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An elastic-collision scheme is developed to achieve slip and semi-slip boundary conditions for lattice Boltzmann methods. Like the bounce-back scheme, the proposed scheme is efficient, robust and generally suitable for flows in arbitrary complex geometries. It involves an equivalent level of computation effort to the bounce-back scheme. The new scheme is verified by predicting wind-driven circulating flows in a dish-shaped basin and a flow in a strongly bent channel, showing good agreement with analytical solutions and experimental data. The capability of the scheme for simulating flows through multiple bodies has also been demonstrated.

A lattice Boltzmann model for the shallow water equations with turbulence modeling (LABSWE^TM) is developed. The flow turbulence is efficiently and naturally taken into account by incorporating the standard subgrid-scale stress model into the lattice Boltzmann equation in a consistent manner with the lattice gas dynamics. The model is applied to solve two flow problems and is verified by comparing numerical predictions with analytical solutions and available experimental data. The results show that the LABSWE^TM is able to provide basic features of flow turbulence and produce good predictions for turbulent shallow water flows.

The lattice Boltzmann model for the shallow water equations (LABSWE) is applied to the simulation of certain discontinuous flows. Curved boundaries are treated efficiently, using either the elastic-collision scheme for slip and semi-slip boundary conditions or the bounce-back scheme for no-slip conditions. The force term is accurately determined by means of the centred scheme. Simulations are presented of a small pulse-like perturbation of the still water surface, a dam break, and a surge wave interaction with a circular cylinder. The results agree well with predictions from alternative high-resolution Riemann solver based methods, demonstrating the capability of LABSWE to predict shallow water flows containing discontinuities.

A multiple-relaxation-time (MRT) collision operator is introduced into the author's rectangular lattice Boltzmann method for simulating fluid flows. The model retains both the advantages and the standard procedure of using a constant transformation matrix in the conventional MRT scheme on a square lattice, leading to easy implementation in the algorithm. This allows flow problems characterized by dominant feature in one direction to be solved more efficiently. Two numerical tests have been carried out and shown that the proposed model is able to capture complex flow characteristics and generate an accurate solution if an appropriate lattice ratio is used. The model is found to be more stable compared to the original rectangular lattice Boltzmann method using the single relaxation time.

We show that a hyperbolic system with a mathematical entropy can be discretized with vectorial lattice Boltzmann schemes using the methodology of kinetic representation of the dual entropy. We test this approach for the shallow water equations in one and two spatial dimensions. We obtain interesting results for a shock tube, reflection of a shock wave and nonstationary two-dimensional propagation. This contribution shows the ability of vectorial lattice Boltzmann schemes to simulate strong nonlinear waves in nonstationary situations.

A type of discrete Boltzmann model for simulating shallow water flows is derived by using the Hermite expansion approach. Through analytical analysis, we study the impact of truncating distribution function and discretizing particle velocity space. It is found that the convergence behavior of expansion is nontrivial while the conservation laws are naturally satisfied. Moreover, the balance of source terms and flux terms for steady solutions is not sacrificed. Further numerical validations show that the capability of simulating supercritical flows is enhanced by employing higher-order expansion and quadrature.

As a continuation of the research on BGK-type schemes, we present in this paper a second-order unsplitting method for the shallow water equations, where the source terms are included explicitly in the time-dependent flux functions across the cell interface.

In this paper, we propose and apply a modified Rusanov scheme for numerical solution of the sediment transport model in one and two dimensions. This model consists of two parts, the first part is modeled by shallow water equations and the second part is described by the bed-load transport equation. The scheme consists of a predictor stage scheme including a local parameter of control. It is responsible for the numerical diffusion. To control this parameter, we use a strategy depending on limiter theory. In the corrector stage, we used special treatment of the bed to get a well-balanced discretization between the flux gradient and source term. Some numerical results are presented for the sediment transport equation in two forms called A-formulation and C-formulation. These results show that the finite volume scheme is accurate and robust for solving the sediment transport equation in one and two dimensions.

In this work, two numerical schemes were developed to overcome the problem of shock waves that appear in the solutions of one/two-layer shallow water models. The proposed numerical schemes were based on the method of lines and artificial viscosity concept. The robustness and efficiency of the proposed schemes are validated on many applications such as dam-break problem and the problem of interface propagation of two-layer shallow water model. The von Neumann stability of proposed schemes is studied and hence, the sufficient condition for stability is deduced. The results were presented graphically. The verification of the obtained results is achieved by comparing them with exact solutions or another numerical solutions founded in literature. The results are satisfactory and in much have a close agreement with existing results.

A full set of conservation laws for the two-layer shallow water equations is presented for the one-dimensional case. We prove that all the conservation laws are linear combination of the equations for the conservation of mass and velocity (in each layer), total momentum and total energy.This result generalizes that of Montgomery and Moodie that found the same conserved quantities by restricting their search to the multinomials expressions in the layer variables. Though the question of whether or not there are only a finite number of these quantities is left as an open question by the authors. Our work puts an end to this: in fact, no more conservation laws are admitted for the two-layer shallow water equations. The key mathematical ingredient of the method proposed leading to the result is the Frobenius problem. Moreover, we present a full set of conservation laws for the classical one-dimensional shallow water model with topography, by using the same techniques.

The section-averaged shallow water model usually applied in river and open channel hydraulics is derived by an asymptotic analysis that accounts for terms up to second order in the vertical/longitudinal length ratio, starting from the three-dimensional Reynolds-averaged Navier–Stokes equations for incompressible free surface flows. The derivation is carried out under quite general assumptions on the geometry of the channel, thus allowing for the application of the resulting equations to natural rivers with arbitrarily shaped cross sections. As a result of the derivation, a generalized friction term is obtained, that does not rely on local uniformity assumptions and that can be computed directly from three-dimensional turbulence models, without need for local uniformity assumptions. The modified equations including the novel friction term are compared to the classical Saint Venant equations in the case of steady state open channel flows, where analytic solutions are available, showing that the solutions resulting from the modified equation set are much closer to the three-dimensional solutions than those of the classical equation set. Furthermore, it is shown that the proposed formulation yields results that are very similar to those obtained with empirical friction closures widely applied in computational hydraulics. The generalized friction term derived therefore justifies a posteriori these empirical closures, while allowing to avoid the assumptions on local flow uniformity on which these closures rely.

We derive consistent shallow water equations (so-called Saint Venant equations) for the superposition of two Newtonian fluids flowing down a ramp. We carry out a complete spectral analysis of steady flows in the low frequency/long wavelength regime and show the occurrence of hydrodynamic instabilities, so-called roll-waves, when steady flows are unstable.

Free-surface water flows over stochastic beds are complex due to the uncertainties in topography profiles being highly heterogeneous and imprecisely measured. In this study, the propagation and influence of several uncertainty parameters are quantified in a class of numerical methods for one-dimensional free-surface flows. The governing equations consist of both single-layer and two-layer shallow water equations on either flat or nonflat topography. For this purpose, the free-surface profiles are computed for different realizations of the random variables when the bed is excited with sources whose statistics are well defined. Many research studies have been dedicated to the development of numerical methods to achieve some order of accuracy in free-surface flows. However, little concern was given to examine the performance of these numerical methods in the presence of uncertainty. This work addresses this specific area in computational hydraulics with regards to the uncertainty generated from bathymetric forces. As numerical solvers for the one-dimensional shallow water equations, we implement four finite volume methods. To reduce the required number of samples for uncertainty quantification, we combine the proper orthogonal decomposition method with the polynomial chaos expansions for efficient uncertainty quantifications of complex hydraulic problems with large number of random variables. Numerical results are shown for several test examples including dam-break problems for single-layer and two-layer shallow water flows. The problem of flow exchange through the Strait of Gibraltar is also solved in this study. The obtained results demonstrate that in some hydraulic applications, a highly accurate numerical method yields an increase in its uncertainty and makes it very demanding to use in an operational manner with measured data from the field. On the other hand, when the complexity of physics increases, these highly accurate numerical methods display less uncertainty compared to the low accurate methods.

We present a new numerical scheme that is a well-balanced and second-order accurate for systems of shallow water equations (SWEs) with variable bathymetry. We extend in this paper the subtraction method (resulting in well-balancing) to the case of unstaggered central finite volume methods that computes the numerical solution on a single grid. In addition, the proposed scheme avoids solving Riemann problems occurring at cell boundaries as it employs intermediately a layer of ghost-staggered cells. The proposed numerical scheme is then implemented and validated. We successfully manage to solve classical SWE problems from the literature featuring steady states and other equilibria. The results of the study are consistent with previous research, which supports the use of the proposed method to solve SWEs.

We investigate the solution of the nonlinear junction Riemann problem for the one-dimensional shallow water equations (SWEs) in a simple star network made of three rectangular channels. We consider possible bottom discontinuities between the channels and possible differences in the channels width. In the literature, the solution of the Riemann problem at the junction is investigated for the symmetric case without bottom steps and channels width variations. Here, the solution is extended to a more general situation such that neither the equality of the channels width nor the symmetry of the flow are assumed in the downstream channels. The analysis is performed under sub-criticality conditions and the results are summarized in a main theorem, while a series of numerical examples are presented and support our conclusions.

The nonlinear shallow water equations and Boussinesq equations have been widely used for analyzing the solitary wave runup phenomenon. In order to quantitatively assess the merits and limitations of these two approaches, a shock-capturing scheme has been applied to numerically solve both sets of equations for predicting the runup processes over plane beaches. The analytical and experimental data available in the literature have been used as references in the assessment. In this study, the uniform sloping beach is preceded by a length of flat seafloor. When incident solitary waves are specified close to the slope, the two approaches are found to produce almost identical results for nonbreaking waves. For breaking waves, the Boussinesq equations give a better representation of the wave evolution prior to the breaking point, whereas they overestimate the short undulations accompanying the breaking process. If incoming solitary waves need to travel a long distance over a flat bed before reaching the slope, the shallow water equations cannot capture the correct waveform transformation, and the predicted runup depends heavily on the length of the flat-bed section. As this length approaches zero, however, the shallow water equations somehow give roughly the same maximum runup heights as those predicted by the Boussinesq model. The bed friction has little effect on the runup for small waves, but becomes important for large waves. The roughness coefficient needs to be calibrated to reproduce the measured runup heights of breaking waves.

The two-dimensional shallow water equations were formulated and numerically solved in an arbitrary curvilinear coordinate system, which offers a relatively high degree of flexibility in representing the natural flow domains with structured meshes. The model employs an efficient TVD-MacCormack scheme, which has second-order accuracy in both time and space. Refinements were made to enhance the model's accuracy and stability in computing the shallow wave dynamics in real-world scenarios, with irregular boundaries and uneven beds. In particular, advanced open boundary conditions have been proposed according to the method of characteristics, and rigorous mass conservation has been enforced during the computation at both the inner-domain and the boundaries. These refinements are necessary when modeling the flood inundation over a large area and the tidal oscillation in a macro-tidal estuary. The effectiveness of the refinements was verified by simulating the forced tidal resonance in an idealized condition and the Malpasset dam-break flood in the Reyran river valley. The application of the refined model in the study of tidal oscillations in the Severn Estuary and Bristol Channel can be found in the companion paper.

This paper presents a 2D model for predicting tsunami propagation on dynamically adaptive grids. In this model, a finite volume Godunov-type scheme is implemented to solve the 2D nonlinear shallow water equations on adaptive grids. The simplified adaptive grid achieves automatic adaptation through increasing or reducing the subdivision level of a background cell according to certain criteria defined by tsunami wave features. The grid system is straightforward to implement and no data structure is needed to store grid information. The present model is validated by applying it to simulate three laboratory-scale test cases of tsunami propagation over uneven beds and finally used to reproduce the 2011 Tohoku tsunami in Japan. The model results confirm the model’s capability in predicting tsunami wave propagation in a reliable and efficient way.

Vegetated areas on the beach can reduce tsunami heights and reduce the loss of life and property damage in coasts. Thick trunks and tangled branches attenuate tsunami waves. In this study, a numerical model is developed based on the finite volume method for simulating tsunami flooding. This model is used to simulate the solitary wave run-up propagation on sloping beaches with and without vegetation. The shallow water equations are used, also the effect of drag force due to vegetation is applied in the momentum equation. The HLLC approximate Riemann solver is selected, and the model is developed to second-order accuracy using the Weighted Average Flux method. After verification of the present model, the model is applied for simulation of solitary wave on a sloping beach. The present model results are compared to the available experimental data and another numerical model. The present numerical results reveal that as forest belt’s width increases, the height, velocity, and force of the tsunami waves decrease. Therefore, to further reduce the tsunami energy, a wider belt is recommended. Also, the effect of different tsunami wave heights on the rate of wave reduction has been investigated. In some areas, the presence of high tsunami waves causes to submerge the vegetation. Consequently, the drag force and the damping rate of the wave decrease. Therefore, the height of the forest zone and the height of the tsunami waves are important parameters.

The aim of this paper is to compare some recent numerical schemes for solving hyperbolic conservation laws. We consider the flux vector splitting finite volume methods, finite volume evolution Galerkin scheme as well as the discontinuous Galerkin scheme. All schemes are constructed using time explicit discretization. We present results of numerical experiments for the shallow water equations for continuous as well as discontinuous solutions and compare accuracy and computational efficiency of the considered methods.