Evaluation of well‐balanced bore‐capturing schemes for 2D wetting and drying processes

We consider numerical solutions of the two‐dimensional non‐linear shallow water equations with a bed slope source term. These equations are well‐suited for the study of many geophysical phenomena, including coastal engineering where wetting and drying processes are commonly observed. To accurately describe the evolution of moving shorelines over strongly varying topography, we first investigate two well‐balanced methods of Godunov‐type, relying on the resolution of non‐homogeneous Riemann problems. But even if these schemes were previously proved to be efficient in many simulations involving occurrences of dry zones, they fail to compute accurately moving shorelines. From this, we investigate a new model, called SURF_WB, especially designed for the simulation of wave transformations over strongly varying topography. This model relies on a recent reconstruction method for the treatment of the bed‐slope source term and is able to handle strong variations of topography and to preserve the steady states at rest. In addition, the use of the recent VFRoe‐ncv Riemann solver leads to a robust treatment of wetting and drying phenomena. An adapted ‘second order’ reconstruction generates accurate bore‐capturing abilities.This scheme is validated against several analytical solutions, involving varying topography, time dependent moving shorelines and convergences toward steady states. This model should have an impact in the prediction of 2D moving shorelines over strongly irregular topography. Copyright © 2006 John Wiley & Sons, Ltd.     

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.     

A modified lattice Boltzmann model for shallow water flows over complex topography

A modified lattice Boltzmann model is proposed to describe shallow water flows over complex topography. In the proposed model, the quadratic depth term is excluded from the equilibrium distribution functions (EDFs), and the hydrostatic pressure term is combined with the bed slope term to be treated as a part of the sourcing term in the lattice Boltzmann equation (LBE). Therefore, it is unnecessary to match the coefficients of the quadratic depth term in the EDFs with those of the bed slope term in the sourcing terms in the LBE. This would bring more flexibility to the treatment of the sourcing terms in the LBE. In order to recover the shallow water equations (SWEs), the basic constraints are redefined, and under these constraints, the coefficients of the EDFs are derived afterwards. Several benchmark problems are used to validate the proposed model, including stationary case, steady flows over a two‐dimensional bump and tidal wave flows over irregular bed elevation. The computed results are in excellent agreement with the results of the other numerical methods and the analytical solutions, indicating that the proposed model is capable of simulating shallow water flows over complex bathymetry. It also proves that the proposed model has potential to produce competitive solutions to shallow water flows over complex bed topography. Copyright © 2014 John Wiley & Sons, Ltd.     

A Godunov method for the computation of erosional shallow water transients

A Godunov method is proposed for the computation of open‐channel flows in conditions of rapid bed erosion and intense sediment transport. Generalized shallow water equations govern the evolution of three distinct interfaces: the water free‐surface, the boundary between pure water and a sediment transport layer, and the morphodynamic bottom profile. Based on the HLL scheme of Harten, Lax and Van Leer (1983), a finite volume numerical solver is constructed, then extended to second‐order accuracy using Strang splitting and MUSCL extrapolation. Lateralisation of the momentum flux is adopted to handle the non‐conservative product associated with bottom slope. Computational results for erosional dam‐break waves are compared with experimental measurements and semi‐analytical Riemann solutions. Copyright © 2003 John Wiley & Sons, Ltd.     

A general Riemann solver for Euler equations

In this paper, we present a general Riemann solver which is applied successfully to compute the Euler equations in fluid dynamics with many complex equations of state (EOS). The solver is based on a splitting method introduced by the authors. We add a linear advection term to the Euler equations in the first step, to make the numerical flux between cells easy to compute. The added linear advection term is thrown off in the second step. It does not need an iterative technique and characteristic wave decomposition for computation. This new solver is designed to permit the construction of high‐order approximations to obtain high‐order Godunov‐type schemes. A number of numerical results show its robustness. Copyright © 2007 John Wiley & Sons, Ltd.     

A finite volume shock‐capturing solver of the fully coupled shallow water‐sediment equations

This paper describes a numerical solver of well‐balanced, 2D depth‐averaged shallow water‐sediment equations. The equations permit variable horizontal fluid density and are designed to model water‐sediment flow over a mobile bed. A Godunov‐type, Harten–Lax–van Leer contact (HLLC) finite volume scheme is used to solve the fully coupled system of hyperbolic conservation laws that describe flow hydrodynamics, suspended sediment transport, bedload transport and bed morphological change. Dependent variables are specially selected to handle the presence of the variable density property in the mathematical formulation. The model is verified against analytical and semi‐analytical solutions for bedload transport and suspended sediment transport, respectively. The well‐balanced property of the equations is verified for a variable‐density dam break flow over discontinuous bathymetry. Simulations of an idealised dam‐break flow over an erodible bed are in excellent agreement with previously published results, validating the ability of the model to capture the complex interaction between rapidly varying flow and an erodible bed and validating the eigenstructure of the system of variable‐density governing equations. Flow hydrodynamics and final bed topography of a laboratory‐based 2D partial dam breach over a mobile bed are satisfactorily reproduced by the numerical model. Comparison of the final bed topographies, computed for two distinct sediment transport methods, highlights the sensitivity of shallow water‐sediment models to the choice of closure relationships. Copyright © 2016 John Wiley & Sons, Ltd.     

A correction for balancing discontinuous bed slopes in two‐dimensional smoothed particle hydrodynamics shallow water modeling

In this paper, a smoothed particle hydrodynamics (SPH) numerical model for the shallow water equations (SWEs) with bed slope source term balancing is presented. The solution of the SWEs using SPH is attractive being a conservative, mesh‐free, automatically adaptive method without special treatment for wet‐dry interfaces. Recently, the capability of the SPH–SWEs numerical scheme with shock capturing and general boundary conditions has been used for predicting practical flooding problems. The balance between the bed slope source term and fluxes in shallow water models is desirable for reliable simulations of flooding over bathymetries where discontinuities are present and has received some attention in the framework of Finite Volume Eulerian models. The imbalance because of the source term resulting from the calculation of the the water depth is eradicated by means of a corrected mass, which is able to remove the error introduced by a bottom discontinuity. Two different discretizations of the momentum equation are presented herein: the first one is based on the variational formulation of the SWEs in order to obtain a fully conservative formulation, whereas the second one is obtained using a non‐conservative form of the free‐surface elevation gradient. In both formulations, a variable smoothing length is considered. Results are presented demonstrating the corrections preserve still water in the vicinity of either 1D or 2D bed discontinuities and provide close agreement with 1D analytical solutions for rapidly varying flows over step changes in the bed. The method is finally applied to 2D dam break flow over a square obstacle where the balanced formulation improves the agreement with experimental measurements of the free surface. Copyright © 2012 John Wiley & Sons, Ltd.     

Simple efficient algorithm (SEA) for shallow flows with shock wave on dry and irregular beds

An explicit Godunov‐type solution algorithm called SEA (simple efficient algorithm) has been introduced for the shallow water equations. The algorithm is based on finite volume conservative discretisation method. It can deal with wet/dry and irregular beds. Second‐order accuracy, in both time and space, is achieved using prediction and correction steps. A very simple and efficient flux limiting technique is used to equip the algorithm with total variation dimensioning property for shock capturing purposes. In order to make sure about the balance between the flux gradient and the bed slope, treatment of the source term has been done using a new procedure inspired mainly by the physical rather than mathematical consideration. SEA has been applied to one‐dimensional problems, although it can equally be applied to multi‐dimensional problems. In order to assess the capability of proposed algorithm in dealing with practical applications, several test cases have been examined. Copyright © 2007 John Wiley & Sons, Ltd.     

Solution of the hyperbolic mild‐slope equation using the finite volume method

A finite volume solver for the 2D depth‐integrated harmonic hyperbolic formulation of the mild‐slope equation for wave propagation is presented and discussed. The solver is implemented on unstructured triangular meshes and the solution methodology is based upon a Godunov‐type second‐order finite volume scheme, whereby the numerical fluxes are computed using Roe's flux function. The eigensystem of the mild‐slope equations is derived and used for the construction of Roe's matrix. A formulation that updates the unknown variables in time implicitly is presented, which produces a more accurate and reliable scheme than hitherto available. Boundary conditions for different types of boundaries are also derived. The agreement of the computed results with analytical results for a range of wave propagation/transformation problems is very good, and the model is found to be virtually paraxiality‐free. Copyright © 2003 John Wiley & Sons, Ltd.     

A robust high‐resolution finite volume scheme for the simulation of long waves over complex domains

The propagation, runup and rundown of long surface waves are numerically investigated, initially in one dimension, using a well‐balanced high‐resolution finite volume scheme. A conservative form of the nonlinear shallow water equations with source terms is solved numerically using a high‐resolution Godunov‐type explicit scheme coupled with Roe's approximate Riemann solver. The scheme is also extended to handle two‐dimensional complex domains. The numerical difficulties related to the presence of the topography source terms in the model equations along with the appearance of the wet/dry fronts are properly treated and extended. The resulting numerical model accurately describes breaking waves as bores or hydraulic jumps and conserves volume across flow discontinuities. Numerical results show very good agreement with previously presented analytical or asymptotic solutions as well as with experimental benchmark data. Copyright © 2007 John Wiley & Sons, Ltd.     

An evaluation of force terms in the lattice Boltzmann models in simulating shallow water flows over complex topography

Proper approximation of the force terms, especially the bed slope term, is of crucial importance to simulating shallow water flows in lattice Boltzmann (LB) models. However, there is little discussion on the schemes of adding force terms to LB models for shallow water equations (SWEs). In this study, we evaluate the performance of forcing schemes coupled with different LB models (LABSWE and MLBSWE) in simulating shallow water flows over complex topography and try to find out their intrinsic characteristics and applicability. Three cases are adopted for evaluation, including a stationary case, a one-dimensional tidal wave flow over an irregular bed, and a steady flow over a two-dimensional seamount. The simulating results are compared with analytical solutions or the results produced by the finite difference method. For LABSWE, all the forcing schemes, except for the weighting factor method, fail to produce accurate solutions for the test cases; this is probably due to the mismatch between the bed slope term in source terms and the quadratic depth term of the equilibrium distribution functions in these forcing schemes. For MLBSWE, all the forcing schemes are capable of simulating flows over the complex topography accurately; furthermore, those schemes taking into account the collision effect τ to eliminate the momentum induced by forces provide more accurate solutions with quicker convergence as the lattice size decreases. In this view, MLBSWE can bring more flexibility in treating the force terms and thus can be a better tool to simulate shallow water flows over complex topography in practical application.     

Well‐balancing issues related to the RKDG2 scheme for the shallow water equations

Discontinuous Galerkin (DG) finite element methods have salient features that are mainly highlighted by their locality, their easiness in balancing the flux and source term gradients and their component‐wise structure. In the light of this, this paper aims to provide insights into the well‐balancing property of a second‐order Runge–Kutta Discontinuous Galerkin (RKDG2) method. For this purpose, a Godunov‐type RKDG2 method is presented for solving the shallow water equations. The scheme is based on local DG linear approximations and does not entail any special treatment of the source terms in order to achieve well‐balanced numerical results. The performance of the present RKDG2 scheme in reproducing conserved solutions for both free surface and discharge over strongly irregular topography is demonstrated by applying to several hydraulic benchmarks. Meanwhile, the effects of different slope limiting procedures on the well‐balancing property are investigated and discussed. This work may provide useful guidelines for developing a well‐balanced RKDG2 numerical scheme for shallow water flow simulation. Copyright © 2009 John Wiley & Sons, Ltd.     

Adaptive Q‐tree Godunov‐type scheme for shallow water equations

This paper presents details of a second‐order accurate, Godunov‐type numerical model of the two‐dimensional shallow water equations (SWEs) written in matrix form and discretized using finite volumes. Roe's flux function is used for the convection terms and a non‐linear limiter is applied to prevent unwanted spurious oscillations. A new mathematical formulation is presented, which inherently balances flux gradient and source terms. It is, therefore, suitable for cases where the bathymetry is non‐uniform, unlike other formulations given in the literature based on Roe's approximate Riemann solver. The model is based on hierarchical quadtree (Q‐tree) grids, which adapt to inherent flow parameters, such as magnitude of the free surface gradient and depth‐averaged vorticity. Validation tests include wind‐induced circulation in a dish‐shaped basin, two‐dimensional frictionless rectangular and circular dam‐breaks, an oblique hydraulic jump, and jet‐forced flow in a circular reservoir. Copyright © 2001 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.     

Shoreline tracking and implicit source terms for a well balanced inundation model

The HyFlux2 model has been developed to simulate severe inundation scenario due to dam break, flash flood and tsunami‐wave run‐up. The model solves the conservative form of the two‐dimensional shallow water equations using the finite volume method. The interface flux is computed by a Flux Vector Splitting method for shallow water equations based on a Godunov‐type approach. A second‐order scheme is applied to the water surface level and velocity, providing results with high accuracy and assuring the balance between fluxes and sources also for complex bathymetry and topography. Physical models are included to deal with bottom steps and shorelines. The second‐order scheme together with the shoreline‐tracking method and the implicit source term treatment makes the model well balanced in respect to mass and momentum conservation laws, providing reliable and robust results. The developed model is validated in this paper with a 2D numerical test case and with the Okushiri tsunami run up problem. It is shown that the HyFlux2 model is able to model inundation problems, with a satisfactory prediction of the major flow characteristics such as water depth, water velocity, flood extent, and flood‐wave arrival time. The results provided by the model are of great importance for the risk assessment and management. Copyright © 2009 John Wiley & Sons, Ltd.     

A well‐balanced upstream flux‐splitting finite‐volume scheme for shallow‐water flow simulations with irregular bed topography

This study extends the upstream flux‐splitting finite‐volume (UFF) scheme to shallow water equations with source terms. Coupling the hydrostatic reconstruction method (HRM) with the UFF scheme achieves a resultant numerical scheme that adequately balances flux gradients and source terms. The proposed scheme is validated in three benchmark problems and applied to flood flows in the natural/irregular river with bridge pier obstructions. The results of the simulations are in satisfactory agreement with the available analytical solutions, experimental data and field measurements. Comparisons of the present results with those obtained by the surface gradient method (SGM) demonstrate the superior stability and higher accuracy of the HRM. The stability test results also show that the HRM requires less CPU time (up to 60%) than the SGM. The proposed well‐balanced UFF scheme is accurate, stable and efficient to solve flow problems involving irregular bed topography. Copyright © 2009 John Wiley & Sons, Ltd.     

Solution of the shallow‐water equations using an adaptive moving mesh method

This paper extends an adaptive moving mesh method to multi‐dimensional shallow water equations (SWE) with source terms. The algorithm is composed of two independent parts: the SWEs evolution and the mesh redistribution. The first part is a high‐resolution kinetic flux‐vector splitting (KFVS) method combined with the surface gradient method for initial data reconstruction, and the second part is based on an iteration procedure. In each iteration, meshes are first redistributed by a variational principle and then the underlying numerical solutions are updated by a conservative‐interpolation formula on the resulting new mesh. Several test problems in one‐ and two‐dimensions with a general geometry are computed using the proposed moving mesh algorithm. The computations demonstrate that the algorithm is efficient for solving problems with bore waves and their interactions. The solutions with higher resolution can be obtained by using a KFVS scheme for the SWEs with a much smaller number of grid points than the uniform mesh approach, although we do not treat technically the bed slope source terms in order to balance the source terms and flux gradients. Copyright © 2004 John Wiley & Sons, Ltd.     

Accuracy of high‐order density‐based compressible methods in low Mach vortical flows

At low Mach numbers, Godunov‐type approaches, based on the method of lines, suffer from an accuracy problem. This paper shows the importance of using the low Mach number correction in Godunov‐type methods for simulations involving low Mach numbers by utilising a new, well‐posed, two‐dimensional, two‐mode Kelvin–Helmholtz test case. Four independent codes have been used, enabling the examination of several numerical schemes. The second‐order and fifth‐order accurate Godunov‐type methods show that the vortex‐pairing process can be captured on a low resolution with the low Mach number correction applied down to 0.002. The results are compared without the low Mach number correction and also three other methods, a Lagrange‐remap method, a fifth‐order accurate in space and time finite difference type method based on the wave propagation algorithm, and fifth‐order spatial and third‐order temporal accurate finite volume Monotone Upwind Scheme for Conservation Laws (MUSCL) approach based on the Godunov method and Simple Low Dissipation Advection Upstream Splitting Method (SLAU) numerical flux with low Mach capture property. The ability of the compressible flow solver of the commercial software, ANSYS FLUENT , in solving low Mach flows is also demonstrated for the two time‐stepping methods provided in the compressible flow solver, implicit and explicit. Results demonstrate clearly that a low Mach correction is required for all algorithms except the Lagrange‐remap approach, where dissipation is independent of Mach number. © 2013 Crown copyright. International Journal for Numerical Methods in Fluids. © 2013 John Wiley & Sons, Ltd.     

1‐D numerical modelling of shallow flows with variable horizontal density

A1‐D numerical model is presented for vertically homogeneous shallow flows with variable horizontal density. The governing equations represent depth‐averaged mass and momentum conservation of a liquid–species mixture, and mass conservation of the species in the horizontal direction. Here, the term ‘species’ refers to material transported with the liquid flow. For example, when the species is taken to be suspended sediment, the model provides an idealized simulation of hyper‐concentrated sediment‐laden flows. The volumetric species concentration acts as an active scalar, allowing the species dynamics to modify the flow structure. A Godunov‐type finite volume scheme is implemented to solve the conservation laws written in a deviatoric, hyperbolic form. The model is verified for variable‐density flows, where analytical steady‐state solutions are derived. The agreement between the numerical predictions and benchmark test solutions illustrates the ability of the model to capture rapidly varying flow features over uniform and non‐uniform bed topography. A parameter study examines the effects of varying the initial density and depth in different regions. Copyright © 2009 John Wiley & Sons, Ltd.     

Godunov-type solution of the shallow water equations on adaptive unstructured triangular grids

A Godunov-type upwind finite volume solver of the non-linear shallow water equations is described. The shallow water equations are expressed in a hyperbolic conservation law formulation for application to cases where the bed topography is spatially variable. Inviscid fluxes at cell interfaces are computed using Roe's approximate Riemann solver. Second-order accurate spatial calculations of the fluxes are achieved by enhancing the polynomial approximation of the gradients of conserved variables within each cell. Numerical oscillations are curbed by means of a non-linear slope limiter. Time integration is second-order accurate and implicit. The numerical model is based on dynamically adaptive unstructured triangular grids. Test cases include an oblique hydraulic jump, jet-forced flow in a flat-bottomed circular reservoir, wind-induced circulation in a circular basin of non-uniform bed topography and the collapse of a circular dam. The model is found to give accurate results in comparison with published analytical and alternative numerical solutions. Dynamic grid adaptation and the use of a second-order implicit time integration scheme are found to enhance the computational efficiency of the model.