallow-water equations were implemented to calculate flow variables, and the equivalent Manning coefficient was used to reflect the effects caused by vegetation patches. New vegetation emerged in regions where the bed shear stress is lower than the critical bed shear stress, whereas the original vegetation was removed in regions where the bed shear stress is higher than the critical value. Two typical cross-sections and two initial Ce were considered to better understand the succession trend of vegetation patches under different external conditions. The findings showed that Ce and Ke increased to a constant value with increasing simulation duration, whereas a higher critical bed shear stress, defined by the threshold value (TV), was linked to a higher final vegetation coverage (Cf) and final longitudinal dispersion coefficient (Kf). Furthermore, different initial vegetation distributions, including coverage and position, caused little effect on the Cf in the rectangular channel, but the Cf in the parabolic channel was linearly affected by the averaged bed shear stress at the initial patches. The maximum Kf in both channels occurred with regularly distributed initial patches on one side of the bank. The vegetation patches in all scenarios evolved from block-shaped to strip-shaped and finally formed a stable vegetated landscape. This conceptual vegetation evolution model will improve our understanding of the influence of hydrodynamic conditions on the vegetation evolution in aquatic ecosystems.

(Multi)wavelets increase both accuracy and efficiency of standard Godunov-type hydrodynamic models: Robust 2D approaches

Multiwavelets (MW) enable the compression, analysis and assembly of model data on a multiresolution grid within Godunov-type solvers based on second-order discontinuous Galerkin (DG2) and first-order finite volume (FV1) methods. Multiwavelet adaptivity has been studied extensively with one-dimensional (1D) hydrodynamic models (Kesserwani et al., 2019), revealing that MWDG2 can be 20 times faster than uniform DG2 and 2 times faster than uniform FV1, while preserving the accuracy and robustness of the underlying formulation. The potential of the MWDG2 scheme has yet to be studied for two-dimensional (2D) modelling, but this requires a design that robustly and efficiently solves the 2D shallow water equations (SWE) with complex source terms and wetting and drying. This paper presents a two-dimensional MWDG2 scheme that: (1) adopts a slope-decoupled DG2 solver as a reference scheme, for its ability to deliver well-balanced piecewise-planar solutions shaped by a simplified 3-component basis; and, (2) adapts the multiresolution analysis of multiwavelets for compatibility with the slope-decoupled DG2 basis. A scaled reformulation of slope-decoupled DG2 is presented alongside two multiwavelet approaches that yield MWDG2 schemes with similar properties, and a Haar wavelet FV1 (HFV1) variant for adapting piecewise-constant model data. The performance of the adaptive HFV1 and MWDG2 solvers is explored alongside their uniform counterparts, while analysing their accuracy, efficiency, grid-coarsening ability, reliability in handling wet-dry fronts across steep bed-slopes, and ability to capture features relevant to practical hydraulic modelling. The results indicate a particular multiwavelet approach that allows the MWDG2 scheme to exploit its grid-coarsening ability for the widest range of flow types. Results also indicate that the proposed (multi)wavelet-based adaptive schemes are even more efficient for the 2D case. Accompanying model software is openly available online.

A remark on finite volume methods for 2D shallow water equations over irregular bottom topography

The 2D shallow water equations are often solved through finite volume (FV) methods in the presence of irregular topography or in non-rectangular channels. The ability of FV schemes to preserve uniform flow conditions under these circumstances is herein analysed as a preliminary condition for more involved applications. The widely used standard Harten-Lax-Van Leer (HLL) and a well-balanced Roe method are considered, along with a scheme from the class of wave propagation methods. The results indicate that, differently from the other solvers, the standard HLL does not preserve the uniform condition. It is therefore suggested to benchmark numerical schemes under this condition to avoid inaccurate hydrodynamic simulations. The analysis of the discretized equations reveals an unbalance of the HLL transversal fluxes in the streamwise momentum equation. Finally, a modification of the HLL scheme, based on the auxiliary variable balance method, is proposed, which strongly improves the scheme performance.

Urban scale flood simulation based on dynamic wave

Currently based on the full 2D shallow water equations (SWEs) model has been widely used in flood simulation, but there are still two problems to be solved in urban inundation simulation: small-scale structure simulation and calculation efficiency of urban scale model. In this paper, the effective representation of small-scale structure and GPU acceleration technology are introduced into the numerical model to establish a high efficiency and high score model. The model is applied to the simulation of urban inundation in Yuhuan. The simulation results show that the model can simulate urban inundation well with a high accuracy.

A 2D finite volume simulation tool to enable the assessment of combined hydrological and morphodynamical processes in mountain catchments

Nowadays, the great power of modern computers allows to develop computational models able to deal with simulations of several coupled phenomena over detailed complex topography. An efficient and properly calibrated computational model represents a useful tool to provide insight into the catchment dynamics at hydrological and geomorphological levels. In addition, it allows to develop detailed risk management and conservation plans. In this work, we present a coupled surface-groundwater distributed flow model with hydrological (rainfall and infiltration) and geomorphological (suspended and bed load sediment transport) components. The coupled model is applied to well characterized experimental catchments that are used as realistic test cases. The calibration of the water flow model response to rainfall is performed by means of the fitting to experimental outlet hydrographs of the results supplied by a coupled formulation of 2D Shallow Water Equations and 2D Darcy’s law for saturated porous media connected via suitable infiltration laws. The calibration of a suspended and bed load model is also addressed by means of the fitting to experimental outlet sedigraphs. The numerical results show a good agreement between numerical and observed hydrographs and sedigraphs, significantly improving previous published simulations. Additionally, the need to repeat the simulations in the calibration processes is no longer an unapproachable problem.

A new criterion for the evaluation of the velocity field for rainfall-runoff modelling using a shallow-water model

The aim of this work is to evaluate the accuracy of physically based models using the 2D Shallow Water Equations (SWE) for the simulation of rain-induced runoff. A key expectation of these models is to be able to reproduce properly both hydrograph at the outlet of the watershed and the associated velocity field within the spatial domain for a given rain event. However, this ability can be very challenging in mountainous environments, which include heterogeneous and high values of bottom slope and transient rain inputs. In this paper, we focused on both the error due to the numerical resolution of the SWE for low water depths runoff on steep slopes and that due to the physical representation of friction on the bottom for various water depth to roughness height ratios. Four test cases have been used, at different scales, to evaluate the model in comparison with experimental data. The first test case (from Kirstetter et al. (2015)) was a rain induced flow in a straight channel. It allows to evaluate the error made for the chosen numerical resolution of the SWE and its impact on the velocity distribution in the domain (metric scale). A criterion defining an acceptable numerical error has been proposed. A second test case (from Cea et al. (2014)), has been chosen for being also metric scaled and in the right range of the numerical error criterion. It allowed to evaluate the ability of a new friction law (adapted from Lawrence (1997)) to represent the measured velocity field in several inundation regimes at the metric scale. The third test case was a 40 m2 real plot studied by Tatard et al. (2008). On this test case, measured velocities have been compared with the simulated ones. Thanks to an infiltration source term added to the mass conservation equation, hydrographs at the outlet of the plot have been well reproduced, as well as the main part (including the highest values) of the measured velocities. In particular, the velocities are well represented when the criterion given in the first test case is satisfied. Finally, the model is applied to a real watershed with an area of 1 km2 on which hydrographs at the outlet have been successfully reproduced for two extreme rainy events (Le Bouteiller et al., 2015). It appeared that both physical and numerical error were low in the hydrographic network and that the local velocities are correctly reproduced there, unlike in the hillslopes where the velocities are not well reproduced even if the mass fluxes are correct.

An adaptive discontinuous Galerkin method for the simulation of hurricane storm surge

Numerical simulations based on solving the 2D shallow water equations using a discontinuous Galerkin (DG) discretisation have evolved to be a viable tool for many geophysical applications. In the context of flood modelling, however, they have not yet been methodologically studied to a large extent. Systematic model testing is non-trivial as no comprehensive collection of numerical test cases exists to ensure the correctness of the implementation. Hence, the first part of this manuscript aims at collecting test cases from the literature that are generally useful for storm surge modellers and can be used to benchmark codes. On geographic scale, hurricane storm surge can be interpreted as a localised phenomenon making it ideally suited for adaptive mesh refinement (AMR). Past studies employing dynamic AMR have exclusively focused on nested meshes. For that reason, we have developed a DG storm surge model on a triangular and dynamically adaptive mesh. In order to increase computational efficiency, the refinement is driven by physics-based refinement indicators capturing major model sensitivities. Using idealised numerical test cases, we demonstrate the model’s ability to correctly represent all source terms and reproduce known variability of coastal flooding with respect to hurricane characteristics such as size and approach speed. Finally, the adaptive mesh significantly reduces computing time with no effect on storm waves measured at discrete wave gauges just off the coast which shows the model’s potential for use as a robust simulation tool for real-time predictions.

Sampling and resolution characteristics in reduced order models of shallow water equations: Intrusive vs nonintrusive

SummaryWe investigate the sensitivity of reduced order models (ROMs) to training data spatial resolution as well as sampling rate. In particular, we consider proper orthogonal decomposition (POD), coupled with Galerkin projection (POD‐GP), as an intrusive ROM technique. For nonintrusive ROMs, we consider two frameworks. The first is using dynamic mode decomposition (DMD), and the second is based on artificial neural networks (ANNs). For ANN, we utilized a residual deep neural network, and for DMD we have studied two versions of DMD approaches; one with hard thresholding and the other with sorted bases selection. Also, we highlight the differences between mean subtracting the data (centering) and using the data without mean subtraction. We tested these ROMs using a system of 2D shallow water equations for four different numerical experiments, adopting combinations of sampling rates and spatial resolutions. For these cases, we found that the DMD basis obtained with hard thresholding is sensitive to sampling rate. The sorted DMD algorithm helps to mitigate this problem and yields more stabilized and converging solution. Furthermore, we demonstrate that both DMD approaches without mean subtraction provide significantly more accurate results than DMD with mean subtracting the data. On the other hand, POD is relatively insensitive to sampling rate and yields better representation of the flow field. Meanwhile, spatial resolution has little effect on both POD and DMD performances. Numerical results reveal that an ANN on POD subspace (POD‐ANN) performs remarkably better than POD‐GP and DMD in capturing system dynamics, even with a small number of modes.

Nonstandard caustics for localized solutions of the 2d shallow water equations with applications to wave propagation and run-up on a shallow beach

The usual semiclassical approximation is not directly applicable to linearized two-dimensional shallow water equations with localized initial data describing long waves (for example, tsunami waves) in a bounded basin of variable depth, because the momentum variables on the geometric optics rays become infinite at the boundary of the region. To obtain asymptotic solutions, one needs to extend the phase space and introduce a modified Maslov canonical operator. The paper gives a brief survey of some results obtained in this direction by the authors and their colleagues.

An Implicit Coupled 1D/2D Model for Unsteady Subcritical Flow in Channel Networks and Embayment

In this study, 1D and 2D shallow-water models were coupled to simulate unsteady flow in channel networks and embayment. The 1D model solved the 1D shallow-water equations (St. Venant) using the Preissmann box method and targeted long narrow reaches of the river networks, while the 2D model targeted broad channels and embayment and solved the 2D shallow-water equations using a semi-implicit scheme applied to an unstructured grid of triangular cells. The 1D and 2D models were solved simultaneously by building a matrix for the free surface elevation at every 1D junction and 2D cell center. Velocities were then computed explicitly based on the results at the previous time step and the updated water level. The originality of the scheme arose from a novel coupling method. The results showed that the coupled 1D/2D model produced identical results as the full 2D model in classical to benchmark problems with considerable savings in computational effort. Application of the model to the Pearl River Estuary in southern China showed that complex patterns of tidal wave propagation could be efficiently modeled.

A novel 1D-2D coupled model for hydrodynamic simulation of flows in drainage networks

Drainage network modelling is often an essential component in urban flood prediction and risk assessment. Drainage network models most commonly use different numerical procedures to handle flows in pipes and junctions. Numerous numerical schemes and models of different levels of complexity have been developed and reported to predict flows in pipes. However, calculation of the flow conditions in junctions has received much less attention and has been traditionally achieved by solving only the continuity equation. This method is easy to implement but it neglects the momentum exchange in the junctions and cannot provide sufficient boundary conditions for the pipe calculation. In this work, a novel numerical scheme based on the finite volume solution to the two-dimensional (2D) shallow water equations (SWEs) is proposed to calculate flow dynamics in junctions, which directly takes into account both mass and momentum conservation and removes the necessity of implementing complicated boundary settings for pipe calculations. This new junction simulation method is then coupled with the widely used two-component pressure approach (TPA) for the pipe flow calculation, leading to a new integrated drainage network model. The new 1D-2D coupled drainage network model is validated against an experimental and several idealised test cases to demonstrate its potential for efficient and stable simulation of flow dynamics in drainage networks.

Experimental and Numerical Investigation of River Closure Project

The success or failure of river closure is directly related to the construction period and project benefit. Therefore, it is very necessary to study the river closure by an appropriate method. In this paper, a 1D–2D coupled river closure model is established to optimize the closure flow rate, closure period, and layout of a real closure project. The 1D transition model between open channel flow and pressurized flow is established by a finite volume scheme. For the 2D model, 2D shallow water equations are solved using an unstructured finite volume scheme. The 1D model and 2D model are coupled by considering the mass and momentum conservation. To validate the model, a physical experiment of a real river closure project is set up according to the gravity similarity criterion with a scale of 1:80. Then, the experimental data obtained by the calibrated physical experiment is compared with the numerical results. Good agreements are achieved in terms of surface elevation, velocity, and flow rate. Finally, the real river closure project is further investigated by the model. The layout, closure flow rate and closure period of this project is analyzed and optimized. The original design of the berm is more suitable to discharge the flow. Moreover, the first stage cofferdam should be removed to floor elevation upstream and downstream of the dam. The river closure flow rate should not exceed 2380 m3/s.

Finite volume model for the simulation of 1D unsteady river flow and water quality based on the WASP

AbstractIn this work, a one-dimensional (1D) finite volume numerical model for the unsteady simulation of the flow hydrodynamics and water quality is developed. The water dynamics is formulated with the 1D shallow water equations, and the water quality evolution is described by the Water Quality Analysis Simulation Program (WASP) model, allowing us to interpret and predict the transport and fate of various biochemical substances along any river reach. This combined system is solved with an explicit finite volume scheme based on Roe's linearization for the advection component of both the flow and the solute transport equations. The proposed model is able to consider temporal variations in tributaries and abstractions occurring in the river basin. This feature is transcendent in order to predict the chemical composition of natural water bodies during winter and summer periods, leading to an improvement in the agreement between computed and observed water quality evolutions. The combined model has been evaluated using literature tests in a steady state and a real-field case of the Ebro river (Spain), characterized by a marked unsteady regime. In the real case, we found that the water temperature was very sensitive to both the solar radiation and the average air temperature, requiring a careful calibration of these parameters. The numerical results are also demonstrated to be reasonably accurate, conservative and robust in real-scale field cases, showing that the model is able to predict the evolution of quality parameters as well as hydrodynamic variables in complex scenarios.

A decision-support framework for emergency evacuation planning during extreme storm events

Developing an effective real-time evacuation strategy during extreme storm events such as hurricanes has been a topic of critical significance to the emergency planning and response community. The spatial and temporal variabilities of inland flooding during hurricanes present significant challenges for robust evacuation planning. In this study, a framework for real-time evacuation planning was developed that combines the results obtained from hydrodynamic modeling and traffic microsimulation. First, a fine-scale hydrodynamic model was developed based on depth-averaged 2D shallow-water equations (SWE) to obtain information pertaining to flood depth and velocity for planning evacuation routes during a storm event. Next, a traffic microsimulation was conducted using time-dependent information from the hydrodynamic model regarding the traffic velocities along evacuation routes during an event. An optimization technique was also implemented to reduce the overall travel time by about 6% from that of the base model. The last component of the framework involves combining the results from both models to generate a time-lapse animation of emergency evacuation based on a geographic information system (GIS). The results obtained using this framework could be easily accessed by the general public and decision-makers to enable efficient evacuation planning during extreme storm events.

Bathymetry Development and Flow Analyses Using Two-Dimensional Numerical Modeling Approach for Lake Victoria

This study explored two-dimensional (2D) numerical hydrodynamic model simulations of Lake Victoria. Several methods were developed in Matlab to build the lake topography. Old depth soundings taken in smaller parts of the lake were combined with more recent extensive data to produce a smooth topographical model. The lake free surface numerical model in the COMSOL Multiphysics (CM) software was implemented using bathymetry and vertically integrated 2D shallow water equations. Validated by measurements of mean lake water level, the model predicted very low mean flow speeds and was thus close to being linear and time invariant, allowing long-time simulations with low-pass filtered inflow data. An outflow boundary condition allowed an accurate simulation to achieve the lake’s steady state level. The numerical accuracy of the linear measurement of lake water level was excellent.

The scaled entropy variables reconstruction for entropy stable schemes with application to shallow water equations

The sign stability of third order WENO reconstruction across jump discontinuities and the sign-preserving SPWENO reconstruction are shown, and extending the reconstruction procedure to systems of conservation laws in multiple space dimensions, with a focus on the frictionless shallow water problems. High resolution entropy stable schemes are constructed by using the SPWENO\\WENO-YC3 reconstructions procedures for cell interface values of the scaled entropy variables in the numerical diffusion term. Numerical simulations of the shallow water equations in both one and two dimensions are presented, and the results demonstrate the stable, low dissipative and the capacity of the proposed schemes to better resolve the solution in the presence of shocks. Furthermore, the proposed ES-SPWENO and ES-WENO-YC3 schemes are close to third-order accurate for 2D shallow water equations.