An unstructured finite volume model for dam‐break floods with wet/dry fronts over complex topography

A robust, well‐balanced, unstructured, Godunov‐type finite volume model has been developed in order to simulate two‐dimensional dam‐break floods over complex topography with wetting and drying. The model is based on the nonlinear shallow water equations in hyperbolic conservation form. The inviscid fluxes are calculated using the HLLC approximate Riemann solver and a second‐order spatial accuracy is achieved by implementing the MUSCL reconstruction technique. To prevent numerical oscillations near shocks, slope‐limiting techniques are used for controlling the total variation of the reconstructed field. The model utilizes an explicit two‐stage Runge–Kutta method for time stepping, whereas implicit treatments for friction source terms. The novelties of the model include the flux correction terms and the water depth reconstruction method both for partially and fully submerged cells, and the wet/dry front treatments. The proposed flux correction terms combined with the water depth reconstruction method are necessary to balance the bed slope terms and flux gradient in the hydrostatical steady flow condition. Especially, this well‐balanced property is also preserved in partially submerged cells. It is found that the developed wet/dry front treatments and implicit scheme for friction source terms are stable. The model is tested against benchmark problems, laboratory experimental data, and realistic application related to dam‐break flood wave propagation over arbitrary topography. Numerical results show that the model performs satisfactorily with respect to its effectiveness and robustness and thus has bright application prospects. Copyright © 2010 John Wiley & Sons, Ltd.     

High resolution Godunov‐type schemes with small stencils

Higher‐order Godunov‐type schemes have to cope with the following two problems: (i) the increase in the size of the stencil that make the scheme computationally expensive, and (ii) the monotony‐preserving treatments (limiters) that must be implemented to avoid oscillations, leading to strong damping of the solution, in particular linear waves (e.g. acoustic waves). When too compressive, limiting procedures may also trigger the instability of oscillatory numerical solutions (e.g. in advection–dispersion phenomena) via the artificial amplification of the shorter modes. The present paper proposes a new approach to carry out the reconstruction. In this approach, the values of the flow variable at the edges of the computational cells are obtained directly from the reconstruction within these cells. This method is applied to the MUSCL and DPM schemes for the solution of the linear advection equation. The modified DPM scheme can capture contact discontinuities within one computational cell, even after millions of time steps at Courant numbers ranging from 1 to values as low as 10^‐4. Linear waves are subject to negligible damping. Application of the method to the DPM for one‐dimensional advection–dispersion problems shows that the numerical instability of oscillatory solutions caused by the over compressive, original DPM limiter is eliminated. One‐ and two‐dimensional shallow water simulations show an improvement over classical methods, in particular for two‐dimensional problems with strongly distorted meshes. The quality of the computational solution in the two‐dimensional case remains acceptable even for mesh aspect ratios Δx/Δy as large as 10. The method can be extend to the discretization of higher‐order PDEs, allowing third‐order space derivatives to be discretized using only two cells in space. Copyright © 2004 John Wiley & Sons, Ltd.     

An upstream flux‐splitting finite‐volume scheme for 2D shallow water equations

An upstream flux‐splitting finite‐volume (UFF) scheme is proposed for the solutions of the 2D shallow water equations. In the framework of the finite‐volume method, the artificially upstream flux vector splitting method is employed to establish the numerical flux function for the local Riemann problem. Based on this algorithm, an UFF scheme without Jacobian matrix operation is developed. The proposed scheme satisfying entropy condition is extended to be second‐order‐accurate using the MUSCL approach. The proposed UFF scheme and its second‐order extension are verified through the simulations of four shallow water problems, including the 1D idealized dam breaking, the oblique hydraulic jump, the circular dam breaking, and the dam‐break experiment with 45° bend channel. Meanwhile, the numerical performance of the UFF scheme is compared with those of three well‐known upwind schemes, namely the Osher, Roe, and HLL schemes. It is demonstrated that the proposed scheme performs remarkably well for shallow water flows. The simulated results also show that the UFF scheme has superior overall numerical performances among the schemes tested. Copyright © 2005 John Wiley & Sons, Ltd.     

Hybrid finite‐volume finite‐difference scheme for the solution of Boussinesq equations

A hybrid scheme composed of finite‐volume and finite‐difference methods is introduced for the solution of the Boussinesq equations. While the finite‐volume method with a Riemann solver is applied to the conservative part of the equations, the higher‐order Boussinesq terms are discretized using the finite‐difference scheme. Fourth‐order accuracy in space for the finite‐volume solution is achieved using the MUSCL‐TVD scheme. Within this, four limiters have been tested, of which van‐Leer limiter is found to be the most suitable. The Adams–Basforth third‐order predictor and Adams–Moulton fourth‐order corrector methods are used to obtain fourth‐order accuracy in time. A recently introduced surface gradient technique is employed for the treatment of the bottom slope. A new model ‘HYWAVE’, based on this hybrid solution, has been applied to a number of wave propagation examples, most of which are taken from previous studies. Examples include sinusoidal waves and bi‐chromatic wave propagation in deep water, sinusoidal wave propagation in shallow water and sinusoidal wave propagation from deep to shallow water demonstrating the linear shoaling properties of the model. Finally, sinusoidal wave propagation over a bar is simulated. The results are in good agreement with the theoretical expectations and published experimental results. Copyright © 2005 John Wiley & Sons, Ltd.     

An eigenvector‐based linear reconstruction approach for time stepping in discontinuous Galerkin scheme used to solve shallow water equations

Discontinuous Galerkin (DG) methods have shown promising results for solving the two‐dimensional shallow water equations. In this paper, the classical Runge–Kutta (RK) time discretisation is replaced by the eigenvector‐based reconstruction (EVR) that allows the second‐order time accuracy to be achieved within a single time‐stepping procedure. Moreover, the EVRDG approach yields stable solutions near drying and wetting fronts, whereas the classical RKDG approach yields instabilities. The proposed EVRDG technique is compared with the original RKDG approach on various test cases with analytical solutions. The EVRDG solutions are shown to be as accurate as those obtained with the RKDG scheme. Besides, the EVRDG scheme is 1.6 times faster than the RKDG method. Simulating dambreaks involving dry beds confirms that EVRDG scheme gives correct solutions, whereas the RKDG method yields instabilities. Copyright © 2012 John Wiley & Sons, Ltd.     

Finite‐volume multi‐stage schemes for shallow‐water flow simulations

A finite‐volume multi‐stage (FMUSTA) scheme is proposed for simulating the free‐surface shallow‐water flows with the hydraulic shocks. On the basis of the multi‐stage (MUSTA) method, the original Riemann problem is transformed to an independent MUSTA mesh. The local Lax–Friedrichs scheme is then adopted for solving the solution of the Riemann problem at the cell interface on the MUSTA mesh. The resulting first‐order monotonic FMUSTA scheme, which does not require the use of the eigenstructure and the special treatment of entropy fixes, has the generality as well as simplicity. In order to achieve the high‐resolution property, the monotonic upstream schemes for conservation laws (MUSCL) method are used. For modeling shallow‐water flows with source terms, the surface gradient method (SGM) is adopted. The proposed schemes are verified using the simulations of six shallow‐water problems, including the 1D idealized dam breaking, the steady transcritical flow over a hump, the 2D oblique hydraulic jump, the circular dam breaking and two dam‐break experiments. The simulated results by the proposed schemes are in satisfactory agreement with the exact solutions and experimental data. It is demonstrated that the proposed FMUSTA schemes have superior overall numerical accuracy among the schemes tested such as the commonly adopted Roe and HLL schemes. Copyright © 2007 John Wiley & Sons, Ltd.     

Well‐balanced positivity preserving central‐upwind scheme for the shallow water system with friction terms

Shallow water models are widely used to describe and study free‐surface water flow. While in some practical applications the bottom friction does not have much influence on the solutions, there are still many applications, where the bottom friction is important. In particular, the friction terms will play a significant role when the depth of the water is very small. In this paper, we study shallow water equations with friction terms and develop a semi‐discrete second‐order central‐upwind scheme that is capable of exactly preserving physically relevant steady states and maintaining the positivity of the water depth. The presence of the friction terms increases the level of complexity in numerical simulations as the underlying semi‐discrete system becomes stiff when the water depth is small. We therefore implement an efficient semi‐implicit Runge‐Kutta time integration method that sustains the well‐balanced and sign preserving properties of the semi‐discrete scheme. We test the designed method on a number of one‐dimensional and two‐dimensional examples that demonstrate robustness and high resolution of the proposed numerical approach. The data in the last numerical example correspond to the laboratory experiments reported in [L. Cea, M. Garrido, and J. Puertas, Journal of Hydrology, 382 (2010), pp. 88–102], designed to mimic the rain water drainage in urban areas containing houses. Since the rain water depth is typically several orders of magnitude smaller than the height of the houses, we develop a special technique, which helps to achieve a remarkable agreement between the numerical and experimental results. Copyright © 2015 John Wiley & Sons, Ltd.     

Solution of the 2‐D shallow water equations with source terms in surface elevation splitting form

A vertex‐centred finite‐volume/finite‐element method (FV/FEM) is developed for solving 2‐D shallow water equations (SWEs) with source terms written in a surface elevation splitting form, which balances the flux gradients and source terms. The method is implemented on unstructured grids and the numerical scheme is based on a second‐order MUSCL‐like upwind Godunov FV discretization for inviscid fluxes and a classical Galerkin FE discretization for the viscous gradients and source terms. The main advantages are: (1) the discretization of SWE written in surface elevation splitting form satisfies the exact conservation property (??‐Property) naturally; (2) the simple centred‐type discretization can be used for the source terms; (3) the method is suitable for both steady and unsteady shallow water problems; and (4) complex topography can be handled based on unstructured grids. The accuracy of the method was verified for both steady and unsteady problems, including discontinuous cases. The results indicate that the new method is accurate, simple, and robust. Copyright © 2007 John Wiley & Sons, Ltd.     

A novel multidimensional solution reconstruction and edge‐based limiting procedure for unstructured cell‐centered finite volumes with application to shallow water dynamics

On unstructured meshes, the cell‐centered finite volume (CCFV) formulation, where the finite control volumes are the mesh elements themselves, is probably the most used formulation for numerically solving the two‐dimensional nonlinear shallow water equations and hyperbolic conservation laws in general. Within this CCFV framework, second‐order spatial accuracy is achieved with a Monotone Upstream‐centered Schemes for Conservation Laws‐type (MUSCL) linear reconstruction technique, where a novel edge‐based multidimensional limiting procedure is derived for the control of the total variation of the reconstructed field. To this end, a relatively simple, but very effective modification to a reconstruction procedure for CCFV schemes, is introduced, which takes into account geometrical characteristics of computational triangular meshes. The proposed strategy is shown not to suffer from loss of accuracy on grids with poor connectivity. We apply this reconstruction in the development of a second‐order well‐balanced Godunov‐type scheme for the simulation of unsteady two‐dimensional flows over arbitrary topography with wetting and drying on triangular meshes. Although the proposed limited reconstruction is independent from the Riemann solver used, the well‐known approximate Riemann solver of Roe is utilized to compute the numerical fluxes, whereas the Green–Gauss divergence formulation for gradient computations is implemented. Two different stencils for the Green–Gauss gradient computations are implemented and critically tested, in conjunction with the proposed limiting strategy, on various grid types, for smooth and nonsmooth flow conditions. Copyright © 2012 John Wiley & Sons, Ltd.     

Unstructured finite volume discretization of two‐dimensional depth‐averaged shallow water equations with porosity

This paper deals with the numerical discretization of two‐dimensional depth‐averaged models with porosity. The equations solved by these models are similar to the classic shallow water equations, but include additional terms to account for the effect of small‐scale impervious obstructions which are not resolved by the numerical mesh because their size is smaller or similar to the average mesh size. These small‐scale obstructions diminish the available storage volume on a given region, reduce the effective cross section for the water to flow, and increase the head losses due to additional drag forces and turbulence. In shallow water models with porosity these effects are modelled introducing an effective porosity parameter in the mass and momentum conservation equations, and including an additional drag source term in the momentum equations. This paper presents and compares two different numerical discretizations for the two‐dimensional shallow water equations with porosity, both of them are high‐order schemes. The numerical schemes proposed are well‐balanced, in the sense that they preserve naturally the exact hydrostatic solution without the need of high‐order corrections in the source terms. At the same time they are able to deal accurately with regions of zero porosity, where the water cannot flow. Several numerical test cases are used in order to verify the properties of the discretization schemes proposed. Copyright © 2009 John Wiley & Sons, Ltd.     

Improved multislope MUSCL reconstruction on unstructured grids for shallow water equations

In shallow water flow and transport modeling, the monotonic upstream‐centered scheme for conservation laws (MUSCL) is widely used to extend the original Godunov scheme to second‐order accuracy. The most important step in MUSCL‐type schemes is MUSCL reconstruction, which calculate‐extrapolates the values of independent variables from the cell center to the edge. The monotonicity of the scheme is preserved with the help of slope limiters that prevent the occurrence of new extrema during reconstruction. On structured grids, the calculation of the slope is straightforward and usually based on a 2‐point stencil that uses the cell centers of the neighbor cell and the so‐called far‐neighbor cell of the edge under consideration. On unstructured grids, the correct choice for the upwind slope becomes nontrivial. In this work, 2 novel total variation diminishing schemes are developed based on different techniques for calculating the upwind slope and the downwind slope. An additional treatment that stabilizes the scheme is discussed. The proposed techniques are compared to 2 existing MUSCL reconstruction techniques, and a detailed discussion of the results is given. It is shown that the proposed MUSCL reconstruction schemes obtain more accurate results with less numerical diffusion and higher efficiency.     

Non‐oscillatory relaxation methods for the shallow‐water equations in one and two space dimensions

In this paper, a new family of high‐order relaxation methods is constructed. These methods combine general higher‐order reconstruction for spatial discretization and higher order implicit‐explicit schemes or TVD Runge–Kutta schemes for time integration of relaxing systems. The new methods retain all the attractive features of classical relaxation schemes such as neither Riemann solvers nor characteristic decomposition are needed. Numerical experiments with the shallow‐water equations in both one and two space dimensions on flat and non‐flat topography demonstrate the high resolution and the ability of our relaxation schemes to better resolve the solution in the presence of shocks and dry areas without using either Riemann solvers or front tracking techniques. Copyright © 2004 John Wiley & Sons, Ltd.     

High‐order semi‐implicit time‐integrators for a triangular discontinuous Galerkin oceanic shallow water model

We extend the explicit in time high‐order triangular discontinuous Galerkin (DG) method to semi‐implicit (SI) and then apply the algorithm to the two‐dimensional oceanic shallow water equations; we implement high‐order SI time‐integrators using the backward difference formulas from orders one to six. The reason for changing the time‐integration method from explicit to SI is that explicit methods require a very small time step in order to maintain stability, especially for high‐order DG methods. Changing the time‐integration method to SI allows one to circumvent the stability criterion due to the gravity waves, which for most shallow water applications are the fastest waves in the system (the exception being supercritical flow where the Froude number is greater than one). The challenge of constructing a SI method for a DG model is that the DG machinery requires not only the standard finite element‐type area integrals, but also the finite volume‐type boundary integrals as well. These boundary integrals pose the biggest challenge in a SI discretization because they require the construction of a Riemann solver that is the true linear representation of the nonlinear Riemann problem; if this condition is not satisfied then the resulting numerical method will not be consistent with the continuous equations. In this paper we couple the SI time‐integrators with the DG method while maintaining most of the usual attributes associated with DG methods such as: high‐order accuracy (in both space and time), parallel efficiency, excellent stability, and conservation. The only property lost is that of a compact communication stencil typical of time‐explicit DG methods; implicit methods will always require a much larger communication stencil. We apply the new high‐order SI DG method to the shallow water equations and show results for many standard test cases of oceanic interest such as: standing, Kelvin and Rossby soliton waves, and the Stommel problem. The results show that the new high‐order SI DG model, that has already been shown to yield exponentially convergent solutions in space for smooth problems, results in a more efficient model than its explicit counterpart. Furthermore, for those problems where the spatial resolution is sufficiently high compared with the length scales of the flow, the capacity to use high‐order (HO) time‐integrators is a necessary complement to the employment of HO space discretizations, since the total numerical error would be otherwise dominated by the time discretization error. In fact, in the limit of increasing spatial resolution, it makes little sense to use HO spatial discretizations coupled with low‐order time discretizations. Published in 2009 by John Wiley & Sons, Ltd.     

Energy relaxation method for chemical non‐equilibrium flow computations

In this paper, an algorithm for chemical non‐equilibrium hypersonic flow is developed based on the concept of energy relaxation method (ERM). The new system of equations obtained are studied using finite volume method with Harten–Lax–van Leer scheme for contact (HLLC). The original HLLC method is modified here to account for additional species and split energy equations. Higher order spatial accuracy is achieved using MUSCL reconstruction of the flow variables with van Albada limiter. The thermal equilibrium is considered for the analysis and the species data are generated using polynomial correlations. The single temperature model of Dunn and Kang is used for chemical relaxation. The computed results for a flow field over a hemispherical cylinder at Mach number of 16.34 obtained using the present solver are found to be promising and computationally (25%) more efficient. The present solver captures physically correct solution as the entropy conditions are satisfied automatically during the computations. Copyright © 2007 John Wiley & Sons, Ltd.     

Extension of a high‐resolution scheme to 1D liquid–gas flow

This paper presents the extension of a high‐resolution conservative scheme to the one‐dimensional one‐pressure six‐equation two‐fluid flow model. Only mixtures of water and air have been considered in this study, both fluids have been characterized using simple equations of state, namely stiffened gas for the liquid phase and perfect gas for the gas phase. The resulting scheme is explicit and first‐order accurate in space and time. A second‐order version of the scheme has also been derived using the MUSCL strategy and slope limiters. Some numerical results show the good capabilities of this type of schemes in the solution of discontinuities in two‐fluid flow problems, all of them are based on water/air numerical benchmarks widely used in the two‐phase flow literature. Copyright © 2005 John Wiley & Sons, Ltd.     

Development of a lattice Boltzmann method for two‐layered shallow‐water flow

In this paper the dynamics of a two‐layered liquid, made of two immiscible shallow‐layers of different density, has been investigated within the framework of the lattice Boltzmann method (LBM). The LBM developed in this paper for the two‐layered, shallow‐water flow has been obtained considering two separate sets of LBM equations, one for each layer. The coupling terms between the two sets have been defined as external forces, acted on one layer by the other. Results obtained from the LBM developed in this paper are compared with numerical results obtained solving the two‐layered, shallow‐water equations, with experimental and other numerical results published in literature. The results are interesting. First, the numerical results obtained by the LBM and by the shallow‐water model can be considered as equivalent. Second, the LBM developed in this paper is able to simulate motion conditions on nonflat topography. Third, the agreement between the LBM (and also shallow‐water model) numerical results and the experimental results is good when the evolution of the flow does not depend on the viscosity, that is, during the initial phase of the flow, dominated by gravity and inertia forces. When the viscous forces dominate the evolution of the flow the agreement between numerical and experimental results depends strongly on the viscosity; it is good if the numerical LBM viscosity has the same order of magnitude of the liquid's kinematic viscosity. Copyright © 2012 John Wiley & Sons, Ltd.     

A spectral approach to inertial confined thin‐film flow

A spectral approach is proposed to determine the flow field of a thin film inside narrow channels of arbitrary shape. Although the method is easily extended to transient flow, only steady flow is considered here. The flow field is represented spectrally in the depthwise direction in terms of orthonormal shape functions, which together with the Galerkin projection lead to a system of ordinary differential equations that can be solved using standard methods. The method is particularly effective for nonlinear flow, including nonlinearities of geometrical or material origins. The validity of the proposed method is demonstrated for a flow with inertia, and, unlike the depth‐averaging method, is not limited to a flow at small Reynolds number. The problem is closely related to high‐speed lubrication flow. The validity of the spectral representation is assessed by examining the convergence of the method, and comparing it with the fully two‐dimensional finite‐element solution, and the widely used depth‐averaging method from shallow‐water theory. It is found that a low number of modes are usually sufficient to secure convergence and accuracy. The influence of inertia is examined on the velocity and pressure fields. The pressure distributions reflect excellent agreement between the low‐order spectral method and the finite‐element solution, even at moderately high Reynolds number. The depth‐averaging solution is unable to predict accurately (qualitatively and quantitatively) the high‐inertia flow. Comparison of the velocity field reflects the expected discrepancy in a boundary layer formulation. Copyright © 2012 John Wiley & Sons, Ltd.     

A high‐order PIC method for advection‐dominated flow with application to shallow water waves

A hybrid Eulerian‐Lagrangian particle‐in‐cell–type numerical method is developed for the solution of advection‐dominated flow problems. Particular attention is given over to the high‐order transfer of flow properties from the particles to the grid. For smooth flows, the method presented is of formal high‐order accuracy in space. The method is applied to solve the nonlinear shallow water equations resulting in a new, and novel, shock capturing shallow water solver. The approach is able to simulate complex shallow water flows, which can contain an arbitrary number of discontinuities. Both trivial and nontrivial bottom topography is considered, and it is shown that the new scheme is inherently well balanced, exactly satisfying the ‐property. The scheme is verified against several one‐dimensional benchmark shallow water problems. These include cases that involve transcritical flow regimes, shock waves, and nontrivial bathymetry. In all the test cases presented, very good results are obtained.     

Boussinesq cut‐cell model for non‐linear wave interaction with coastal structures

Boussinesq models describe the phase‐resolved hydrodynamics of unbroken waves and wave‐induced currents in shallow coastal waters. Many enhanced versions of the Boussinesq equations are available in the literature, aiming to improve the representation of linear dispersion and non‐linearity. This paper describes the numerical solution of the extended Boussinesq equations derived by Madsen and Sørensen (Coastal Eng. 1992; 15 :371–388) on Cartesian cut‐cell grids, the aim being to model non‐linear wave interaction with coastal structures. An explicit second‐order MUSCL‐Hancock Godunov‐type finite volume scheme is used to solve the non‐linear and weakly dispersive Boussinesq‐type equations. Interface fluxes are evaluated using an HLLC approximate Riemann solver. A ghost‐cell immersed boundary method is used to update flow information in the smallest cut cells and overcome the time step restriction that would otherwise apply. The model is validated for solitary wave reflection from a vertical wall, diffraction of a solitary wave by a truncated barrier, and solitary wave scattering and diffraction from a vertical circular cylinder. In all cases, the model gives satisfactory predictions in comparison with the published analytical solutions and experimental measurements. Copyright © 2007 John Wiley & Sons, Ltd.