Numerical Simulation of Bed Load and Suspended Load Sediment Transport Using Well-Balanced Numerical Schemes

Sediment transport can be modelled using hydrodynamic models based on shallow water equations coupled with the sediment concentration conservation equation and the bed con-servation equation.The complete system of equations is made up of the energy balance law and the Exner equations.The numerical solution for this complete system is done in a seg-regated manner.First,the hyperbolic part of the system of balance laws is solved using a finite volume scheme.Three ways to compute the numerical flux have been considered,the Q-scheme of van Leer,the HLLCS approximate Riemann solver,and the last one takes into account the presence of non-conservative products in the model.The discretisation of the source terms is carried out according to the numerical flux chosen.In the second stage,the bed conservation equation is solved by using the approximation computed for the system of balance laws.The numerical schemes have been validated making comparisons between the obtained numerical results and the experimental data for some physical experiments.The numerical results show a good agreement with the experimental data.

含源项双曲守恒方程的保平衡HLL格式

针对含源项的双曲守恒方程给出了一种新的有限体积格式.经典的有限体积格式不能正确地模拟对流通量项和外力之间的平衡所产生的动力学问题.为解决这个问题,仿照经典的HLL近似Riemann求解器设计思路设计了含源项的近似Riemann求解器.针对含重力源项的一维流体Euler方程和理想磁流体方程,通过对通量计算格式的修正得到了保平衡HLL格式(WB-HLL),并给出了保平衡的证明.针对一维Euler方程和理想磁流体给出了两个算例,比较了传统HLL格式和提出的WB-HLL格式的计算精度.计算结果表明,WB-HLL格式精度更高,收敛更快.

Moving-Water Equilibria Preserving HLL-Type Schemes for the Shallow Water Equations

We construct new HLL-type moving-water equilibria preserving upwind schemes for the one-dimensional Saint-Venant system of shallow water equations with nonflat bottom topography.The designed first-and secondorder schemes are tested on a number of numerical examples,in which we verify the well-balanced property as well as the ability of the proposed schemes to accurately capture small perturbations of moving-water steady states.

A Well-Balanced Numerical Model for the Simulation of Long Waves over Complex Domains

This paper presents a well-balanced two-dimensional (2D) finite volume model to simulate the propagation, runup and rundown of long wave. Non-staggered grid is adopted to discretize the governing equation and the intercell flux is computed using a central upwind scheme, which is a Riemann-problem-solver-free method for hyperbolic conservation laws. The nonnegative reconstruction method for water depth is implemented in the present model to treat the appearance of wet/dry fronts, and the friction term is solved by a semi-implicit scheme to ensure the stability of the model. The Euler method is applied to update flow variable to the new time level. The model is verified against two experimental cases and good agreements are observed between numerical results and observed data.

求解具有复杂地形二维浅水方程的修正HLL格式

采用非结构网格的有限体积方法,对具有复杂地形的二维浅水方程进行数值模拟.采用HLL近似Riemann解计算界面数值通量,基于三角形网格,底坡源项采用简单的斜底模型离散,保证了地形的离散精度,摩阻源项采用全隐方式求解以保证格式的稳定性.采用多维重构及多维限制器的方法获得空间二阶精度的格式,时间离散采用三阶Runge-Kutta法以获得高阶时间精度.为保证格式的和谐性,对经典的HLL格式计算的数值通量中的静水压力项进行了修正.数值计算的结果验证此格式具有良好的高精度捕捉间断的能力,可以应用到地形复杂的二维浅水问题计算中去.

A Well-Balanced FVC Scheme for 2D Shallow Water Flows on Unstructured Triangular Meshes

This paper aims to present a new well-balanced,accurate and fast finite volume scheme on unstructured grids to solve hyperbolic conservation laws.It is a scheme that combines both finite volume approach and characteristic method.In this study,we consider a shallow water system with Coriolis effect and bottom friction stresses where this new Finite Volume Characteristics(FVC)scheme has been applied.The physical and mathematical properties of the system,including the C-property,have been well preserved.First,we developed this approach by preserving the advantages of the finite volume discretization such as conservation property and the method of characteristics,in order to avoid Riemann solvers and to enhance the accuracy without any complexity of the MUSCL reconstruction.Afterward,a discretization was applied to the bottom source term that leads to a well-balanced scheme satisfying the steady-state condition of still water.A semi-implicit treatment will also be presented in this study to avoid stability problems due to source terms.Finally,the proposed finite volume method is verified on several benchmark tests and shows good agreement with analytical solutions and experimental results;moreover,it gives a noteworthy accuracy and rapidity improvement compared to the original approaches.

A New Well-Balanced Finite Volume CWENO Scheme for Shallow Water Equations over Bottom Topography

In this article,we develop a new well-balanced finite volume central weighted essentially non-oscillatory(CWENO)scheme for one-and two-dimensional shallow water equations over uneven bottom.The well-balanced property is of paramount importance in practical applications,where many studied phenomena can be regarded as small perturbations to the steady state.To achieve the well-balanced property,we construct numerical fluxes by means of a decomposition algorithm based on a novel equilibrium preserving reconstruction procedure and we avoid applying the traditional hydrostatic reconstruction technique accordingly.This decomposition algorithm also helps us realize a simple source term discretization.Both rigorous theoretical analysis and extensive numerical examples all verify that the proposed scheme maintains the well-balanced property exactly.Furthermore,extensive numerical results strongly suggest that the resulting scheme can accurately capture small perturbations to the steady state and keep the genuine high-order accuracy for smooth solutions at the same time.

Application of ADER Scheme in MHD Simulation

The Arbitrary accuracy Derivatives Riemann problem method(ADER) scheme is a new high order numerical scheme based on the concept of finite volume integration,and it is very easy to be extended up to any order of space and time accuracy by using a Taylor time expansion at the cell interface position.So far the approach has been applied successfully to flow mechanics problems.Our objective here is to carry out the extension of multidimensional ADER schemes to multidimensional MHD systems of conservation laws by calculating several MHD problems in one and two dimensions: (ⅰ) Brio-Wu shock tube problem,(ⅱ) Dai-Woodward shock tube problem,(ⅲ) Orszag-Tang MHD vortex problem.The numerical results prove that the ADER scheme possesses the ability to solve MHD problem,remains high order accuracy both in space and time,keeps precise in capturing the shock.Meanwhile,the compared tests show that the ADER scheme can restrain the oscillation and obtain the high order non-oscillatory result.

High Order Finite Difference WENO Methods for Shallow Water Equations on Curvilinear Meshes

A high order finite difference numerical scheme is developed for the shallow water equations on curvilinear meshes based on an alternative flux formulation of the weighted essentially non-oscillatory(WENO)scheme.The exact C-property is investigated,and comparison with the standard finite difference WENO scheme is made.Theoretical derivation and numerical results show that the proposed finite difference WENO scheme can maintain the exact C-property on both stationarily and dynamically generalized coordinate systems.The Harten-Lax-van Leer type flux is developed on general curvilinear meshes in two dimensions and verified on a number of benchmark problems,indicating smaller errors compared with the Lax-Friedrichs solver.In addition,we propose a positivity-preserving limiter on stationary meshes such that the scheme can preserve the non-negativity of the water height without loss of mass conservation.

Well-Balanced Central Scheme for the System of MHD Equations with Gravitational Source Term

A well-balanced second order finite volume central scheme for the magnetohydrodynamic(MHD)equations with gravitational source term is developed in this paper.The scheme is an unstaggered central scheme that evolves the numerical solution on a single grid and avoids solving Riemann problems at the cell interfaces using ghost staggered cells.A subtraction technique is used on the conservative variables with the support of a known steady state in order to manifest the well-balanced property of the scheme.The divergence-free constraint of themagnetic field is satisfied after applying the constrained transport method(CTM)for unstaggered central schemes at the end of each time-step by correcting the components of the magnetic field.The robustness of the proposed scheme is verified on a list of numerical test cases from the literature.

An Efficient High Order Well-Balanced Finite Difference WENO Scheme for the Blood Flow Model

The blood flow model admits the steady state,in which the flux gradient is non-zero and is exactly balanced by the source term.In this paper,we present a high order well-balanced finite difference weighted essentially non-oscillatory(WENO)scheme,which exactly preserves the steady state.In order to maintain the wellbalanced property,we propose to reformulate the equation and apply a novel source term approximation.Extensive numerical experiments are carried out to verify the performances of the current scheme such as the maintenance of well-balanced property,the ability to capture the perturbations of such steady state and the genuine high order accuracy for smooth solutions.

A Well-Balanced Weighted Compact Nonlinear Scheme for Pre-Balanced Shallow Water Equations

It is well known that developing well-balanced schemes for the balance laws is useful for reducing numerical errors.In this paper,a well-balanced weighted compact nonlinear scheme(WCNS)is proposed for shallow water equations in prebalanced forms.The scheme is proved to be well-balanced provided that the source term is treated appropriately as the advection term.Some numerical examples in oneand two-dimensions are also presented to demonstrate the well-balanced property,high order accuracy and good shock capturing capability of the proposed scheme.

Boussinesq水波方程新型数值解法

为建立高效的Boussinesq类水波数值模型,提出了一种新型的、基于有限差分和有限体积方法的混合数值格式。针对守恒形式的一维控制方程,在等间距矩形控制体内对其进行积分并离散,采用有限体积方法计算界面数值通量,剩余源项采用有限差分方法计算。其中,采用MUSTA格式并结合高精度状态插值方法计算控制体界面数值通量。时间积分则采用具有TVD性质的三阶龙格-库塔多步积分法进行。除验证模型外,重点对MUSTA格式和广泛使用的HLL格式进行了比较。结果表明,MUSTA格式可用于Boussinesq类水波方程数值求解,综合考虑数值精度、计算效率、程序编制和实际应用这几个方面,其较HLL格式更具有优势。

求解二维浅水波方程的旋转混合格式

针对二维浅水波方程数值求解问题,构造了一种旋转通量混合格式.空间方向上,该算法利用浅水波方程通量函数的旋转不变性,在单元界面法线方向及单元界面切线方向上采用可消除红斑现象的HLL与满足热力学第二定律的熵稳定加权混合数值通量函数,时间方向上采用三阶强稳定Runge-Kutta法.数值结果表明,该混合格式对于二维浅水波方程数值求解具有分辨率高的良好特性.

求解二维Euler方程的旋转通量混合格式

为提高求解二维Euler方程数值结果的分辨率,提出了一种旋转通量混合格式.该算法采用旋转通量法的类一维处理思想,通量函数选用满足热力学第二定律的熵稳定数值通量和具有良好鲁棒性的HLL数值通量耦合的混合格式,时间方向采用三阶强稳定Runge-Kutta方法进行推进.该旋转通量混合格式具有结构简单、分辨率高的优点,数值结果表明了该算法的良好特性.

Numerical investigation on tsunami wave mitigation on forest sloping beach

An explicit one-dimensional model based on the shallow water equations(SWEs) was established in this work to simulate tsunami wave propagation on a vegetated beach. This model adopted the finite-volume method(FVM)for maintaining the mass balance of these equations. The resistance force caused by vegetation was taken into account as a source term in the momentum equation. The Harten–Lax–van Leer(HLL) approximate Riemann solver was applied to evaluate the interface fluxes for tracing the wet/dry transition boundary. This proposed model was used to simulate solitary wave run-up and long-periodic wave propagation on a sloping beach. The calibration process suitably compared the calculated results with the measured data. The tsunami waves were also simulated to discuss the water depth, tsunami force, as well as the current speed in absence of and in presence of forest domain. The results indicated that forest growth at the beach reduced wave energy loss caused by tsunamis. A series of sensitivity analyses were conducted with respect to variable parameters(such as vegetation densities, wave heights, wave periods, bed resistance, and beach slopes) to identify important influences on mitigating tsunami damage on coastal forest beach.

A well-balanced positivity preserving two-dimensional shallow flow model with wetting and drying fronts over irregular topography

This paper presents an improved well-balanced Godunov-type 2-D finite volume model with structured grids to simulate shallow flows with wetting and drying fronts over an irregular topography. The intercell flux is computed using a central upwind scheme, which is a Riemann-problem-solver-free method for hyperbolic conservation laws. The nonnegative reconstruction method for the water depth is implemented to resolve the stationary or wet/dry fronts. The bed slope source term is discretized using a central difference method to capture the static flow state over the irregular topography. Second-order accuracy in space is achieved by using the slope limited linear reconstruction method. With the proposed method, the model can avoid the partially wetting/drying cell problem and maintain the mass conservation. The proposed model is tested and verified against three theoretical benchmark tests and two experimental dam break flows. Further, the model is applied to predict the maximum water level and the flood arrival time at different gauge points for the Malpasset dam break event. The predictions agree well with the numerical results and the measurement data published in literature, which demonstrates that with the present model, a well-balanced state can be achieved and the water depth can be nonnegative when the Courant number is kept less than 0.25.

一种健壮的TV通量分裂格式

随着计算流体力学的快速发展,设计精确、高效并且健壮的数值格式变得尤为重要。Toro等 [8] 提出的TV通量分裂格式表现出简单、高效和精确分辨接触间断等优点,但是在计算一些多维算例时会出现数值激波不稳定现象。两波近似的HLL格式在计算中非常高效和健壮,但是不能分辨接触间断大大地限制了其应用。本文对TV通量分裂格式进行稳定性分析,据此提出一种混合格式来消除TV格式的数值激波不稳定性。数值试验表明,本文构造的混合格式不仅保留了原始TV格式的优点,而且具有更好的健壮性,在计算二维问题时不会出现数值激波不稳定现象。

A Well-Balanced Runge-Kutta Discontinuous Galerkin Method for Multilayer ShallowWater Equations with Non-Flat Bottom Topography

A well-balanced Runge-Kutta discontinuous Galerkin method is presented for the numerical solution of multilayer shallow water equations with mass exchange and non-flat bottom topography.The governing equations are reformulated as a non-linear system of conservation laws with differential source forces and reaction terms.Coupling between theflow layers is accounted for in the system using a set of ex-change relations.The considered well-balanced Runge-Kutta discontinuous Galerkin method is a locally conservativefinite element method whose approximate solutions are discontinuous across the inter-element boundaries.The well-balanced property is achieved using a special discretization of source terms that depends on the nature of hydrostatic solutions along with the Gauss-Lobatto-Legendre nodes for the quadra-ture used in the approximation of source terms.The method can also be viewed as a high-order version of upwindfinite volume solvers and it offers attractive features for the numerical solution of conservation laws for which standardfinite element methods fail.To deal with the source terms we also implement a high-order splitting operator for the time integration.The accuracy of the proposed Runge-Kutta discontinuous Galerkin method is examined for several examples of multilayer free-surfaceflows over bothflat and non-flat beds.The performance of the method is also demonstrated by comparing the results obtained using the proposed method to those obtained using the incompressible hydrostatic Navier-Stokes equations and a well-established kinetic method.The proposed method is also applied to solve a recirculationflow problem in the Strait of Gibraltar.

一种能准确模拟波浪复杂演化过程的非静力学数值模型

选取σ坐标系下的不可压缩Navier-Stokes方程组作为控制方程。将速度变量定义在计算单元的中心位置,同时将非静力学压力定义在计算单元竖直方向界面位置以便于结合Godunov型格式。采用有限体积和有限差分混合方法结合Godunov型格式对控制方程进行空间离散。利用HLL黎曼求解器求解计算单元之间的通量以实现激波捕捉,并采用二阶非线性强稳定性保持龙格库塔(SSP Runge-Kutta)格式进行时间步迭代。基于以上数值方法,文中发展了一种能准确模拟波浪复杂演化过程的非静力学数值模型。将非静力学数值模型用于数值模拟淹没式堤坝上波浪传播、孤立波沿斜坡爬高和海底滑坡诱发海啸三种波浪复杂演化过程,数值计算结果与相应的实验结果较为吻合。