Well-balanced unstaggered central schemes for one and two-dimensional shallow water equation systems

We propose a new well-balanced unstaggered central finite volume scheme for hyperbolic balance laws with geometrical source terms. In particular we construct a new one and two-dimensional finite volume method for the numerical solution of shallow water equations on flat/variable bottom topographies. The proposed scheme evolves a non-oscillatory numerical solution on a single grid, avoids the time consuming process of solving Riemann problems arising at the cell interfaces, and is second-order accurate both in space and time. Furthermore, the numerical scheme follows a well-balanced discretization that first discretizes the geometrical source term according to the discretization of the flux terms, and then mimics the surface gradient method and discretizes the water height according to the discretization of the water level. The resulting scheme exactly satisfies the C-property at the discrete level. The proposed scheme is then applied and classical one and two-dimensional shallow water equation problems with flat or variable bottom topographies are successfully solved. The obtained numerical results are in good agreement with corresponding ones appearing in the recent literature, thus confirming the potential and efficiency of the proposed method.     

A robust central scheme for the shallow water flows with an abrupt topography based on modified hydrostatic reconstructions

We study a second-order central scheme for the shallow water flows with a discontinuous bottom topography based on modified hydrostatic reconstructions (HRs). The first HR scheme was proposed in Audusse et al, which may be missing the effect of the large discontinuous bottom topography. We introduce a modified HR method to cope with this numerical difficulty. The new scheme is well-balanced for still water solutions and can guarantee the positivity of the water depth. Finally, several numerical results of classical problems of the shallow water equations confirmed these properties of the new scheme. Especially, the new scheme yields superior results for the shallow water downhill flow over a step.     

A simple finite volume method for the shallow water equations

We present a new finite volume method for the numerical solution of shallow water equations for either flat or non-flat topography. The method is simple, accurate and avoids the solution of Riemann problems during the time integration process. The proposed approach consists of a predictor stage and a corrector stage. The predictor stage uses the method of characteristics to reconstruct the numerical fluxes, whereas the corrector stage recovers the conservation equations. The proposed finite volume method is well balanced, conservative, non-oscillatory and suitable for shallow water equations for which Riemann problems are difficult to solve. The proposed finite volume method is verified against several benchmark tests and shows good agreement with analytical solutions.     

Well-balanced central finite volume methods for the Ripa system

We propose a new well-balanced central finite volume scheme for the Ripa system both in one and two space dimensions. The Ripa system is a nonhomogeneous hyperbolic system with a non-zero source term that is obtained from the shallow water equations system by incorporating horizontal temperature gradients. The proposed numerical scheme is a second-order accurate finite volume method that evolves a non-oscillatory numerical solution on a single grid, avoids the process of solving Riemann problems arising at the cell interfaces, and follows a well-balanced discretization that ensures the steady state requirement by discretizing the geometrical source term according to the discretization of the flux terms. Furthermore the proposed scheme mimics the surface gradient method and discretizes the water height according to the discretization of the water level. The proposed scheme is then applied and classical one and two-dimensional Ripa problems with flat or variable bottom topographies are successfully solved. The obtained numerical results are in good agreement with corresponding ones appearing in the recent literature, thus confirming the potential and efficiency of the proposed method.     

Two-dimensional numerical modelling of shallow water flows over multilayer movable beds

The two-dimensional modelling of shallow water flows over multi-sediment erodible beds is presented. A novel approach is developed for the treatment of multiple sediment types in morphodynamics. The governing equations include the two-dimensional shallow water equations for hydrodynamics, an Exner-type equation for morphodynamics, a two-dimensional transport equation for the suspended sediments, and a set of empirical equations for entrainment and deposition. Multilayer sedimentary beds are formed of different erodible soils with sediment properties and new exchange conditions between the bed layers are developed for the model. The coupled equations yield a hyperbolic system of balance laws with source terms. As a numerical solver for the system, we implement a fast finite volume characteristics method. The numerical fluxes are reconstructed using the method of characteristics which employs projection techniques. The proposed finite volume solver is simple to implement, satisfies the conservation property and can be used for two-dimensional sediment transport problems in non-homogeneous isotropic beds without need of complicated three-dimensional equations. To assess the performance of the proposed models, we present numerical results for a wide variety of shallow water flows over sedimentary layers. Comparisons to experimental data for dam-break problems over movable beds are also included in this study.     

Regularized Equations for Numerical Simulation of Flows in the Two-Layer Shallow Water Approximation

Regularized equations describing hydrodynamic flows in the two-layer shallow water approximation are constructed. A conditionally stable finite-difference scheme based on the finitevolume method is proposed for the numerical solution of these equations. The scheme is tested using several well-known one-dimensional benchmark problems, including Riemann problems.     

求解浅水波方程的熵相容格式

提出了一种求解浅水波方程组的熵相容格式．在熵稳定通量中添加特征速度差分绝对值的项来抵消解在跨过激波时所产生的熵增,从而实现熵相容．新的数值差分格式具有形式简单、计算效率高、无需添加任何的人工数值粘性的特点．数值算例充分说明了其显著的优点．利用新格式成功地模拟了不同类型溃坝问题的激波、稀疏波传播及溃坝两侧旋涡的形成,是求解浅水波方程组较为理想的方法.     

A staggered-grid numerical algorithm for the extended Boussinesq equations

A second-order accurate numerical scheme is developed to solve Nwogu’s extended Boussinesq equations. A staggered-grid system is introduced with the first-order spatial derivatives being discretized by the fourth-order accurate finite-difference scheme. For the time derivatives, the fourth-order accurate Adams predictor–corrector method is used. The numerical method is validated against available analytical solutions, other numerical results of Navier–Stokes equations, and experimental data for both 1D and 2D nonlinear wave transformation problems. It is shown that the new algorithm has very good conservative characteristics for mass calculation. As a result, the model can provide accurate and stable results for long-term simulation. The model has proven to be a useful modeling tool for a wide range of water wave problems.     

Finite volume and finite element methods for solving a one-dimensional space-fractional Boussinesq equation

In this paper, a new one-dimensional space-fractional Boussinesq equation is proposed. Two novel numerical methods with a nonlocal operator (using nodal basis functions) for the space-fractional Boussinesq equation are derived. These methods are based on the finite volume and finite element methods, respectively. Finally, some numerical results using fractional Boussinesq equation with the maximally positive skewness and the maximally negative skewness are given to demonstrate the strong potential of these approaches. The novel simulation techniques provide excellent tools for practical problems. These new numerical models can be extended to two- and three-dimensional fractional space-fractional Boussinesq equations in future research where we plan to apply these new numerical models for simulating the tidal water table fluctuations in a coastal aquifer.     

Balance-characteristic scheme as applied to the shallow water equations over a rough bottom

The CABARET scheme is used for the numerical solution of the one-dimensional shallow water equations over a rough bottom. The scheme involves conservative and flux variables, whose values at a new time level are calculated by applying the characteristic properties of the shallow water equations. The scheme is verified using a series of test and model problems.     

Application of a fractional steps method for the numerical solution of the two-dimensional modeling of the Lake Mariut

A fractional steps technique for the numerical solution of the shallow water equations is applied to study the water velocity in Lake Mariut, its concentration and the distribution of the temperature along it. Lake Mariut is considering the most productive natural systems in Egypt. The current configuration of this lake is changing rapidly, due to people’s activities and natural processes. Most of its water supply comes from polluted agricultural drains. Several problems affect the conservation of the Lake Mariut, mainly pollution, land reclamation, intensive aquatic vegetation, over fishing and coastal erosion. The shallow water equations for this lake are discretized on a fixed grid and time stepped with the fractional steps method, where the Riemann invariants of the equations are interpolated at each time step along the characteristics of the equations using a cubic spline interpolation. The method is efficient and simple, since it evolves the equations without the iterative steps involved in the multi-dimensional interpolation problem. The absence of iterative steps in the present technique makes it very suitable for the problems in which small time steps and grid sizes are required and the simplicity of the method makes it very suitable for parallel computer. Therefore, the method provides numerical algorithms which are more efficient than other classical schemes.     

A Chebyshev spectral method for solving Korteweg–de Vries equation with hydrodynamical application

It is well known that many hydrodynamical problems appearing in the study of shallow water theory or the theory of rotating fluids, can be reduced to Korteweg–de Vries equation subject to certain initial and boundary conditions. In this work, a Chebyshev spectral method for obtaining a semi-analytical solution to such equation is presented. One numerical application is considered to show how we can apply the presented proposed method. A comparison between our results and the numerical results obtained by the Hopscotch method are made.     

正变坐标系下完全深度平均湍流方程组

本文针对宽浅型水域，对三维湍流时均方程组逐项进行深度平均，推导出包含自由水面和地形影响的深度平均流动控制方程组．本文还同时获得了深度平均形式的ｋ－ε湍流模型方程组．因计入了水流的三维效应，该模型称为完全深度平均模型．考虑到天然水域几何边界复杂，本文运用较简便的方法，将上述模型方程组交换至正交坐标系下．所得控制方程组可以直接运用于对实际问题的数值模拟．     

Relaxation approximation to bed-load sediment transport

In this work we propose and apply a numerical method based on finite volume relaxation approximation for computing the bed-load sediment transport in shallow water flows, in one and two space dimensions. The water flow is modeled by the well-known nonlinear shallow water equations which are coupled with a bed updating equation. Using a relaxation approximation, the nonlinear set of equations (and for two different formulations) is transformed to a semilinear diagonalizable problem with linear characteristic variables. A second order MUSCL-TVD method is used for the advection stage while an implicit–explicit Runge–Kutta scheme solves the relaxation stage. The main advantages of this approach are that neither Riemann problem solvers nor nonlinear iterations are required during the solution process. For the two different formulations, the applicability and effectiveness of the presented scheme is verified by comparing numerical results obtained for several benchmark test problems.     

A numerical computation of a non-dimensional form of stream water quality model with hydrodynamic advection–dispersion–reaction equations

Mathematical models of water quality assessment problems often arise in environmental science. The modelling often involves numerical methods to solve the equations. In this research, two mathematical models are used to simulate pollution due to sewage effluent in the nonuniform flow of water in a stream with varied current velocity. The first is a hydrodynamic model that provides the velocity field and elevation of the water flow. The second is a dispersion model, where the commonly used governing factor is the one-dimensional advection–dispersion–reaction equation that gives the pollutant concentration fields. In the simulation processes, we used the Crank–Nicolson method system of a hydrodynamic model and the backward time central space scheme for the dispersion model. Finally, we present a numerical simulation that confirms the results of the techniques.     

Central unstaggered finite volume schemes for hyperbolic systems: Applications to unsteady shallow water equations

A class of central unstaggered finite volume methods for approximating solutions of hyperbolic systems of conservation laws is developed in this paper. The proposed method is an extension of the central, non-oscillatory, finite volume method of Nessyahu and Tadmor (NT). In contrast with the original NT scheme, the method we develop evolves the numerical solution on a single grid; however ghost cells are implicitly used to avoid the resolution of the Riemann problems arising at the cell interfaces. We apply our method and solve classical one and two-dimensional unsteady shallow water problems. Our numerical results compare very well with those obtained using the original NT method, and are in good agreement with corresponding results appearing in the recent literature, thus confirming the efficiency and the potential of the proposed method.     

A hybrid numerical method for the shallow water equations with source term

In coastal oceanography there is interest in problems modeled by the shallow water equations, where variations in channel depth are accounted for by the presence of source terms. A numerical treatment for the solution of such problems is presented here, in terms of a hybrid approach, which combines a second-order TVD scheme for conservation law equations (assuming no source terms) with an eigenvector projection scheme that incorporates the effects of nonzero source terms (in regions where the bottom is not flat). For the case where an initially sharp wave profile is assumed, the progress of a wave as it traverses an estuary whose channel depth varies is calculated. Excellent numerical results are obtained. © 1996 John Wiley & Sons, Inc.     

竖直平面波动问题的水动力系数

本文阐述了一种用以计算深水中竖直平面波动问题水动力系数的数值方法.其过程是,以格林公式为基础导出了波源法,应用该方法计算了受线性波动作用时的半圆及矩形截面的物体,并将计算结果与沃兹(Vugts)早期的试验结果进行了比较.     

Tribalj dam-break and flood wave propagation

In this paper we present numerical simulations for the dam-break flood wave propagation from Tribalj accumulation to the town of Crikvenica (Croatia). The mathematical models we used were the one-dimensional open channel flow and the two-dimensional shallow water equations. They were solved with the well-balanced finite volume numerical schemes which additionally include special numerical treatment of the wetting/drying front boundary. These schemes were tested on CADAM test problems. The aim of this study was to assess potential damage in the village of Tribalj and the town of Crikvenica. Results of these simulations were used as the basis for urban planning and micro-zoning of the flood-risk areas. Several different dam-break scenarios were considered, ranging from sudden dam disappearance to partial and dynamic breach formation.